Passive VHF Radar Interferometer Implementation, Observations, and Analysis

Transcription

1 Passive VHF Radar Interferometer Implementation, Observations, and Analysis Melissa G. Meyer A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering University of Washington 2003 Program Authorized to Offer Degree: Electrical Engineering

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3 University of Washington Graduate School This is to certify that I have examined this copy of a master s thesis by Melissa G. Meyer and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. Committee Members: John D. Sahr Donald B. Percival Robert H. Holzworth Robert S. Hiers Rachel A. Yotter Date:

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5 In presenting this thesis in partial fulfillment of the requirements for a Master s degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of this thesis is allowable only for scholarly purposes, consistent with fair use as prescribed in the U.S. Copyright Law. Any other reproduction for any purpose or by any means shall not be allowed without my written permission. Signature Date

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7 University of Washington Abstract Passive VHF Radar Interferometer Implementation, Observations, and Analysis by Melissa G. Meyer Chair of Supervisory Committee: Professor John D. Sahr Electrical Engineering In this work, we describe a new method for remote sensing of ionospheric plasma turbulence: an extension of the passive radar technique to include interferometry. We discuss the implementation of a passive radar interferometer, and show many observations of varied targets, including ground clutter, aircraft, and meteor trails, as well as plasma density irregularities in the E-region ionosphere. Because of the very fine resolution of our instrument (as fine as 0.06 in azimuth, or 1.5 km 2 at a distance of 1000 km), we can form two dimensional images of these targets, and we are able to show that many E-region irregularities exist in compact scattering volumes (as narrow as 30 km in the transverse dimension), and that significant structure exists on very fine scales. We also describe in detail the passive radar interferometer cross-correlation estimator and its statistical properties, and perform an analysis of the resolution capability of the instrument. Finally, we demonstrate how the interferometer can be used to measure geophysical information, such as electric fields and velocity shears across scattering volumes in the ionosphere.

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9 TABLE OF CONTENTS List of Figures List of Tables Glossary Notation Conventions v ix x xii Chapter 1: Introduction The Manastash Ridge Radar Interferometry The Manastash Ridge Radar Interferometer What can be Learned Chapter 2: System Description An Example of Target Detection with MRR Hardware Software Chapter 3: Interferometer Observations Ground Clutter Airplanes and Compact Targets Geophysical Targets i

10 3.3.1 Meteor Trails Auroral Observations Analysis of Data Products Two-Dimensional Interferometer Images Azimuth Aliasing Other Issues Chapter 4: Signal Processing Preliminary Assumptions MRR Signal Processing Motivation for Statistical Calculations The Interferometer Estimator Expected Value of the Interferometer Estimator A Model for the Received Scatter A More Complicated Scatter Model The Expected Value after Coherent Integration Variance of the Interferometer Estimator Variance with the Simple Scattering Model Variance after Coherent Averaging Interferometer Implementation Issues Statistics of Other Estimators and Additional Comments The Periodogram (FFT-Based Spectrum Estimation) Application to MRR Signal Processing Periodogram Variance ii

11 Chapter 5: Simulations A Simulated Target Empirical Estimation of Cross Spectrum Variance Evaluation of Interferometer Resolution Target Detection by Phase Compactness Chapter 6: Analysis of Ionospheric Events with Interferometry Description of Irregularity Backscatter February 2, Natural Progression to Two-Dimensional Images March 24, Velocity Vector Measurements Electric Field Measurements March 30, Chapter 7: Conclusions Summary Future Work Bibliography 116 Appendix A: Variance Derivation 120 A.1 Isserlis Gaussian Moment Theorem Applied to Analytic Radar Signals 120 A.2 Variance Derivation, Continued A.3 Analysis of 8 th -order Correlations A.3.1 Term A.3.2 Term iii

12 A.4 The Full Variance Expression Appendix B: Simulating Appropriate Passive Radar Signals 130 B.1 Simulation of an FM Signal B.2 Simulation of a Target Signal B.3 Creating the Scattered Signal iv

13 LIST OF FIGURES 1.1 An illustration of radar interferometry. The radial and transverse dimensions are shown at bottom right An illustration of the MRR system A range-doppler diagram, the typical data product of MRR. Reflections from the Cascade mountain range can be seen at zero Doppler near 100 km; at a range of 1000 km, echoes due to ionospheric plasma turbulence are present A scatter plot of in-phase vs. quadrature components of a typical reference FM transmission. The station is 96.5 MHz; this is 1 second of data taken at 200 khz. Note that the samples are zero-mean, and the inphase and quadrature components are uncorrelated. The narrowband time series can be considered an analytic signal A range-doppler display from MRR. Examining the near-ranges reveals ground clutter and airplanes An interferometer image with the same ground clutter and airplanes shown in figure Interferometer data from the Mt. Rainier echo. Both antennas show a narrow, DC spike, and the correlation coefficient between them is well above the 95% significance level at zero Doppler Interferometer data and power spectra for the airplane with Doppler shift -50 m/s (also shown in figure 3.1). Again, we include the 95% significance level for the coherence spectrum An illustration of the wavevectors involved in Bragg scatter. A backscatter radar can detect the scattered signal due to k s when θ = A range-doppler diagram showing an echo from a meteor trail (790 km) during the 2002 Quadrantids meteor shower Cross- and self-spectra for the range containing the meteor echo shown in figure 3.6. The 95% significance level for the coherence is also shown. 29 v

14 3.8 A meteor trail image, produced with MRR interferometer data Individual range spectra for the meteor trail shown in figure 3.8. (Observed during the Geminids meteor shower, December 14, 2001.) Selected meteor echoes observed by MRR during the year 2002 with corresponding interferometer data. For each group of plots, the top panel is a power spectrum, the middle panel is the coherence, and the bottom panel shows the cross-spectrum phase. In each case, the Doppler velocities range from -400 to 400 m/s. Some of the echoes are very low SNR An MRR range vs. Doppler diagram showing an echo from E-region turbulence between 600 and 800 km A two-dimensional interferometer image of the same auroral echo shown in figure Histogram and sorted samples of simulated coherence values. In the bottom panel, dotted lines are shown for the 50 th, 75 th, 95 th, and 99 th percentiles. These graphs represent a sample size of 500, An irregularity observed on 18 April, 2002, at UT 06:03. Power spectra from both antennas as well as an interferometer image are shown. The color scale applies to all panels An irregularity observed on 24 March, 2002, at UT 05:07. Power spectra from both antennas as well as an interferometer image are shown. The color scale applies to all panels Normalized histograms showing the trend toward Gaussian distribution after coherent integration. Gaussian probability density functions with the same mean and variance as the data have been overlayed in the right panels. Each figure represents the real part of 10 6 complex data points, sampled at a rate of 100 khz A typical autocorrelation function for an FM transmission (96.5 MHz). The signal is mostly decorrelated after 10 µseconds, indicating that samples taken at 100 khz are independent. These samples were taken at 200 khz, roughly the bandwidth of the signal A comparison of 2-D image resolutions obtained by range averaging and varying the number of phase bins used in the interferometer program (essentially phase averaging ). The data shown here is an auroral echo observed on February 2, 2002, UT 05: vi

15 4.4 Cross-correlation magnitude of the meteor trail echo shown in figure 3.6. The noise, clutter, and interference contribution is contained in lag zero A comparison of bias removal techniques. The top plot shows the uncorrected spectrum; the next two demonstrate different correction methods discussed in the text. The data is from the meteor trail observed on January 3, 2002 at UT 19: A simulated target, with -13 db SNR An illustration of the variance estimation method used here The top three panels show different representations of cross-spectrum variance versus SNR. The bottom panel shows the number of variance estimates that were averaged together to form the final value Cross-spectrum phase variance plotted as the spectrum SNR increases A comparison of transverse resolutions in interferometer data. From left to right, the phase is separated into consecutively fewer bins, resulting in degrading transverse resolution. The range resolution in each figure is 1.5 km Power spectra at 3 different ranges for the auroral echo we observed on February 2, 2002, at UT 05: An example of cross-spectrum phase compactness where a (simulated) target exists. The standard deviation of phase estimates can be used to detect targets, in some cases more reliably than power spectrum magnitude The MRR field of view, over southwestern Canada, shown with contours of constant range and aspect angle. The edges of the figure are labeled with geographic latitude (N) and longitude (E). Credit for creating the figure goes to Dr. Frank Lind A range-doppler display from MRR showing an auroral echo near 1000 km; an area of high intensity and large Doppler shift is flanked by a larger, diffuse type 2 echo, which we show to be spread across the transverse dimension, indicating a shear vii

16 6.3 The cross-spectrum and self-spectra for one range in the auroral echo from figure 6.2. Due to the highly organized phase, we observe that the echo seems limited to one interferometer lobe; the phase width implies a scattering volume extent of approximately 30 km. The 95% significance level for the coherence is also shown in the middle panel Cross spectrum at range 984 km for the auroral echo on February 2, 2002, UT 05: Cross spectrum at range 989 km for the auroral echo on February 2, 2002, UT 05: Cross spectrum at range 992 km for the auroral echo on February 2, 2002, UT 05: Cross spectrum at range 1005 km for the auroral echo on February 2, 2002, UT 05: Power spectra (from one antenna) and interferometer phase vs. range for the auroral echo on February 2, 2002, UT 05: The February 2, 2002 auroral echo, represented in a 2-D image of range vs. transverse width. The resolution in this plot is 3 5 km (range transverse dimension) D interferometer images from March 24, 2002, UT 05:07 05: Measured velocity vectors for the drifting irregularity shown above in figure An example of Doppler power spectrum versus range from irregularity scatter obtained on March 30, A sequence of 5 consecutive interferometer images from the March 30 event, 4 minutes apart in time. The first three frames indicate a mean transverse drift speed of 70 m/s. The final frame clearly shows that the scattering volume has split, a feature unrecognizable from the range- Doppler spectra in figure B.1 Scatter plots of actual (left) and simulated FM data B.2 Power spectral densities of actual (left) and simulated FM data B.3 Autocorrelation functions (real part) of actual (left) and simulated FM data. Each sample is marked with a dot B.4 A comparison of histograms of actual FM samples and samples from a simulated FM signal viii

17 LIST OF TABLES 2.1 Latitude/longitude coordinates for some locations relevant to MRR Interferometer transverse resolutions appropriate for a given SNR ix

18 GLOSSARY / FREQUENTLY USED ACRONYMS AR: Autoregressive. Describing a random time series which can be written in terms of a linear combination of its previous values, plus a random component. ARMA: An autoregressive moving average process (see MA). CUPRI: The Cornell University Portable Radar Interferometer. CW: Continuous wave. A continuous (non-pulsed) transmission. DFT: Discrete Fourier transform. ERP: Effective radiated power. FFT: Fast Fourier transform. FM: Frequency modulation. A scheme for encoding information into the frequency of a carrier signal. GPS: Global Positioning System. IMF: Interplanetary magnetic field. The sun s magnetic field, carried outwards by the solar wind. MA: Moving average. Describing a random time series which can be written in terms of a mean value plus a series of perturbations based on a white noise process. MATLAB: A commercial numerical computation and visualization software tool. MRO: Manastash Ridge Observatory, MRR s namesake and where its scatterreceiving antennas are located. x

19 MRR: The Manastash Ridge Radar, designed and developed at the University of Washington. PLL: Phase-locked loop. A circuit with an oscillator that locks on to the frequency of an input signal. SCR: Signal-to-Clutter Ratio. SHERPA: The Système HF d Etudes Radar Polaires et Aurorales, an HF coherent radar operated from Schefferville, Québec. SNR: Signal-to-Noise Ratio. Typically measured in db, this is the ratio of signal power to noise power. In this document, we obtain SNR by a peak to floor measurement. STARE: The Scandinavian Twin Auroral Radar Experiment. RF: RX: Radio frequency. Receiver. UW: University of Washington, where this work was conducted, and where one of the MRR receivers is located. xi

20 NOTATION CONVENTIONS C( ): C : C N : A magnitude-normalized cross spectrum (complex coherency). Clutter contribution from all ranges. Clutter contribution from a single range. ˆK( ): Cross correlation estimator for MRR. R XY ( ): A correlation function between signals x and y, defined as x(t)y (t τ). S( ): A spectral density function. S (ρ) ( ): The periodogram, a spectral estimator. : Denotes expected value of the argument. xii

21 ACKNOWLEDGMENTS First and foremost, I want to thank my advisor, mentor, and friend, John Sahr. He has opened a whole world of opportunities to me, and it s in no small part thanks to him that I am having the time of my life in grad school. Hearty thanks also goes to the other members of my committee, Don Percival, Bob Holzworth, and Robert Hiers, and my many other excellent mentors over the years Frank Vanzant, John Hopkins, and Paul Crilly: you all have helped and inspired me more than you realize. To Frankye Jones especially, as well as the rest of the EE, space science, and CS communities here at UW: you have been like families to me since I came here. I do so much enjoy working and the talking that goes along with it with you! A huge hug goes to my bestest 1 friend, Andrew, without whom I would certainly have gone insane by this time; my wonderful housemates Lena, Adam, Matt, and Shanaz, who are very dear friends despite the fact that I never see them due to being on campus so much; and last but not least, my loving family, who gave me up so that I could follow my dreams 3000 miles out here to Seattle. 1 It s a technical term. xiii

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23 1 Chapter 1 INTRODUCTION Always beware when someone with a Ph.D. calls something interesting. Dr. Steve Keeling We begin with an appeal to the reader s sense of curiosity. The primary motivation for this work is to develop ways to study the reasons and mechanisms behind the aurora borealis, or northern lights. A fascinating phenomenon that has intrigued Earth s inhabitants for thousands of years, the aurora are ultimately caused by reactions in the Earth s atmosphere to the sun s energy in various forms. The outermost layer of the atmosphere is called the ionosphere because it is composed partly of ionized particles (plasma) rather than completely of neutral particles. At lower altitudes in the ionosphere (termed the D- and E- regions) the electrons and ions are kept too energetic to recombine during daylight hours by the constant radiation from the sun; at night these layers disappear (become neutral). The outermost layer, called the F-region, is much more sparsely populated and remains ionized all the time. Thus, the Earth is surrounded by a sheath of plasma which absorbs the most energetic radiation from the sun. Since these particles are charged, they respond to electromagnetic forces, and are strongly influenced by both the Earth s magnetic field as well as any electric fields that may exist (due, for example, to the much higher mobility of electrons as compared to ions). A very complicated and intricate

24 2 structure of currents results from the coupling between the ionosphere, the magnetosphere (the area within Earth s magnetic force-field ), and the interplanetary solar wind (which carries the sun s magnetic field frozen into it). In short, the northern (and southern) lights are caused by energy pouring into the ionosphere from currents traveling along magnetic field lines. These field-aligned currents excite particles in the atmosphere, which emit photons (of different colors, depending on their chemical makeup) as they decay back down to their previous lower energy levels. The entire system can become very perturbed when the radiation from the sun is highly energetic for some reason (for example, after a solar flare) or when the interplanetary magnetic field (IMF) carried by the solar wind lines up so as to cancel out part of the Earth s magnetic field. During events such as these, the field-aligned currents causing the aurora can be pushed southward into lower latitudes, where their effects can be seen by, for example, plasma physicists at the University of Washington in Seattle 1. As interesting as the physics responsible for the aurora is, it is a challenge to study experimentally due to the awkward altitude at which the interesting activity takes place. Ionospheric heights (typically km for the E-region) are too low for satellite orbits, yet too high for balloons; rockets with scientific payloads can be (and have been the ERRRIS campaign, for example [24]) flown through the area of interest, but these are expensive and short-lived experiments. Remote sensing makes the most sense as a strategy for extended study of ionospheric processes, and much work has been done toward developing robust and reliable radar techniques toward this end. In this work, we present a new technique for passive radar remote sensing of plasma turbulence in the ionosphere: passive radar interferometry. 1 Provided the sky is not overcast.

25 3 1.1 The Manastash Ridge Radar The Manastash Ridge Radar (MRR) is a project at the University of Washington [27] designed and developed as an affordable way to do remote sensing of ionospheric E-region field-aligned irregularities (plasma turbulence usually correlated with the visual aurora, and sometimes called the radar aurora ). Located in the northwestern United States, the MRR field of view covers a region over southwestern Canada in the sub-auroral zone, corresponding approximately to the geomagnetic latitudes of (50 57 geographic latitude). Therefore, it detects coherent scatter from auroral irregularities only during disturbed ionospheric conditions. For example, in the year 2002, we observed E-region irregularities with MRR on 27 separate days 2. The most unique feature of MRR is that it does not have a dedicated transmitter. Conventional active radars transmit successive pulses (or coded pulse sequences) and measure the time of flight of the reflected pulses to detect and determine the range of reflecting targets. MRR, however, takes advantage of existing commercial transmitters as a source of illumination; for this reason we refer to it as a passive radar. The system, described more completely in chapter 2, consists of two receivers: one to provide a reference copy of the transmitted signal, and one to collect the signal after it has been scattered from various targets. 1.2 Interferometry In coherent scatter studies of the ionosphere, interferometric techniques have frequently been used to resolve the transverse structure of scattering volumes and to provide information about the location of scatterers in the field of view [5, 25, 28]. The interferometer we have created has very fine resolution, and this is useful for 2 This is still more often than we expected to detect auroral scatter, based on our sub-auroral field of view.

26 4 determining transverse structures. The basic idea behind interferometry is that the phase difference between the signals generated by a wave arriving on two (or multiple, but we consider only the two-antenna case here) closely spaced 3 antennas provides information about the angle of arrival of the wave. In particular, the phase difference between the signals on the two antennas, φ, is related to the physical angle of arrival θ by φ = kd cos θ = 2π λ d cos θ (1.1) where k is the wave number, λ is the wavelength, and d is the distance between the antennas. Figure 1.1 illustrates this idea; it is convenient to think of φ as the phase accumulated by the wave as it travels the extra distance d cos θ to the second antenna. The interferometer will be able to uniquely map one wavelength of phase to a full 180 field of view (a front-to-back ambiguity remains) if the baseline is less than or equal to half a wavelength; d λ 2 this will cause φ to vary between π and π. If the baseline is any larger, the interferometer will have multiple lobes (one for each multiple of 2π radians of phase between the antennas), and the physical angle information will be aliased. We can write equation 1.1 for the general case by adding a few terms: φ + φ 0 + n2π = kd cos(θ + θ 0 ) (1.2) Here the n2π term represents any aliasing (n is an integer); φ 0 is a net phase correction term, necessary because of arbitrary phase delays in the system, such as transmission lines between the antennas and receiver; and the θ 0 term is a correction term for the 3 Here closely spaced means that the evolution of a wave traveling between them can be completely described in terms of phase accumulation; i. e., the time of flight between the two antennas is less than the correlation time of the wave.

27 5 Reflecting target in far field Plane wave with Uniform phase front Extra distance to antenna 2 d cos θ θ d strikes antenna 1 first 2 1 transverse dir. radial dir. Figure 1.1: An illustration of radar interferometry. The radial and transverse dimensions are shown at bottom right. angle of arrival, and could be used to compensate for heterogeneous antenna patterns, for instance. If the antennas are oriented such that their gain patterns are directed broadside to the baseline connecting them, the interferometer will emphasize azimuth angle information, whereas a configuration with the antennas oriented in an endfire position relative to the baseline will respond better to the elevation angle of the target. We will consider the azimuth-emphasizing orientation here, as this is the type of interferometer we have implemented.

28 The Manastash Ridge Radar Interferometer In order to perform interferometry at MRR, we need data from three antennas: two separated by a known distance d to collect scattered signals, and one copy of the reference FM transmission. The two antennas we refer to when describing the MRR interferometer are the scatter-collecting antennas. The baseline between them is approximately 47 meters long, or roughly sixteen wavelengths (16 λ) at MRR s carrier frequency of 100 MHz. Thus, the interferometer has 32 lobes, and the width of a single lobe is approximately 5.6 in azimuth angle (100 km wide at a range 1000 km). The antenna configuration we have used is due mainly to the geometry of the space available at the site where we keep our antennas. The large baseline was available and convenient when we began this project, so we used it, and made plans to set up additional antennas in the future. The method we use to extract the phase difference between the antennas is a complex cross-correlation (or cross-spectrum, in the Fourier domain) between the target signals on the two antennas. We refer to the correlation coefficient between the two antennas (or the cross-spectrum magnitude normalized by the power on each individual antenna) as coherence [5]. The coherence provides information about the transverse size of scatterers, while the phase provides information about their locations. We describe in detail our passive radar interferometer cross-correlation estimator and determine its statistical properties in chapters 4 and What can be Learned If transverse structure information is available, we can learn many things about the scattering volume. First, we can create two-dimensional images of the scatterer (where the two dimensions are the radial direction away from the radar and the direction perpendicular to that, which is described by azimuth angle). Given the time evolution of

29 7 scatterers in these two dimensions, we can estimate their transverse velocities in addition to their Doppler shifts. Finally, by making approximations about the scatterer motion based on known behaviors of plasmas, we can use interferometer information to measure geophysical properties of the scattering volume, such as electric field. Chapter 3 shows interferometer observations of many different types of targets, the various data products we can create, and their interpretations. In chapter 6 we demonstrate the use of our interferometer on radar data containing scatter from E-region irregularities, which is the primary focus of this work.

30 8 Chapter 2 SYSTEM DESCRIPTION The party line is that more data are a blessing though bedraggled grad students might disagree! Dr. Don Percival The Manastash Ridge Radar (MRR) is a unique instrument that utilizes ambient radio illumination in the environment to detect and determine the characteristics of sound waves and other density perturbations in the ionosphere. Because of its passive nature, it does not require a dedicated transmitter, and this is an advantage for many reasons. Most notably, radar transmitters are very expensive and powerful instruments, and they therefore require a significant amount of maintenance and caution. Furthermore, appropriate licenses must be obtained to operate high-powered transmitters. In our case, we wished to operate a VHF radar in the 100 MHz frequency range, but this band is allocated for FM radio broadcasts. Therefore, we designed MRR to listen to the broadcasts themselves. As it happens, FM radio provides extremely useful radar waveforms for studies of the ionosphere. The transmitters are powerful and omnidirectional, and the effective radiated power is high (typically 100 kw) since they are CW, not pulsed. The 100 MHz carrier frequency is nearly immune from ionospheric refraction and atmospheric absorption effects, yet scatters readily from plasma turbulence. Most importantly, the typical FM waveform has an excellent ambiguity function in the average sense [8], often completely free of range and Doppler aliasing (of course, it changes with

31 9 time, so occasionally we experience bad ambiguity ). The ambiguity function of a complex-valued radar waveform envelope u(t) is defined as the output of its matched filter, χ(τ, ν) = u(t)u (t τ) exp(j2πνt) (2.1) and describes the radar signal s ability to resolve targets in range (time lag, τ) and Doppler (frequency, ν) space [19]. Typical ambiguity functions from MRR data show perfect range resolution (in the sense that there is no clutter due to range ambiguity) at a scale of 1.5 km, corresponding to a 100 khz receiver sampling rate, and Doppler sidelobes that are quite low, especially when compared to other coded radar pulses, such as the barker codes [8]. Essentially, the FM transmission acts like a stochastically coded long pulse [11], which is very useful for overspread 1 targets such as turbulent events in the ionosphere. The bandwidth of the FM transmission is large compared to the timescales of fluctuations in the ionospheric plasma, and this permits overspread target pulse compression. Pulse compression is a signal processing technique for achieving fine range resolution without sacrificing sensitivity. Narrow pulse widths require a large receiver bandwidth, letting a lot of noise into the system and degrading detectability. Also, the amount of power a transmitter can expel in an extremely short pulse is limited, while a longer pulse can thoroughly illuminate a target. Therefore, longer pulses are modulated (usually in a bipolar phase scheme) by pulse sequences chosen specifically for their ambiguity function performance. After matched filtering, these long pulses perform as though they had very narrow widths, but with the sensitivity advantages of long pulses. To extract useful information from our radar, we need both the original FM transmission (for matched filtering) as well as any radiation scattered from targets. How- 1 An overspread target is one which is simultaneously too distant and moving too fast to unambiguously resolve its range and frequency characteristics with a pulsed radar.

32 10 E-region Irregularities and Meteors Reference Receiver x(t) Commercial Transmitter km ground clutter km Remote Receivers y(t) 1 y(t) 2 30 km 150 km Figure 2.1: An illustration of the MRR system. ever, the transmitter can be over 100 db louder than the faint scatter returning from ionospheric targets, and this makes using a single receiver very difficult due to the required dynamic range. Of existing passive radars, MRR solves this problem by locating the transmitter recording and scatter collecting receivers in separate places, approximately 150 km apart. This distance, as well as the Cascade mountain range which lies between the two receivers, effectively prevents most, if not all, radiation entering the scatter receivers directly from the transmitter. Therefore, MRR is a bistatic (arguably tristatic ) passive radar system. An illustration of MRR is shown in figure 2.1; its field of view and the locations of the receivers and transmitter are also shown in figure 6.1 (page 96). In table 2.1, we provide the exact locations, in latitude/longitude coordinates, of the radar features indicated in these figures. While separating the receivers solves the dynamic range problem, it causes problems of its own. For example, the receivers must be synchronized very precisely: an

33 11 Table 2.1: Latitude/longitude coordinates for some locations relevant to MRR. Site Latitude Longitude Comments UW N W reference RX location MRO N W scatter RX location; elevation 3960 KYPT N W FM station at 96.5 MHz; 100 kw ERP KWJZ N W FM station at 98.9 MHz; 52 kw ERP Mt. Rainier N W prominent ground clutter offset of 10 microseconds will cause an error in range greater than our current range resolution. We synchronize the receivers with a time and frequency reference from GPS receivers at each end (the references are accurate to 100 nanoseconds). Also, we are now required to send large amounts 2 of data over a sometimes tenuous internet link. For a 10-second period every 4 minutes (currently the normal radar operation), we sample the scatter receivers on four channels 3 at 100 khz, and the reference receiver on two channels (a single antenna, but two frequencies). The scatter and reference datastreams must be combined to extract useful information from the system. The network load is not unreasonable, however, since in most cases we only need to send a single file from the scatter receiver to a central location to be processed with the reference data. The resulting internet traffic is 4 megabytes every 4 minutes, which is sustainable. If this single antenna-frequency combination indicates an interesting event in our data, however, we transfer the remaining data for processing and storage, which often takes a long time and puts significant stress on the connecting network. 2 The default operation of MRR generates over 8 gigabytes in raw receiver samples per day. It takes a full-time grad student to keep up with it all! 3 Two frequencies on each of two antennas. Soon there will be more channels, when we erect new antennas for the interferometer.

34 12 Range information is obtained from the MRR system by correlating the original FM transmission against successively delayed copies of the received scatter. Target signals are boosted above the background noise floor when these two time series align correctly. We wish to detect targets and determine their range as well as their Doppler characteristics. Figure 2.2 shows a typical data product from MRR: a range- Doppler diagram. These diagrams are composed of power spectra (of the target signal), stacked up over multiple ranges. The vertical axis indicates Doppler shift, or frequency information, while the horizontal axis shows the distance of the scatterer from the radar. The greyscale is in db (uncalibrated units). In this particular plot we can see ground clutter features at near ranges (caused by the mountains which lie between the transmitter and scatter receiver), as well as an example of scatter from E-region turbulence at about 1000 km. The ground clutter is centered at zero Doppler, as we expect, although ambiguity sidelobes in the Doppler dimension are apparent. Occasionally we also detect aircraft (with appropriate Doppler shifts) at ranges less than about 200 km. The E-region echoes are moving much faster, typically with Doppler shifts near 450 m/s. This figure represents a 10-second time integration, a typical length for MRR. 2.1 An Example of Target Detection with MRR Using the latitude/longitude information in table 2.1, we can calculate the distances between important features of the radar. For example, the total bounce distance from the KYPT commercial FM transmitter to Mt. Rainier and then to the scatter receiver at MRO is 160 km. At the speed of light, it takes 530 µsec to traverse this distance; if our receivers take samples every 10 µsec (a 100 khz sampling rate), then the echo from Mt. Rainier should lag the transmitter waveform by 53 samples. However, since the distance between the KYPT transmitter and our reference receiver at the

35 April 20, 2002 UT 09:47 db 232 Velocity (m/s) Ground clutter, with ambiguity sidelobes visible E region turbulence Range (km) 218 Figure 2.2: A range-doppler diagram, the typical data product of MRR. Reflections from the Cascade mountain range can be seen at zero Doppler near 100 km; at a range of 1000 km, echoes due to ionospheric plasma turbulence are present. University of Washington is 20 km, the apparent time of flight for all scatter received by the system is 7 lags fewer than the true time of flight. Because of the elapsed time between a transmission and when our receivers start recording, the phrase Warning: Targets are further away than they appear applies, and should be compensated for in the radar signal processing. Thus, we expect the echo from Mt. Rainier to appear around lag 46, as it does in figure 2.2 (we also show individual power spectra from a ground clutter echo at this range in figure 3.3, page 21). In the inverse problem, the number of lags can be converted into a physical distance. Backscatter radars determine the target range by halving the time-of-flight range; however, bistatic distance is not quite as straightforwardly understood as the circles of increasing radii one imagines for a monostatic radar. In our case, contours

36 14 of constant range are elliptically shaped (or rather, shells of constant range form ellipsoids), with the transmitter and scatter-receiver as the foci of the ellipsoids. Thus, for close-range echoes seen by MRR, the one half time-of-flight times the speed of light approximation is not useful. However, since the transmitter and scatter receivers are still close together (150 km) with respect to many of the far-away echoes we detect ( km), the approximation is sufficient for our purposes. Finally, we note that the total distance traveled by the radar transmission is also affected by altitude (constrained to km for our E-region targets of interest). For this reason, radar range estimates are often referred to as slant ranges. 2.2 Hardware In the absence of a transmitter, the most prominent hardware components in the system are its antennas, receivers, and the computing resources used to process the data. The two antennas used to collect scatter from targets (with which we do interferometry) are a Yagi (mounted on a 40 foot tower) and a log periodic, mounted on the roof of a building. Both of these antennas are kept at the Manastash Ridge Observatory (MRO), which is situated at a relatively high altitude (3960 feet). The gain patterns of these two antennas are very different; the Yagi has a higher gain (14 db) with a narrow field of view directed toward the northeast, but the log periodic has a very wide field of view and a more modest gain (6 db). The antenna used to receive the transmitter signal is a simple half-wave dipole, kept at the University of Washington in the Electrical Engineering Building. The first generation MRR receivers were implemented with analog components [20]. In late 2001, however, a second generation of MRR was implemented, making use of the quickly developing digital receiver technology. We now use the Echotek ECDRGC214/TS digital receiver for both the reference and scattered signals. These

37 15 x In Phase / Quadrature Scatter Plot of FM Transmission Imaginary Part Real Part x 10 4 Figure 2.3: A scatter plot of in-phase vs. quadrature components of a typical reference FM transmission. The station is 96.5 MHz; this is 1 second of data taken at 200 khz. Note that the samples are zero-mean, and the in-phase and quadrature components are uncorrelated. The narrowband time series can be considered an analytic signal. receivers sample the incoming RF signals directly at 56 MHz (thus, we require antialiasing bandpass filters) with 14-bit ADCs. They connect to PCI slots in desktop computers and then implement further downconversion, mixing, and filtering digitally. An example of the resulting in-phase and quadrature receiver samples is shown in figure 2.3; the receivers produce very high-quality samples. Zhou [32] discusses this receiver in more detail. The computational capacity required to process the MRR receiver samples is high, but easily handled by today s over the counter computing resources. We

38 16 currently use a 750 MHz single processor desktop PC for all our computational needs. Other hardware includes GPS receivers, which are needed at each digital receiver to synchronize the nodes in the system; PLLs, which make use of the frequency reference provided by the GPS; and passive filters and amplifiers for preconditioning the received signal before digitization. Finally, MRR is, inherently, a distributed system, an instrument composed of widely separated components connected by computer networks. Therefore, in an exhaustive list of hardware required for the system, we must include all the network hardware involved in transferring our raw data files from their respective receiver locations to the final storage area where the data is processed. 2.3 Software With the exception of the antennas and digital receivers (the latter of which arguably could be referred to as software), the majority of a passive radar such as MRR is implemented in software. After both the reference and scatter receiver samples are available (for a particular increment of time), the next task toward creating diagrams like that shown in figure 2.2 is signal processing: we must deconvolve the transmitter signal from the received scatter, and then perform spectrum estimation on the resulting time series. Currently the signal processing algorithms are such that the radar can keep up with data processing in real-time (e. g., it takes approximately 10 seconds to process a 10-second burst of data). However, this does not include the time required to transfer raw data from remote receiver locations. The signal processing itself is described in more detail in chapter 4; here we list the basic software components of the system which accomplish the task. The main signal processors that do the majority of the number crunching are written in the C and C++ languages. These processors are invoked and controlled on a

39 17 regular basis (during the real-time operation of the radar ) by several scripts written in bash, csh, perl, and python (linux shell commands and programming languages). The diversity of software tools is mainly a weakness in the system caused by the preferences of the different programmers developing the system. However, there are many diverse tasks which must be accomplished in the day-to-day operation of the radar, including: data acquisition, which involves both driving the digital receivers to take data as well as determining when data needs to be taken; scripts for moving, storing, and transferring data from computer to computer; processing the data to create the range-doppler diagrams (or other data products of interest); and finally, creating images in a useful form from the data and publishing these images to the internet. Additionally, we have developed interferometer signal processing programs along with an array of other programs and scripts for plotting and displaying this data. We often find that the most difficult part of collecting and dealing with experimental data is determining how best to display it. In the next chapter we show several examples of our experimental results with the interferometer.

40 18 Chapter 3 INTERFEROMETER OBSERVATIONS There s nothing new under the sun but you could look under a rock. Dr. John Hopkins In this chapter we present several examples of data from the MRR interferometer and provide general commentary on our results. Although we will be mostly interested in upper-atmospheric targets (in particular, volumetric scatter from turbulence in the E-region), we also investigate more conventional targets, such as aircraft, for diagnostic purposes and so that our data may be readily compared with the products of other radars. An obvious first target at which to direct our attention is ground clutter from mountains in the area, a predictable feature present in all of our data. Next, we consider aircraft; these are point targets and are useful because they have simple, well-understood Doppler characteristics. We also show examples of scatter from meteor trails, and demonstrate the use of our interferometer in detecting meteor tubes, from which wind shears at mesosphere altitude can be inferred. We fully describe the signal processing involved in producing these images later in chapter 4; in chapter 6 we discuss in detail several observations of E-region irregularities, our primary purpose for remote sensing. 3.1 Ground Clutter While ground clutter is generally undesirable and has a negative impact on target detectability, it nevertheless provides a useful way to gauge the health of our system.

41 19 For example, in chapter 2, we determine the time lag after which we expect the echo from Mt. Rainier to arrive at the scatter receiver. Indeed, in almost every range- Doppler diagram MRR generates, the most noticeable feature is the ground clutter due to Mt. Rainier. We use this strong echo as a crude system diagnostic: its shape in range-doppler space gives us an idea of the FM waveform ambiguity (at the time the data was taken), and we know that the radar has become insensitive if we are ever unable to identify Mt. Rainier. For this reason, the detection of ground clutter is a convenient sanity check, and we briefly consider it with the interferometer. First, to clarify our meaning, we use the term clutter to refer to echoes from unwanted delays, which contribute energy at all ranges since the transmitter is CW. However, we can look specifically at a ground clutter echo from, say, Mt. Rainier, and determine its power spectrum. In the case of a mountain, we would of course expect a narrow spike at zero Doppler, since the target is (hopefully!) not moving or spewing rocks and ash into the air at different velocities. We would also expect the cross-correlation between the two interferometer antennas to be large, provided that the reflecting surface on the mountain is not larger than an interferometer lobe (a limitation discussed in section 3.4.2). Figure 3.1 shows a range-doppler diagram of the first 300 km of the MRR field of view. Several ranges show echoes on the zero-doppler line, which we presume to be ground clutter from the Cascade mountain range; a few aircraft are visible as well (the most prominent ones have been marked with arrows). An image derived from interferometer data is also shown in figure 3.2 for corresponding ranges; we are able to compare targets in range-velocity space with their representation in a range vs. transverse dimension image. The strongest ground clutter shows up in the black pixels, while aircraft and weaker ground or mountain echos appear in dark grey. We will discuss the generation of plots like figure 3.2 later in section Other features

42 20 April 20, 2002 UT 04:23 db Velocity (m/s) Range (km) Figure 3.1: A range-doppler display from MRR. Examining the near-ranges reveals ground clutter and airplanes. Transverse Position (km) April 20, 2002 UT 04: Range (km) Figure 3.2: An interferometer image with the same ground clutter and airplanes shown in figure 3.1.

43 21 Cross Spectrum Phase and Magnitude at range 69 km Phase Normalized Magnitude Self Spectra (db) Doppler (m/s) Log Periodic Yagi Figure 3.3: Interferometer data from the Mt. Rainier echo. Both antennas show a narrow, DC spike, and the correlation coefficient between them is well above the 95% significance level at zero Doppler. visible due to the strong echoes in figure 3.1 are the ambiguity sidelobes of the FM waveform. These are frequently seen with high SNR echoes, and are in general not confusing and easily recognizable as ambiguity sidelobes. In figure 3.3, we show interferometer data from the specific range (lag 46) at which we expect the Mt. Rainier echo, plotted against Doppler velocity. As we expect, the power spectra on both individual antennas (bottom panel) show a large, narrow spike at zero Doppler. The middle panel shows the normalized cross-spectrum magnitude (coherence), which grows to nearly a perfect correlation of 1, again at zero Doppler. We also plot the 95% significance level for the coherence, which turns out to be 0.51 in

44 22 this case, obtained through the numeric simulation of 500,000 independent instances of a random variable with the same distribution as our coherence estimates (discussed further in section 3.4). As with conventional interferometry, we find that the cross-spectrum phase (top panel of figure 3.3) becomes organized in the same area that the coherence is large. This is because the strongest signal on both antennas is arriving from the same direction. The cross-spectrum phase, then, is related to the angle of arrival of the Mt. Rainier echo (see section 1.2), and it appears that the transverse size of the reflecting surface of Mt. Rainier is significantly smaller than the width of one interferometer lobe at this range (roughly 10 km). However, even though the phase in the top panel of figure 3.3 is compact, the average interferometer phase of Mt. Rainier can change drastically 1 on a minute-length time scale due to propagation paths and channel effects. Therefore, we are unable to determine an actual angle of arrival for the echo. Additionally, our data suffers from azimuth aliasing due to the multiple, closely-spaced interferometer lobes. We would require antennas set closer together (wider interferometer lobes) to deemphasize channel-related fluctuations and uniquely determine the bearing of the target. Finally, we note that the phase remains concentrated over a Doppler extent much larger than that of the actual narrow-band echo. This is due to the ambiguity sidelobes, which reach Doppler shifts far from the true spectral peak. These Doppler features appear to arise from the same direction, and generate phase similar to that of the mountain reflection itself. 1 Since the reflection from Mt. Rainier most likely enters both antennas through a sidelobe, we believe the large phase variation in this case to be caused by antenna sidelobe effects, which also confuse attempts at using averaging to obtain a stable value. For unmoving targets in the main beam of both antennas, such large variation should not be a problem.

45 23 Cross Spectrum Phase and Magnitude at range 105 km Phase Normalized Magnitude Self Spectra (db) Doppler (m/s) Log Periodic Yagi Figure 3.4: Interferometer data and power spectra for the airplane with Doppler shift -50 m/s (also shown in figure 3.1). Again, we include the 95% significance level for the coherence spectrum. 3.2 Airplanes and Compact Targets Next, we direct our attention back to figure 3.1, and consider the Doppler-shifted echoes visible in the range-doppler diagram. Spectral peaks, presumably from aircraft, exist at 180 m/s, +50 m/s, and -50 m/s. MRR routinely sees echoes like this, and upon examining successive segments of radar time series, we find that the range migrations of targets such as these are consistent with their Doppler velocities. We show interferometer data for the range containing the plane with 50 m/s Doppler shift in figure 3.4. Again, we see that the individual power spectra and the

46 24 cross-spectrum magnitude are consistent with a narrow-band target at 50 m/s; also, the interferometer phase is concentrated in the same velocity bins where the spectral peak exists. In principle, it is possible to estimate the angular width of the scattering surface (or volume) from the coherence [5]; however, it is also possible to use the phase estimates within those Doppler bins where the coherence is significant to infer properties of the target. For instance, we can interpret the range of phase values covered in the concentrated area as a measure of the angular width of the target. In this case, the range of the phase estimates (denoted φ) over the area of concentration is about 0.6 radians (by inspection of the plot). We can approximate the azimuth extent with θ φ/(kd) if we assume the target is located broadside to the interferometer axis (using φ = kd sin θ and sin θ θ for small θ). Then, with an antenna baseline of 47 m, kd = 47 2π/λ 98; thus, the azimuth extent θ 0.6/ , and the transverse extent of the reflecting surface is approximated by r θ 600 meters, where we have used r 100 km from a-priori knowledge. Here, 600 meters is the transverse distance covered by the airplane during the 10 second duration in which data was taken. Thus, we know the total velocity vector of the airplane: 50 m/s in the radial direction away from the radar, and 60 m/s perpendicular to the radar line of sight 2, resulting in a speed of approximately 78 m/s, or 150 miles per hour. From the multi-range interferometer data in figure 3.2, we can also detect the plane at 132 km (with Doppler shift 180 m/s), although in most cases the mixture of ground clutter and aircraft is difficult to interpret without additional Doppler information. Again, even though the target phase is compact, we are unable to unambiguously determine an angle of arrival due to the aliasing of the cross-spectrum phase. 2 Determining whether this transverse direction is eastward or westward is possible, and would require examining the phase on each antenna individually.

47 Geophysical Targets We now consider targets in the upper atmosphere that scatter radio waves by mechanisms other than reflection off a tangible surface 3. Some radars that observe volumetric scatter from deep targets in the atmosphere rely on a phenomenon called Bragg resonance to produce echoes that can be detected by the radar receiver(s). MRR is one of these; we wish to detect irregularities in the ionospheric plasma density distribution. Actually, we are interested in the plasma processes, space weather, and magnetosphere-ionosphere coupling of which the irregularities are a symptom. We are able to make measurements of various parameters related to these irregularities remotely with MRR. Without going into great detail or speculation about the means by which ionospheric density perturbations arise, we assume some mechanism exists that results in a force on a portion of the plasma fluid [21]. The disturbance will typically propagate outwards as a wave 4 with a certain periodic structure (wave number), causing successive plasma density enhancements and rarefications, or, alternatively, successive areas of high and low electrical impedance. When a radar wave attempts to propagate through such a periodically-structured area, each layer interface (in a layered-media approximation) will reflect a small amount of energy. In order for significant backscatter to occur, the plasma density structures must exist with an appropriate spatial scale to produce phase delayed-reflections that will constructively interfere; this process is known as Bragg scatter [29]. However, a perfectly periodic, wavelike plasma distribution is not necessary for 3 There are other scattering mechanisms which we do not discuss here. For instance, many weather radars rely on Rayleigh scattering from volumes filled with many point targets of size much smaller than the radar wavelength (i. e., raindrops). 4 For E-region plasmas at frequencies in which we are interested, this will usually be an ion acoustic wave.

48 26 kscatter ks kr θ Figure 3.5: An illustration of the wavevectors involved in Bragg scatter. A backscatter radar can detect the scattered signal due to k s when θ = 0. Bragg scatter to occur. Arbitrary turbulent density fluctuations can be decomposed into spatial Fourier components that, if matched appropriately with the illuminating waveform, will produce a reflection. The radar backscatter is thus due to the component of the plasma spatial frequency spectrum which resonates with the radar wavelength. We can describe this relationship for arbitrary scattering angles (or, equivalently, angles of incidence) by λ r λ s = 2 cos θ (3.1) or (3.2) k s = 2 sin θk r (3.3) Here λ s (k s ) is the wavelength (wavevector) of the plasma spatial structure component; λ r (k r ) is the radar wavelength (wavevector); and θ is the angle of incidence, measured as shown in figure 3.5. In the case where there is backscatter (when θ = 0 ), we have λ s = 1λ 2 r, or in terms of wavevectors, k s = 2k r. Thus, the radar wavelength selects very specific spatial scales, and the nature of the scatter returned to the receivers is extremely dependent on the frequency of the illuminating wave. For targets at far ranges, the bistatic geometry of MRR can be approximated by a monostatic

49 27 (backscattering) system. MRR operates at VHF frequencies (100 MHz), so λ r = 3 meters; it therefore detects structures on a spatial scale of 1.5 meters Meteor Trails Although the exact physical processes causing radio wave scatter associated with meteors are not completely understood, it is widely believed that small meteoroids 5 cause tubes of plasma to form in the upper atmosphere (near mesosphere heights) by ionizing gas as they burn up during their entry. These plasma tubes can extend over a large range of altitudes, and are subject to wind shears from different layers of the atmosphere. Radars such as MRR can detect them via Bragg scattering processes. The tubes dissipate and the plasma recombines quickly, so the majority of meteor echoes are short-lived (they last approximately 1 second); however, they occur often and are usually easily identified. Radar observations of meteor trails are interesting for several reasons. For instance, their Doppler shifts reveal neutral wind velocities in the upper atmosphere, and their spectral widths depend on the decay time constant of returned power, which in turn is related to the relevant diffusion coefficient for the plasma [6]. Interferometer observations of meteor trails can show their transverse extent, and when combined with Doppler information, neutral wind shears at mesosphere altitude can be detected. By creating plots such as the one in figure 3.8, we are even able to form rudimentary images of the plasma tubes caused by the meteors. In figure 3.6, we show a typical range-doppler display from MRR during the 2002 Quadrantids shower (January 3, 2002) when a meteor is present; the corresponding two-element interferometer data for range 790 km (where the meteor appears) can be seen in figure 3.7. The spectral peak is nearly centered at zero Doppler velocity; the 5 Of sizes on the order of a grain of sand.

50 28 January 3, 2002 UT 19:23 db Velocity (m/s) Range (km) Figure 3.6: A range-doppler diagram showing an echo from a meteor trail (790 km) during the 2002 Quadrantids meteor shower. trail is not moving quickly. However, the Doppler width covers 50 or 60 m/s, and the interferometer phase shows a clear downward linear trend over the Doppler bins with significant coherence. By examining the phase estimates in the concentrated linear area, we find that φ is approximately 1 radian. At a range of 790 km, this indicates that the meteor trail is roughly 8 km wide in the radar field of view. Thus, the scattering volume appears to contain a wind shear across at least 8 km in the transverse direction. We present another multiple-range interferometer image in figure 3.8; the total backscattered power from a meteor trail has been plotted on a two-dimensional grid (frequency information has been integrated out). MRR observed this echo during the Geminids meteor shower on December 14, The transverse span of the plasma

51 29 Cross Spectrum Phase and Magnitude at range 790 km 2 Phase Normalized Magnitude Self Spectra (db) Doppler (m/s) Doppler (m/s) Log Periodic Yagi Figure 3.7: Cross- and self-spectra for the range containing the meteor echo shown in figure 3.6. The 95% significance level for the coherence is also shown. tube appears to be km; it also extends through 2-4 range cells (3-6 km in slantaltitude). In figure 3.9 we show Doppler spectra from the range 585 km; we see that this meteor trail has a net positive Doppler shift (it is blueshifted, or moving towards the radar). It also has a large Doppler width: nearly 100 m/s. However, unlike the meteor trail shown in figures 3.6 and 3.7, the phase estimates do not vary linearly over the Doppler bins with large magnitude. They are still compact, but bunched up, and not organized according to Doppler velocity. This indicates that the meteor trail is present over a wide area, revealing different mesospheric wind velocities. From the bottom panel of figure 3.9, we also see that the meteor trail spectrum

52 30 Meteor Trail: 14 December 2001, UT 04:39 One Interferometer Lobe (5 6 o azimuth) Range (km) Transverse Position, Measured from Lobe Center (km) Figure 3.8: A meteor trail image, produced with MRR interferometer data. has roughly three peaks. We speculate that the scattering volume (area of ionization) is due to a meteoroid which broke into three (or more) pieces during its entry into the atmosphere. Each of these pieces had a different velocity relative to the radar line of sight; thus, the resulting spectrum has three separate areas with significant magnitude. Finally, in figure 3.10 we show several power spectra from other meteor trails observed by MRR, along with corresponding interferometer coherence and phase. In many cases the coherence has a very high variance; this is due to the low SNR of the meteor trail backscatter.

53 31 Cross Spectrum Phase and Magnitude at range 585 km Phase (rad) Normalized Magnitude Self Spectra (db) Doppler (m/s) Log Periodic Yagi Figure 3.9: Individual range spectra for the meteor trail shown in figure 3.8. (Observed during the Geminids meteor shower, December 14, 2001.) Auroral Observations The majority of our interest in geophysical targets lies in auroral E-region irregularities, or, as we mentioned at the beginning of this section, turbulence and waves in the ionospheric plasma caused by various interactions with the magnetospheric plasma and solar wind. Because of the location of MRR (in the northwest United States), we focus on high latitude phenomena, and because of the structure of Earth s magnetic field and the aspect angle dependency of the plasma irregularities, we are unable to detect F-region phenomena. Therefore, we concentrate on echoes from E- region plasma turbulence in this work. In particular, we are interested in what the

54 32 Selected Meteor Echoes Figure 3.10: Selected meteor echoes observed by MRR during the year 2002 with corresponding interferometer data. For each group of plots, the top panel is a power spectrum, the middle panel is the coherence, and the bottom panel shows the crossspectrum phase. In each case, the Doppler velocities range from -400 to 400 m/s. Some of the echoes are very low SNR.

55 33 radar detects when the auroral oval expands into the radar field of view at lower latitudes, sometimes called the radar aurora [28]. On April 20, 2002, the Kp index 6 grew to above 7.0, indicating high geomagnetic activity, and MRR recorded many examples of E-region turbulence. One such auroral echo is shown in figures 3.11 (range vs. Doppler velocity) and 3.12 (two-dimensional interferometer image). By examining these figures, we see that the scattering volume extends for many range cells (over 100 km), and that its transverse width is approximately 20 km. The net Doppler shift of the entire echo is near 400 m/s (redshifted); we might classify the echo as type 2 because of its large Doppler width [28]. The ranges with narrower spectra ( km, km) could possibly be type 3. When the data is of high quality (e. g., it does not suffer from interference or a momentarily-ambiguous transmitter waveform), much can be learned about these irregularity echoes from the MRR interferometer. We will present other observations and more extensive analyses in chapter Analysis of Data Products We desire a complete description of the probability density function of the interferometer coherence estimates in order to determine what level of coherence is statistically significant, indicating the presence of a target. We outline a possible approach for analytically determining the probability density function of the coherence estimates, and then describe a numeric experiment we conducted to estimate confidence levels via simulated data. The target signal, denoted s(t), is Gaussian, as is its Fourier transform S(f), since 6 A measure of geomagnetic disturbance designed to assess solar particle radiation by its magnetic effects as seen from the Earth at mid-latitudes [1]. The index value is the aggregate of 13 groundbased magnetometer measurements from around the globe and is scaled from 0 to 9. The name Kp originates from planetarische Kennziffer (planetary index).

56 April 20, 2002 UT 06:27 db Velocity (m/s) Range (km) Figure 3.11: An MRR range vs. Doppler diagram showing an echo from E-region turbulence between 600 and 800 km. a linear combination of Gaussian-distributed random variables remains Gaussian. Its coherence spectrum is formed by the complex conjugate multiplication of the Fourier transforms of the signal on two antennas (p and q), normalized by the self power from each antenna, as follows: S p (f)s q (f) Sp (f)s p(f)s q (f)s q (f) = S p (f) Sp (f)s p(f) S q (f) Sq (f)s q (f) (3.4) A large number of receiver samples allows us to form a final coherence spectrum from the summation of M individual spectra. Therefore, a single coherence estimate (in a single Doppler bin) is obtained by summing M K-distributed random variables, where the K-distribution results from the multiplication of two unit variance Gaussians [28]. The functional form of the K-distribution is a Modified Bessel Function of the second kind, denoted with a K, giving the distribution its name. A scheme is now needed

57 35 April 20, 2002 UT 06: Range (km) Transverse Position (km) Figure 3.12: A two-dimensional interferometer image of the same auroral echo shown in figure for dealing with the large sum (up to 100) of random variables; a moment generating function approach seems the most feasible. However, now we turn to numeric simulation for an approximate result. Using the MATLAB software (including its Gaussian random number generator), we simulated 500,000 instances of the sum of M complex, zero mean, unit variance Gaussian random variables 7. The resulting histogram for M = 78 is shown in the top panel of figure We also sort the resulting coherences in order to determine their percentile, as shown in the bottom panel. We have plotted the 95% significance level on 7 M = 78, 39, and 19 for initial coherent integrations of 50, 100, and 200, respectively.

58 36 Histogram of Sample Coherence Distribution Number of Samples Sorted Coherence Samples 95 Percentile (samples below coherence value) Coherence Figure 3.13: Histogram and sorted samples of simulated coherence values. In the bottom panel, dotted lines are shown for the 50 th, 75 th, 95 th, and 99 th percentiles. These graphs represent a sample size of 500,000. each of our experimental coherence spectra for reference. Close observation of the cross spectra presented here will reveal occasional imperfections in the interferometer data and/or signal processing. For example, in the middle panel of figure 3.7, the normalized magnitude exceeds unity, a physically impossible result which is caused by the removal of bias in the cross spectrum, a quantity which must be estimated (see section 4.7). A second flaw remains in the phase (for example, in the top panel of figure 3.7); the phase at Doppler shifts far from the meteor peak is not distributed uniformly over [ π, π]. This is because the ambiguity

59 37 sidelobes of the echo reach to Doppler shifts far from the true spectral peak. These Doppler features appear to arise from the same part of the sky, and generate phase similar to that of the meteor peak. In figure 3.9, the coherence shows a false spike (at least, a spectral feature that does not correspond with a high SNR feature in the individual antenna spectra in the panel below). Since the interferometer coherence is a correlation between the time series on two antennas, very low-snr features can show up prominently, and can potentially cause problems with interpretation Two-Dimensional Interferometer Images While much can be learned from interferometer cross-spectra at individual ranges, we find it useful to create plots like those shown in figures 3.2, 3.8, and 3.12 for easily determining the large-scale structure of radar targets. These two-dimensional plots of range vs. transverse position can be considered true images, since the pixels are total power received (Doppler information is integrated out), organized on a grid by physical position. To produce these images, we separate the total scattered power from each range into a quantized set of positions across the transverse dimension. Using interferometer phase information, we group each point in the cross-spectrum into a certain (user specified) number of phase bins, then sum the (un-normalized) spectrum magnitudes in each phase bin. Essentially, we create a phase histogram, weighted by the cross-spectrum magnitude. We then stack these transverse dimension spectra according to range. In general, we try to keep the range resolution and transverse resolution comparable, although this is not necessary. The transverse resolution is determined by the number of phase bins we specify for the interferometer signal processing, and we consider this choice in great detail in chapters 4 and 5. In creating the two-dimensional images, we have made some simplifying assumptions. First, we have taken the azimuthal beamwidth of the interferometer to be

60 ; this value assumes 32 equally-spaced lobes in a 180 field of view, which is not entirely accurate. Also, while transverse size may be inferred using range information together with the 5.6 beamwidth approximation above, the transverse dimension in the plots remains, at best, phase information rather than a true physical dimension. Second, we have taken contours of constant range to be circles, which is accurate for a monostatic radar, but in our bistatic case this is another approximation. Angle aliasing is also an issue, and we discuss this below Azimuth Aliasing Since our interferometer baseline is several wavelengths long (16λ), the interferometer has several lobes (32) within its field of view, and our data is strongly aliased in angle. Therefore, we have no information about the absolute arrival angle of the radar scatter. Much of this ambiguity may be removed by considering a-priori information we have about the underlying physical processes causing the irregularities and various elements of the system (such as the antenna beam patterns). As we will see, in many cases the probable scattering volume location is constrained to an area only as large as two interferometer lobes; this allows us to speculate about transverse structures of irregularity echoes with some confidence. For example, it is well known [29, 28, 17, 20] that these field-aligned irregularities are strongly damped when traveling in directions parallel to Earth s magnetic field (or, at small aspect angles 8 ), and this limits the possible location of any such echoes detected by MRR, as shown in figure 6.1. Also, because of the geometry of the MRR field of view and the magnetic field s dip angle at high latitudes, we detect only irregularities in the E-region of the ionosphere; thus, in our case, scattering volumes are confined to exist at E-region heights. Finally, because our Yagi antenna has a much 8 We have defined 0 aspect angle as a line of sight parallel to Earth s magnetic field.

61 39 narrower field of view (directed toward the northeast) than our log periodic antenna does, we occasionally detect echoes with the log periodic that the Yagi does not sense. In these cases, we speculate that the scattering volume lies in the westernmost solution area (where the appropriate range and perpendicular aspect angle contours intersect). Alternatively, we know the local time of the echo observation with respect to magnetic midnight, which tells us whether to expect (in general) westward or eastward convection, and thus the blue- or red-shiftedness of the echo can give us information about whether the scattering volume lies to the east or the west of the radar. Therefore, by considering the range, local time, and antenna gain pattern associated with an irregularity observation as well as the constraints of E-region altitude and perpendicular aspect angle, the location of scattering volumes can be estimated to within a reasonable area with a good amount of confidence. Another initial concern about our large baseline was that it might completely prevent us from detecting auroral scatter with the interferometer. At first, we had expected the scattering volume of most irregularities to exceed the 5.6 of angular width available in each lobe, causing the phase to vary over an entire wavelength and the coherence to remain low despite the existence of a strong echo. However, we did not find this to be the case; in fact, in the year 2002 we collected interferometric measurements from E-region phenomena on 27 separate days that show transverse scattering sizes as narrow as 30 km (at a range of 1000 km). Occasionally we observe irregularities at very close range, and in these cases the interferometer lobes are extremely closely spaced, and too narrow to resolve even moderately sized scattering volumes. For example, the right panels of figure 3.14 show individual antenna views of a close scatterer at 400 km; the echo also has an uncommonly high SNR on both antennas. Yet in the corresponding interferometer image on the left, the echo can barely be seen. We attribute this low detectability to

62 Log Periodic Antenna Range (km) Velocity (m/s) Yagi Antenna Velocity (m/s) Transverse Position (km) Range (km) Figure 3.14: An irregularity observed on 18 April, 2002, at UT 06:03. Power spectra from both antennas as well as an interferometer image are shown. The color scale applies to all panels. a scattering volume which spans multiple interferometer lobes and thus shows little coherence. In contrast, figure 3.15 shows an echo that in many ways is more representative of the type of irregularities MRR detects. First, the SNR of the echo on both antennas is much lower than in the previous example (note that the greyscale is the same for all panels in both figures 3.14 and 3.15). Next, the echo exists at a range of approximately 1000 km, much farther away than the one in figure Thus, the interferometer

63 Log Periodic Antenna Range (km) Yagi Antenna Velocity (m/s) Velocity (m/s) Transverse Position (km) Range (km) Figure 3.15: An irregularity observed on 24 March, 2002, at UT 05:07. Power spectra from both antennas as well as an interferometer image are shown. The color scale applies to all panels. lobes are necessarily wider at this range, and are able to accommodate larger targets. Though the received scatter in this case has a much lower SNR, the interferometer still clearly detects a scatterer. This scatterer is almost certainly contained within a single interferometer lobe, its transverse width spanning km. A consequence of phase compactness and the method by which we generate the two-dimensional interferometer images can also be seen in the left panel of figure Since all the phase estimates for the auroral echo are concentrated around the appropriate value for the irregularity direction of arrival, the area adjacent to the

64 42 echo in the image is almost completely devoid of noise. This emptiness may be regarded as a sign of local high SNR (at a particular range); nevertheless, we must realize that the effect is a byproduct of our signal processing. Currently, though we are often able to make useful measurements (of transverse structure) with our interferometer despite its angle aliasing, we would still like to determine scatterer locations. We have recently constructed two shorter baselines (by adding a third antenna) to alleviate this problem; however, so far we have been unable to make many measurements with the new antenna Other Issues Other remaining nonideal elements of the system that will be characterized and either changed in the future or calibrated and compensated for in the signal processing include the orientations and positions of the antennas, both vertically and horizontally, and differing cable lengths between the antennas and the receiver. Finally, the antennas currently in use in our system are quite inhomogeneous; we obtained the data presented above from one Yagi and one log periodic antenna. The recently-constructed (and even more recently deconstructed) third antenna is a half-wave dipole. While these differences in gains and beam patterns are not necessarily a drawback (see discussion of irregularity echo location at the beginning of chapter 7), further analysis of the field of view and other antenna-related properties of the system will be conducted to ensure proper treatment. 9 The scatter-receiving antennas face many hardships that are difficult to simulate in MATLAB, including, but not limited to, severe weather on Manastash Ridge and rural Washington natives with shotguns.

65 43 Chapter 4 SIGNAL PROCESSING Their only advantage would seem to be that they would allow one to whom the autocorrelation is sacred to apply it. Drs. Bogert, Healy, and Tukey (In The Quefrency Alanysis of Time Series for Echoes: Cepstrum, Pseudo-Autocovariance, Cross-Cepstrum and Saphe Cracking [2]) The basic signal processing algorithm employed at MRR is designed to efficiently extract power spectral density measurements from the receiver samples we collect. Our primary data product is a range vs. Doppler diagram (figure 2.2, for example). However, we will be mainly concerned here with the slightly different problem of estimating the cross-correlation (or cross-spectrum) between antennas, which is useful in interferometry for determining angle of arrival and the transverse structure of targets. Before we begin, we state a few facts and make some assumptions about the nature of the data produced by the passive radar system. 4.1 Preliminary Assumptions We denote many signals using a functional notation with a continuous time argument: f(t). However, to the extent that our analysis is meant to describe signal processing implemented on a digital computer, it is convenient to think of t as quantized by the associated sampling period 1/f s.

66 44 The most important assumption we have made is that all the stochastic signals we are concerned with are zero-mean, Gaussian (in amplitude) processes. Although the FM waveform in particular is definitely not Gaussian, we make this assumption in order to simplify the discussion and make analytic analyses of the radar signal processing tractable. When we consider the effect of a coherent integration on the received time series, however, the Gaussian assumption is vastly more valid, thanks to the Central Limit Theorem. To illustrate this, we compare histograms of radar time series before and after a coherent integration of 50 consecutive points. Figure 4.1 shows (in the top two frames) relative-frequency histograms of the real part of 10 6 samples from an FM transmission. The right panel, after the coherent integration, is clearly closer to a Gaussian distribution than the left; for longer integration times we only expect this approximation to improve. The histograms have been normalized by the number of data points and their bin widths so that they have unit area and may be compared to probability density functions; Gaussian probability densities with the same mean and variance as the data have been overlayed in the post-coherent integration plots. The bottom two panels show the same information for the radar detected time series, which is the result of matched filtering. The detected time series (with no target present) appears to have an exponential distribution, but after the coherent integration, it too appears Gaussian. Also, odd moments of order 3 and above should tend towards zero as a distribution function approaches that of a Gaussian random variable. We measured the skewness (normalized third central moment) of the time series shown above both before and after the integration. We found that, in both cases, the skewness of the decimated time series was less than 10% of the skewness of the original time series. The scattering signals themselves can be assumed Gaussian 1, and they are zero- 1 At VHF frequencies, the wavelength of the illuminating waveform is often shorter than the

67 45 Transmitter Signal Probability Densities of Radar Samples Before and After Coherent Integration 8e 5 8e 5 4e 5 4e 5 0 2e4 0 2e4 0 5e5 0 5e5 Detected Time Series 2e 8 1e 8 0 2e8 0 2e8 Before Coherent Integration 1.5e e 9 0 2e9 0 2e9 With 50 Point Coherent Integration Figure 4.1: Normalized histograms showing the trend toward Gaussian distribution after coherent integration. Gaussian probability density functions with the same mean and variance as the data have been overlayed in the right panels. Each figure represents the real part of 10 6 complex data points, sampled at a rate of 100 khz. mean as long as the time average we use is long enough. Of course, we would like to keep our time averages as short as possible, because we also assume that our radar targets are statistically stationary over the time we observe them. Depending on the time resolution used and the nature of the target, this assertion is questionable (see discussion of E-region targets in chapter 6). However, to proceed with a tractable theoretical analysis, it is a necessary assumption, so we make it. Next, we consider the characteristics of the transmitter signal, an FM commercial correlation length of the scatterer. For longer wavelengths, or shorter correlation lengths, the received spectrum should be Lorentzian, according to collective wave scattering theory [10].

68 46 1 FM Transmission Autocorrelation Magnitude Correlation Coefficient Lag (microseconds) Figure 4.2: A typical autocorrelation function for an FM transmission (96.5 MHz). The signal is mostly decorrelated after 10 µseconds, indicating that samples taken at 100 khz are independent. These samples were taken at 200 khz, roughly the bandwidth of the signal. radio broadcast. An FM process has a bandwidth around its VHF carrier of about 200 khz. Frequency modulation does not affect the carrier amplitude, but it has a nonlinear effect on the phase. Thus, the transmitted signal has constant modulus and, from the point of view of an outside observer, essentially random phase. Properties of FM broadcasts and their potential as radar waveforms are described in detail by Hansen [8], Hall [7], and Lind [20]. Here we are mostly interested in the correlation time, τ x (shown in figure 4.2), which is typically 10µs. Thus, it s true that the FM signal autocorrelation R x (τ) is a sharply peaked

69 47 function on time scales of tens of microseconds. Sahr and Lind [27] use a Gaussian model, R x (τ) = e πτ 2 /τx 2 (4.1) where, in general, τ τ x (τ x is small compared to time lags in which we are interested). Given that our usual sampling frequency at MRR is 100 khz, however, the transmitter time series can be modeled as white, i.e., each sample in the time series is independent from the next, sampled after the waveform decorrelates. Then we can write R x (τ) = R x (0)δ(τ), where, to avoid problems with higher-order powers of R x (τ), we use the Kronecker delta function so that δ(0) = 1 (which implies that δ 2 (τ) = δ(τ)). This model will vastly reduce the complexity in the derivations to come. Armed with these tools, we proceed to a description and analysis of the signal processing. 4.2 MRR Signal Processing With our passive radar, we first perform a matched filter operation by correlating the received scatter (denoted y) with the reference transmitted signal (the original FM transmission, denoted x). Suppose we model the scattered signal purely as a delayed copy of a perfect FM transmission modulated by a signal of interest s(t): y(t) = s(t)x(t r 0 ) (4.2) Then, since an FM process ideally has constant modulus, the matched filtering operation in the ideal case will recover s(t) exactly: z 1 (t, r 0 ) = y(t)x (t r 0 ) (4.3) = s(t)x(t r 0 )x (t r 0 ) (4.4) = s(t) (4.5)

70 48 From this point forward in this document we treat both x and s (and therefore y also) as stochastic signals. We call the time series z 1 (t, r 0 ) emerging from the matched filter the detected signal. It is a function of time, and is proportional to the scattering amplitude of a target at range r 0. The range r 0 is actually a sample number with units of time quantized by the sampling period 1 f s, which we use as a delay in the transmitter signal x(t), but in terms of the target, r 0 corresponds to the bistatic slant distance r 0 c 2 With a sampling frequency f s = 100 khz, each sample is 10µs apart and each range delay is 1.5 km. During this detection step in the MRR signal processing, we also do a coherent integration, typically of 50 points or greater: z(t, r) = D 1 49 t =0 z 1 (t + t, r), t = 0, D, 2D,... (4.6) The time t now evolves on a much slower time scale than the samples t. are able to do this because the correlation time of the target is very large with respect to the correlation time of the reference signal. Thus, this integration step accomplishes a reduction in the data rate from the raw sample rate (100 khz) to the much lower typical target bandwidth ( 2 khz). This eases the burden of computing power spectra with the data, and puts appropriate limits on the range of Doppler velocities computed. Furthermore, it reduces the power from clutter arriving at ranges other than the range of interest, since x(t) becomes uncorrelated quickly during this interval. Also, the voltage integration gives us a processing gain (or, alternatively, the variance of the target time series is reduced). Processing gain (for a lowpass operation We

71 49 with real-valued coefficients) is defined by G P ( f[k]) 2 (f[k]) 2 (4.7) where f[k] is the window associated with the signal processor impulse response. In the above integration and decimation step, we achieve a processing gain of D per sample; the expression would become slightly more complicated if we used a data taper or overlapped/sliding windows. When D = 50, our processing gain at this step is 50, or approximately 17 db. Finally, the coherent integration step allows us to appeal to the Central Limit Theorem. The sum of many independent samples of the transmitter waveform allows us to approximate it as Gaussian distributed, even though it certainly is not. However, if we can assume that x is Gaussian, then we can use Isserlis Gaussian moment theorem [16], upon which we lean heavily in the estimator derivations below. Thus, during typical MRR operation, we perform matched filtering and coherent integration on the data, usually followed by FFT-based spectrum estimation. Below, we consider the radar signal processing in great detail in order to learn more about its effects on the statistics of the observed target. 4.3 Motivation for Statistical Calculations A major motivation for these derivations is to determine an appropriate resolution to expect from our instrument. In chapter 3, we showed several 2D interferometer images which were obtained by separating the cross-spectrum phase into bins, and plotting the total power in each bin over several ranges. The number of bins used in this operation is variable, and we essentially specify the transverse resolution of the interferometer by setting this parameter. Figure 4.3 shows a comparison of interferometer images with different range resolutions and transverse resolutions.

72 50 Range Resolution / Transverse Resolution Comparison 1.5 km / 1 km 3 km / 2 km 6 km / 5 km 12 km / 10 km 1100 Range (km) Transverse Position (km) Figure 4.3: A comparison of 2-D image resolutions obtained by range averaging and varying the number of phase bins used in the interferometer program (essentially phase averaging ). The data shown here is an auroral echo observed on February 2, 2002, UT 05:11. We claim that the range resolution value of 1.5 km (the finest resolution in the grid, and the one used at MRR currently) is appropriate; with this range resolution, each sample in the 100 khz time series represents a separate range, and if the samples from the transmitter are independent for each range, ideally there should be no range ambiguity. Advantages in spectrum computation and SNR are available if rangeaveraging is employed, but from a theoretical point of view, we assert that the intrinsic range resolution of a radar waveform with correlation time τ x = 10µs is c τ 2 x = 1.5 km. We would like to get a similar idea for an appropriate transverse resolution to

73 51 use in our interferometer images. The idea of resolution is related to the uncertainty associated with a measurement, which is in turn directly dependent on the variance of the estimate being displayed as a data product. Error bars are a common way to display information about the uncertainty of a measurement; often error bars are displayed as ± one standard deviation of an estimate. Tarantola [30] uses an instructive description of experimental data: while an estimated parameter can be completely described by its probability density function, an estimator essentially makes a decision about the true data value based on the distribution moments it estimates from experimental data. For example, in a grid of data displayed as an image, each pixel represents the sample mean of the true value; in the case of the interferometer, the location at which to plot each pixel is also an estimate. Therefore, the image resolution should be based upon the variance in the interferometer data products. In the following sections we derive expressions for the expected value and variance of our interferometer data product, the cross-correlation of a radar target between two scatter-receiving antennas. Our intent is to quantify the statistical properties of the estimate so that we can display our data in the most useful and least misleading manner possible. Following Sahr and Lind [27] and Hall [7], who derived similar expressions for the MRR estimator of the target autocorrelation, we identify sources of bias and the factors contributing to the variance in our interferometer estimates. We also describe the effects of signal degradation, such as interference and receiver noise, on the estimator. 4.4 The Interferometer Estimator To extract interferometric information from radar data, we estimate the complex cross-correlation of the target signal as it arrives on multiple antennas in a known

74 52 spatial configuration. We consider only the two-antenna case here. Our goal is to compute R pq ss(τ) = s p (t)s q(t τ) (4.8) which is the complex cross-correlation of the target signal s as it arrives on antennas p and q. With a conventional (pulsed) radar, the natural estimator takes the form v p (t)v q(t τ) where v p and v q are voltages on the scatter-receiving antennas. However, in the passive radar case, we first need to deconvolve the reference signal from the received scatter, as shown above in section 4.2. Therefore, we must use the detected signal z in place of the antenna voltages for our estimator (which we have named K). The additional z = yx correlation step results in a much larger number of terms to consider when evaluating the statistics of the passive radar interferometer estimator: ˆK pq (r 0, τ) = z p (t, r 0 )z q(t τ, r 0 ) (4.9) = y p (t)x (t r 0 )(y q (t τ)x (t r 0 τ)) (4.10) = y p (t)x (t r 0 )y q(t τ)x(t r 0 τ) (4.11) The estimate ˆK pq (r 0, τ) represents a single sample of the cross-correlation between antennas p and q of the scatterer at range r 0. For convenience we will often drop the clarifying subscripts pq and arguments r 0 and τ, when these are clear from context. In the next section we evaluate the expected value of this estimator. We show that it is biased by clutter and interference, although at long lags (where the transmitter signal becomes uncorrelated), it becomes unbiased.

75 Expected Value of the Interferometer Estimator The expected value of this estimator is a 4 th order correlation involving the reference transmission and the scattered signal observed from two perspectives (each antenna in the interferometer): ˆK = yp (t)x (t r 0 )y q(t τ)x(t r 0 τ) (4.12) However, the scattered signal is itself formed from the interaction of the transmitter signal x and any target signals it encounters. Before proceeding, then, we introduce a model for the voltages on the two scatter-receiving antennas A Model for the Received Scatter In the ideal case, which we treat first, we call the scattered signal y(t) a superposition of signals from all contributing ranges, modulated by successively delayed copies of the transmitter waveform. Models for y(t) on two antennas p and q (assumed to have the same gain) are given below. y p (t) = y q (t) = R s i (t)x(t i) i=1 R s i (t)e jkd cos θ i x(t i) i=1 (4.13) Here k is the wavenumber of the transmitter carrier frequency, d is the distance between the antennas, and θ i is the azimuth angle of the target (see figure 1.1 on page 5). The signals s i (t) are the target signals; we assume them in general to be zero-mean, complex Gaussian processes (s i (t) includes any Doppler information as well as radar cross section). For simplicity, we assume all the scattering signals are point targets in azimuth, so that θ i is a single value with a probability distribution that we model as Gaussian. However, superposition allows us to sum many targets

76 54 at one range in order to model targets that are distributed in angle, if we wish. R is the number of contributing ranges, equal to approximately 800, or 1200 km in the current implementation of MRR; after this distance, E-region targets fall below the horizon, and we cannot detect them with our line-of-sight system. (At high latitudes the magnetic field dip angle is such that VHF radars are unable to detect F-region scatter.) We must include all R ranges, because the transmitter operates continuously. As we will see, the continuous wave nature of the transmitter causes clutter from all ranges to be present in every correlation (or power spectrum) estimate; this is one striking difference between passive radars and conventional radars. There are advantages associated with the 100% duty cycle, however: the transmitter effective radiated power is much larger than that available with pulsed-mode radars. Finally, we note that the possibility for direct illumination of the scatter receivers by the transmitter has been included in the model, since we sum over all contributing ranges, including range zero (direct illumination would be at range number 51, as this is the amount of time it takes for light to travel directly from the transmitter to the scatter receivers). We expect ranges 1 through 50 to be empty after matched filtering, as the scatter and transmitter waveform should be completely uncorrelated.

77 55 Using the model for received scatter, then, we write the expected value of the cross-correlation estimator for a target at range r 0 as: ˆK = yp (t)x (t r 0 )y q(t τ)x(t r 0 τ) (4.14) = = = ( R ) x (t r 0 ) s n (t)x(t n) n=1 ( R ) s m(t τ)e +jkd cos θm x (t m τ) m=1 R R sn (t)s m(t τ)e +jkd cos θm n=1 m=1 R n=1 m=1 x(t r 0 τ) (4.15) x (t r 0 )x(t n)x(t r 0 τ)x (t m τ) (4.16) R e +jkd cos θ m sn (t)s m(t τ) x (t r 0 )x(t n)x(t r 0 τ)x (t m τ) (4.17) where the last equality follows from the fact that the transmitter signal x and the target signals s are zero mean and uncorrelated. We now find ourselves with a 4 th order correlation in x. As stated above, we assume that x is Gaussian distributed, which allows us to apply Isserlis Theorem to break higher-order even correlations into combinations of 2 nd order correlations (see discussion in section A.1 of appendix A). Under the Gaussian assumption, we have: ˆK = R R e +jkd cos θ m Rs,n (τ)δ nm n=1 m=1 [ x(t n)x (t r 0 ) x(t r 0 τ)x (t m τ) + x(t n)x (t m τ) x(t r 0 τ)x (t r 0 ) ] (4.18)

78 56 In writing s n (t)s m(t τ) = R s,n (τ)δ nm, we have made the assumption that target signals at different ranges are independent. Given our range resolution of 1.5 km, this assumption seems completely reasonable 2. This Kronecker delta function will select m = n, and dispense with one of the sums: R ˆK = e +jkd cos θ n Rs,n (τ) [ R x (r 0 n)rx(r 0 n) + R x (τ)rx(τ) ] (4.19) = n=1 R n=1 e +jkd cos θ n Rs,n (τ) [ R x (r 0 n) 2 + R x (τ) 2 ] (4.20) In order to better interpret this expression, we make the simplifying assumption that R x (τ) = R x (0)δ(τ). This is equivalent to saying that x is a white process with power (spectral level) R x (0). We justified this in section 4.1, and making the substitution here leads to the following simplification: ˆKpq (r 0, τ) = R e +jkd cos θ n Rs,n (τ) [ Rx(0)δ(r 2 0 n) + Rx(0)δ(τ) ] 2 (4.21) n=1 = Rx(0) [ 2 e +jkd cos θr 0 Rs,r0 (τ) + R δ(τ) e +jkd cos θ n Rs,n (0) ] (4.22) n=1 We see that the cross-correlation estimator is unbiased for nonzero lags (or, for a conservative estimate, lags greater than 20µs or so). At τ = 0, the delta function (due to the assumed whiteness of the transmitter signal) turns on a large clutter term containing the energy and interferometer phase from signals at all ranges. Thus, this lag is essentially useless 3, but otherwise we have the values we seek: the target 2 E-region scattering structures typically have a lifetime (correlation time) of a few milliseconds, which corresponds to less than a 10 meter extent when drifting at speeds near the ion acoustic speed. 3 However, we make use of it as an estimate of the noise floor in section 4.7.

79 57 autocorrelation at the range of interest, r 0, and the interferometer phase, φ = kd cos θ. The entire expression is multiplied by a constant factor (which we can estimate) of the transmitter signal energy squared. While identifying a condition under which the the correlation estimate is unbiased is certainly a good thing, it is not particularly useful in the situation where the primary radar data product is a power spectrum. This is currently the case with the normal operation of the Manastash Ridge Radar. Although the correlation estimates are only biased at short lags, the corresponding power spectra exhibit a nearly flat increase in the noise/clutter/interference floor due to the clutter term. To show this, we take the Fourier transform of equation 4.22: Ŝpq (r 0, f) = R 2 x(0) [ e +jkd cos θ r0 Ss,r0 (f) + ] R e +jkd cos θ n Rs,n (0) n=1 (4.23) This result is the expected value of the cross-spectrum between antennas p and q (not considering any effects of the cross-spectrum estimator itself, discussed in section 4.8.1). The desired target power spectrum is given by S s,r0 (f), and the interferometer phase remains the same. The zero-lag bias acts like a flat spectrum in the frequency domain, reducing the detectability of targets in which we are interested. Now, following Farley et al. [5], we normalize the magnitude by dividing by the individual power spectra on both antennas, to obtain the complex coherency: C pq (r 0, f) = Ŝpq (r 0, f) Ŝpp(r 0, f) Ŝ qq(r 0, f) (4.24) The phase difference does not appear in the self-power terms Ŝpp and Ŝqq (since the signals are multiplied by their own complex conjugates). Therefore, they are purely real, and we have [ ] R Ŝpp = Ŝqq = R 2 x (0) S s,r0 (f) + R s,n (0) (4.25) n=1

80 58 so we can write C pq (r 0, f) = S s,r0 (f) + e jkd(cos θn cos θr 0 ) R s,n (0) e +jkd cos θr n 0 S s,r0 (f) + R s,n (0) n (4.26) At frequencies where S r0 is large (in the interesting case where there is a target with significant amplitude), we can make the approximation C pq (r 0, f) e +jkd cos θr 0 (4.27) which gives us a coherence of unity and the phase accumulated over the extra distance traveled by the wave before it hits the second antenna. This is what we hope for and expect. Otherwise, in the absence of a strong scatterer, the coherence is small 4, and random fluctuations due to the clutter contribution in equation 4.26 dominate both the phase and magnitude spectra. The above approximation is useful because it demonstrates the ideal behavior of the interferometer in the presence of a strong scatterer. However, in practice, the coherence and phase estimates are severely affected by the random fluctuations caused by clutter at other ranges. For this reason, we have devised methods for estimating and subtracting the clutter-bias from our spectra. This will be addressed in section 4.7. Finally, although we have arrived at equations 4.26 and 4.27 by modeling only point targets in azimuth, the angular extent of the target also affects both the coherence and the interferometer phase [5]. In particular, if the angular extent is as large or larger than one interferometer lobe (2π of interferometer phase), the coherence will be low, the phase will appear unorganized (randomly distributed over the entire 2π of interferometer phase, depending on the target characteristics), and the interferometer will not detect a target, though the individual power spectra may clearly show one. 4 We briefly describe the statistical behavior of the coherence estimates in section 3.4.

81 59 In chapters 3 and 6 we show several experimental measurements of cross-spectrum estimates, and attempt to offer interpretations of the data, drawing from these analytic results A More Complicated Scatter Model In the previous section, we derived the expected value of an estimate of the crosscorrelation (and cross-spectrum) between two antennas in an interferometer. However, in our simple model for the received scatter (equations 4.13), we did not consider any non-ideal effects, such as receiver noise and interference. Here we briefly consider a new model for the received scatter which contains receiver noise and interference, which we define as any signal appearing on both antennas p and q at the frequency of interest (other than delayed copies of the reference FM signal). Interference, therefore, includes cosmic noise and other ambient radio-frequency energy in the environment which is either in the FM band or aliased into the FM band by our receiver 5. With this new model, we will find that the cross-correlation estimate contains a new bias term (at zero lag) consisting of the transmitter and interference signal energies. This type of analysis is useful in addressing larger questions such as whether the radar sensitivity is noise-limited or clutter-limited. Of course, we are still neglecting to model an endless number of nonideal elements in the system. Among these are the transmitter and receiver filters; receiver nonlinearity; antenna gain patterns; the propagation channel of the radio waves (weather, multipath, and fading effects); and imperfections in the reference copy of the transmitter waveform. These last two are discussed by Chucai Zhou [32] in his Ph.D. dissertation; we assume them all to be ideal here. 5 The receivers sample the FM band at 56 MHz, which causes a lot of energy to be aliased into our data. We use anti-aliasing filters; however, we still find interference from frequencies outside the FM band to be a problem.

82 60 The new model for the received scatter on an antenna p is: I [ ] y p (t) = n rx, p (t) + nsky, i (t)e jk p Ω R [ i + sn (t)e jk p Ω n x(t n) ] (4.28) i=1 n=1 As before, the signals s(t) contain amplitude and Doppler information about the targets at each range. We have written the phase term due to the angle of arrival separately for each antenna to illustrate the action of the interferometer estimator. The vector p represents the location of antenna p and the unit vector Ω represents the direction of the target with respect to the origin (here, the midpoint between the two antennas). Also present are a sum over all interference noise sources, n sky, and a receiver noise term, n rx, p, which is specific to antenna p. By specific, we mean that the receiver noise associated with the signal from one antenna is uncorrelated with that associated with other antennas. With new digital radio technology, however, the lines between receiver channels are becoming blurred. Our current system (see chapter 2) samples RF signals directly, implementing most of the preliminary signal processing (filtering, downconversion, amplification, resampling) digitally. This can result in separate channels producing identical data, if they are tuned to the same antenna at the same frequency and share other settings. This raises questions about the independence of separate receiver channels. We still use, however, separate analog amplifiers and passive filters on each antenna signal prior to digitization, so for now our assumption of uncorrelated receiver noise is safe.

83 61 To find the expected value of the K estimator using the model in equation 4.28, we again write the cross-correlation between the detected signal on antennas p and q: ˆKpq (r 0, τ) = y p (t)x (t r 0 )y q(t τ)x(t r 0 τ) (4.29) = ( xx n rx, p + ( n rx, q + ) I R n sky, i e jk p Ω i + s n xe jk p Ω n i=1 n=1 ) I R n sky, i e jk q Ω i + s n xe jk q Ω n i=1 n=1 (4.30) On the second line we have dropped the complex conjugate signs and the time arguments of the signals for compactness, mimicking the shorthand notation found in Sahr and Lind s description of the MRR target autocorrelation estimator [27]. This looks messy, but in fact, when we consider that the receiver noises n rx, p and n rx, q are zero-mean and uncorrelated with each other as well as every other signal in the expression above, the equation simplifies rapidly. Many of the cross-terms average to zero and drop out of the expectation. This is an advantage we have since our goal is the cross-correlation between antennas rather than the autocorrelation of a signal received on only one antenna. As Paul Hall shows [7], the noise term survives in the self-correlation and, in the end, contributes a value of noise power scaled by the transmitter signal power to the zero-lag of the target autocorrelation estimate.

84 62 We also note that the interference sources n sky are uncorrelated with the transmitter signal x and the target signals s. This only leaves two terms: ˆK = ( xx n sky, i n sky, j e jk( p Ω i q Ω j ) + s n s m xxe jk( p Ω n q Ω m)) (4.31) i j n m = xxn sky, i n sky, j e jk( p Ω i q Ω j ) + s n s m xxxxe jk( p Ω n q Ω m) (4.32) i j n m = xx n sky, i n sky, j e jk( p Ω i q Ω j ) + i j s n s m xxxx e jk( p Ω n q Ω m) (4.33) n m Finally, we reason that two separate sources of interference will be uncorrelated, and repeat our assumption that targets at different ranges are uncorrelated. This allows us to remove two of the sums: ˆK = δ ij R sky, i (τ)rx(τ) e jk( p q) Ω i + i j δ nm R s,n (τ) xxxx e jk( p q) Ω n (4.34) n m = R sky, i (τ)rx(τ) e jk( p q) Ω i + R s,n (τ) xxxx e jk( p q) Ω n (4.35) i n The second term in equation 4.35 is the same as the cross-correlation estimate with the ideal model for y(t), and simplifies to the expression in equation 4.22 above. The first term is a new bias contribution due to interference. If we use the approximation R x (τ) = R x (0)δ(τ) again, we find that the new bias term is I δ(τ)r x (0) R sky, i (0) e jk( p q) Ω i (4.36) i=1 which, like clutter, only contributes to the zero lag of the cross-correlation. Thus, the noise floor in the cross-spectrum is raised by an additional amount equal to the

85 63 sum of the signal power of all interfering signals, times the power in the transmitter signal The Expected Value after Coherent Integration We have not yet considered a coherent integration of the radar time series in our analysis, though we discussed this as part of the default MRR signal processing in section 4.2. If we use a coherent integration of D points according to equation 4.6, our estimator of the cross-correlation (using the basic scatter model in equations 4.13) becomes ˆK (d) pq (r 0, τ) = = ( D 1 ) ( D 1 ) z p (t + a, r 0 ) zq(t + b τ, r 0 ) a=0 b=0 D 1 a=0 D 1 b=0 (4.37) y p (t + a)x (t + a r 0 )y q(t + b τ)x(t + b r 0 τ) (4.38) (where the (d) superscript is for decimation, named for the decimation in the raw data rate due to coherent integration). We note that successive time lags (τ) are now further apart by a factor of D, although the time (t) still advances at the raw sampling rate. The expected value of the interferometer cross-correlation using the estimator given above is ˆK(d) pq (r 0, τ) = R 2 x(0) D 1 a=0 D 1 b=0 [ e +jkd cos θ r0 Rs,r0 (τ + a b) + δ(τ + a b) R e +jkd cos θ n Rs,n (0)] n=1 (4.39) Essentially, the coherent integration causes each point of the target autocorrelation (now sampled more sparsely) to be a weighted sum (a triangular window, actually) over the adjacent closely-spaced lags. The equivalent frequency-domain operation

86 64 is the filtering of the power spectrum by a sinc 2 ( ) type smoothing window. Since R s,r0 (τ) remains nearly constant over the interval ±D, the resulting correlation function of the target is much stronger with respect to any noise or interference that may be present (these do not show up in our expected values since we assume they are perfectly uncorrelated for nonzero lags, but in practice they do affect the radar data). R x (τ), on the other hand, becomes decorrelated quickly within this same interval, so there is no clutter price to pay. We will see that this effective smoothing will show up as a reduction in variance in later sections. 4.6 Variance of the Interferometer Estimator Variance with the Simple Scattering Model We now work out the variance of the cross-correlation estimator using the model for the scattered signal given in equations The definition of variance is Var ˆK = ( ( ˆK ˆK ) ˆK ˆK ) (4.40) = ˆK ˆK ˆK ˆK (4.41) (which is just the second central-moment). To ease the presentation, we first tackle the ˆK ˆK term, using the expression for ˆK in equation 4.20: ˆK ˆK = ( R e +jkd cos θ n Rs,n (τ) [ R x (r 0 n) 2 + R x (τ) 2 ]) n=1 ( R e jkd cos θ l R s,l (τ) [ R x (r 0 l) 2 + R x (τ) 2 ]) (4.42) = l=1 R R n=1 l=1 e +jkd(cos θ n cos θ l ) R s,n (τ)r s,l(τ) [ Rx (r 0 n) 2 + R x (τ) 2 ][ R x (r 0 l) 2 + R x (τ) 2 ] (4.43) Note that in writing e +jkd cos θn e jkd cos θ l = e jkd(cos θ n cos θ l ), we have assumed

87 65 that the scatterer angular distributions (θ) are uncorrelated at different ranges. Next, ˆK ˆK = y p (t)x (t r 0 )y q(t τ)x(t r 0 τ) = yp(t)x(t r 0 )y q (t τ)x (t r 0 τ) (4.44) x (t r 0 )x(t r 0 )x(t r 0 τ)x (t r 0 τ) ( R ) ( R ) s n (t)x(t n) s i (t τ)e +jkd cos θ i x (t i τ) n=1 i=1 ( R ) ( R ) s m(t)x (t m) s l (t τ)e jkd cos θ l x(t l τ) (4.45) m=1 l=1 ssss xxxxxxxx e jkd(cos θ i cos θ l ) (4.46) = n m l i The last equality follows from the fact that s and x are zero-mean and uncorrelated (and assumed to be Gaussian). Now we will handle the ssss and xxxxxxxx terms separately. As before, we rely on Isserlis Gaussian moment theorem to break up the large correlations: ssss = sn (t)s m(t) s l (t τ)s i (t τ) + sn (t)s i (t τ) s m(t)s l (t τ) (4.47) = ( )( ) R s,n (0)δ nm Rs,l (0)δ li + ( )( ) R s,n (τ)δ ni R s,l (τ)δ lm }{{}}{{} term 1 term 2 (4.48) Again we have assumed that the targets are uncorrelated in range. The Kronecker delta functions in term 1 will cause m = n and i = l, while in term 2 they force i = n and m = l. Since these different conditions cause different correlations of the x terms, we handle each piece separately:

88 66 ˆK ˆK = R s,n (0)R s,l (0) xxxxxxxx + n l }{{} term 1 R s,n (τ)rs,l(τ) e jkd(cos θn cos θ l) xxxxxxxx n l } {{ } term 2 (4.49) Next, we must deal with the 8 th -order correlation in x. Using the Gaussian moment theorem, this will reduce to a total of 24 2 nd -order terms. For details, please refer to appendix A. For brevity, we skip to the final result after the substitution R x (τ) = R x (0)δ(τ) has been applied: Var ˆK = [ Rx(0) 4 Rs,r 2 0 (0) + 6δ(τ)Rs,r 2 0 (0) + δ(τ) Rs,n(0) 2 n + R s,r0 (0) R s,n (0) ( 2 + 4δ(τ) + 2δ(τ + r 0 n) + 2δ(τ + n r 0 ) ) n + R s,n (0)R s,l (0) ( 1 + δ(τ) + δ(τ + r 0 n) + δ(τ + l r 0 ) n l + δ(τ + l n) + δ(τ + l r 0 )δ(τ + r 0 n) ) + 2 R s,r0 (τ) 2 + 8δ(τ) R s,r0 (τ) 2 + 3δ(τ)R s,r0 (τ) n e jkd(cos θ r0 cos θ n) R s,n(τ) + 2δ(τ)Rs,r 0 (τ) e jkd(cos θ n cos θ r0 ) R s,n (τ) n e jkd(cos θ n cos θ l ) R s,n (τ)rs,l(τ) + δ(τ) n l + n R s,n (τ) 2 ( 1 + δ(τ) + δ(τ + r 0 n) + δ(τ + n r 0 ) )] (4.50) To make this expression easier to interpret, we introduce the notation C (τ), which we define to be the clutter contribution from all ranges (thus the subscript).

89 67 The argument τ indicates that the clutter term is the sum of all target autocorrelation functions at lag τ. In particular, C (0) = R R s,n (0) (4.51) n=1 is the sum of all reflected power from every range. Likewise, we define C n (τ) = R s,n (τ) to be the clutter contribution from only one range, n. Using this notation, we can approximately 6 write equation 4.50 as [ Var ˆK Rx(0) 4 Cn(0) 2 + 2C n (0)C (0) + C (0) 2 + 2Cn(τ) 2 + n=1 [ R + δ(τ)rx(0) 4 6Cn(0) 2 + Cn(0) 2 + 4C n (0)C (0) n=1 + 2Cn(0) 2 + 2Cn(0) 2 + C (0) 2 + C n (0)C (0) + C n (0)C (0) + C n (0)C (0) + C 2 n(0) R ] Cn(τ) 2 + 8Cn(τ) 2 + 3C n (τ)c (τ) + 2C n (τ)c (τ) R ] + C (τ) + Cn(τ) 2 + Cn(τ) 2 + Cn(τ) 2 n=1 [ ] Rx(0) 4 C (0) 2 + 2C n (0)C (0) + Cn(0) 2 + (R + 2)Cn(τ) 2 [ + δ(τ)rx(0) 4 C (0) 2 + 7C n (0)C (0) + (R + 11)Cn(0) 2 ] + C (τ) 2 + 5C n (τ)c (τ) + (R + 10)Cn(τ) 2 (4.52) (4.53) where we have also absorbed the autocorrelation function of the target of interest (R s,r0 ) into the clutter at only one range term, C n. We see that the variance of a single estimate of the cross-correlation at lag τ (the particular range of interest is now indistinguishable) depends on the signal power, 6 We ignore phase terms and complex conjugates; we do not distinguish between individual clutter ranges. We defend the approximation by claiming that the new notation is merely a crutch for making the variance expression quickly meaningful; uniformly representing all clutter with the C symbol and disregarding phase information can only cause us to overestimate the variance.

90 68 or radar cross section, of all scattering targets. It is also strongly affected by the transmitter signal power. While part of the expression for the variance applies only for short lags (the part multiplied by δ(τ)), the remaining portion is still quite large. Finally, for a very over-simplified interpretation, if we assume that the clutter arriving from only one range is much smaller than the clutter arriving from all ranges (i.e., C n (τ) C (τ)), and we avoid looking at very small correlation lags (τ > 20µs), we obtain the following very simple approximation: Var ˆK R 4 x(0)c 2 (0) (4.54) From the derivation of equation 4.50 in appendix A, we note that the fast decorrelation of the transmitter waveform helps immensely in reducing the variance due to clutter, though it remains significant, and probably would prevent us from noticing all but the strongest echoes. However, we actually do much better than this. Note that in equation 4.50 (or any subsequent statement of the variance), there is no dependence on N (a number of data points used in the estimation). This would imply that we have an inconsistent estimator: one whose variance properties we cannot improve by providing more data. However, we have considered only a single cross-correlation estimate formed from three voltage measurements at an instant in time (one from the reference antenna and two from the scatter-collecting antennas). In practice, averaging is used on many time series voltages before the correlation or spectrum estimation is performed. Also, many individual cross-correlation estimates are incoherently averaged to form a final result. The latter step serves to reduce the variance from the expression in equation 4.50 by a factor of 1/ N, where N is the number of estimates averaged together. We discuss both of these options in more detail below.

91 Variance after Coherent Averaging Again, we note that we have not yet considered any coherent integration of the target time series in our analysis. If a coherent average is used, i.e. z(t, r) = 1 D D 1 49 t =0 z 1 (t + t, r), t = 0, D, 2D,... (4.55) we can achieve a result with smaller variance than what is indicated above. This is because equation 4.55 is a lowpass-filtering operation, so the variance in the target time series itself is reduced by a factor of 1/D, an improvement of 17 db when D = 50. Including the coherent average in the derivation of the variance immediately introduces a factor of 1/D 4 ; however, it also introduces 4 additional sums and vastly increases the complexity of the expression. Therefore, we do not work out an analytic expression for the variance after coherent averaging here. We note that Hall [7] works out a full expression for the single antenna case (autocorrelation estimator), and Sahr and Lind [27] also show an expression, organized as a polynomial in transmitter correlation function (degree of clutter suppression). At this point, we lean on other arguments to get an analytic idea of the variance, and we show numerical estimates of variance in simulated data in chapter Interferometer Implementation Issues As we mentioned earlier, the interferometer cross correlation estimator is unbiased for lags greater than the correlation time of the FM waveform. This is a useful result, yet if one prefers to work in the frequency domain (the preferred form of most MRR data products), the energy contained in lag zero of the correlation function still appears over all frequencies as a noise and clutter floor. Also, the interferometer phase is significantly affected by the clutter at all ranges. Therefore, we require a method of

92 70 estimating the interferometer bias (mainly due to clutter), and removing it from the cross-spectrum. A preliminary technique for estimating the bias is to calculate the cross-spectrum at a distant range 7 assumed not to contain any signal of interest, but which will still contain any interference as well as the clutter from all ranges. We then subtract this bias estimate from the cross-spectrum at the range of interest. Point-by-point subtraction allows us to distinguish between the bias that may be present in different Doppler bins, which will be useful in a frequency-selective fading environment, or in the presence of narrowband interference sources. This technique is not perfect, and it can be shown that the distant range bias estimate will miss errors that enter into the spectrum as cross-terms between a target signal and clutter. To see this, consider the following: (clutter + target) 2 = clutter 2 + 2(clutter)(target) + target 2 (clutter + nothing) 2 = clutter 2 where the top line is the desired cross-spectrum and the bottom line is the bias estimate. Alternatively, we can use the zero lag of the cross-correlation function as a (Doppler independent) estimate of the bias due to clutter, noise, and interference. We show an example of a target cross-correlation function in figure 4.4; the data is from the same meteor trail echo we showed in figure 3.6 on page 28. Comparing the zero-lag value of the meteor correlation function with the spectrum floor value (bottom panel of figure 3.7), we find that they are consistent. But this method, too, has the disadvantage that lag zero of the correlation function also contains energy from the target 7 We have also tried averaging spectra from many empty ranges together for a bias estimate; this does not noticeably improve the correction-technique performance. It does, however, significantly increase computational load.

93 71 Cross Correlation Magnitude 2.5 x Correlation Function of Meteor Trail Time Series Noise + Clutter Level ~163 db Lag (ms) Figure 4.4: Cross-correlation magnitude of the meteor trail echo shown in figure 3.6. The noise, clutter, and interference contribution is contained in lag zero. of interest itself. In figure 4.5 we demonstrate the two techniques described above on the meteor trail time series, and compare the resulting coherence and phase spectra to the uncorrected cross-spectrum. We conclude that some method of bias removal is needed, based mostly on our experience viewing many examples of interferometer data. In many cases, the target phase is indistinguishable from the background clutter unless one of the above methods is used. Although in this particular example the zero lag correlation method produces a more readable magnitude spectrum (the variance appears smaller), we have chosen to use the alternate range estimator for the bias (mainly due to its success with the phase spectrum).

94 72 1 No Bias Removal 0.5 Normalized Cross Spectrum Magnitude and Phase With Alternate Range Estimate of Bias With Zero Lag Correlation Estimate of Bias Doppler (m/s) Figure 4.5: A comparison of bias removal techniques. The top plot shows the uncorrected spectrum; the next two demonstrate different correction methods discussed in the text. The data is from the meteor trail observed on January 3, 2002 at UT 19:23.

95 Statistics of Other Estimators and Additional Comments The Periodogram (FFT-Based Spectrum Estimation) Since we have worked out most of our formulas in the correlation (time lag) domain, we have not yet considered one of the most widely used methods of spectral estimation (indeed, the one that MRR currently implements): the periodogram. The unaltered periodogram (in discrete frequency) is the magnitude squared of the DFT of the data, Ŝ (p) [k] 1 N 1 2 x[n]e j2πkn/n N n=0 (4.56) Using this method for estimating radar power spectra is computationally efficient, since we are able to use the FFT algorithm. We have shown that clutter and interference cause bias in our power spectrum estimates by contributing to the noise floor. By considering the periodogram as a power spectrum estimator, we discover another type of bias due to finite data length. As shown in many texts which deal with spectral estimation (Percival and Walden [23] and Oppenheim and Schafer [22], for example), supplying the DFT with only a finite sequence of data is equivalent to supplying it with (what must be assumed to be) an infinite time series, multiplied by a rectangular window. In the frequency domain, the result is the desired spectrum convolved with a sinc function, which, depending on the length of data supplied, will have a mainlobe of a certain width (causing blurring of the spectrum) and sidelobes of a certain height which persist into adjacent frequencies, causing bias and further blurring at those frequencies. The problem might be better understood as one of frequency resolution; the function that is convolved with the true spectrum is called a spectral window. As the length of data fed into the DFT becomes larger (the rectangle becomes wider), the mainlobe width of the spectral window decreases, giving better frequency resolution. Other techniques called data-tapering have been developed to alleviate the leakage of power from one frequency to another due to the sidelobes of

96 74 the spectral window. These involve pre-multiplying the time series to be analyzed by functions that taper the beginning and end of the times series more gradually toward zero (forcing down the sidelobes in the frequency domain). There is a trade-off, however, between resolution and leakage (bias), since these data-taper spectral windows also have increased mainlobe width. Much work has been done in exploring various data-tapering and spectral estimation techniques and their associated trade-offs ([23] and references therein, for example) Application to MRR Signal Processing Experimentation with data tapering at MRR has indicated that our power spectrum estimates are not significantly impacted by sidelobe leakage effects, so we only consider the above in the context of the trade-off between frequency (Doppler) resolution and variance in our radar data products. Given a finite amount of data N from which to estimate a power spectrum, if we use an M-point FFT, we are able to average approximately N/M spectra together for a final estimate. The incoherent averaging reduces the variance in our estimate by a factor of 1/ N/M = M, which we would N like to be as small as possible; however, the larger M is, the better Doppler resolution we achieve. Thus, we must balance the classical trade-off between frequency resolution and variance. We now name some specific values for the parameters above that are in use at MRR now. In order to determine the amount of time for which we can take data (i.e., the timeresolution of our power spectra), we consider the statistical behavior of our intended target. We would like to take data continuously for the longest length of time possible while the target remains statistically stationary. A time resolution T stationary could be inferred for E-region irregularities by considering inherent time scales of plasma processes, wave growth and decay mechanisms, etc. We do not discuss this particular

97 75 problem here, however. The current time resolution we use at MRR is 10 seconds 8. Next, we need to determine a sampling frequency. There are two considerations here: we want to capture the majority of the power in the illuminating signal, and we would also like to not oversample so much that we have many samples during the correlation time τ x of the transmitter waveform. In our case, τ x was shown above (section 4.1) to be approximately 10µs, which corresponds to a receiver bandwidth of 100 khz, which fortunately is also sufficient to capture about 90% of the FM signal s power. Therefore, we sample our receivers with an output data rate of 100 khz. Now we must consider the correlation time of the target. Since we d like to end up with a time series for analysis composed of independent samples, we decimate the 100 khz time series, typically by 50 (see equation 4.6) to obtain a new time series which evolves at the much lower bandwidth of the target ( 2 khz). This gives us an approximate Doppler range of ±1500 m/s; a longer decimation would give us a smaller Doppler window. This decimation (actually a coherent integration of 50 samples, over which the target is correlated but the transmitter waveform is not) not only achieves a reduction in the data rate and produces a time series of independent target samples, but also suppresses clutter from other ranges and reduces the variance in the radar data products (see section 4.6.2). Finally, we are left with (10 seconds) (100 ksamples) decimation rate of 50 = 20, 000 samples with which to produce a spectrum. If we use a 256-point FFT, we get 78 individual spectra which we can average for a final estimate, giving us a reduction in variance by 8 Long observation times (2 minutes of continuous data) have revealed field-aligned irregularities that appear to be unchanging during this interval; however, we have also observed irregularities that do not even last for the typical 10 second duration. The choice of 10 seconds is somewhat arbitrary, but serves us well.

98 76 1/ 78, or approximately 9.5 db. The 256-point FFT gives us a Doppler resolution of f s λ 2 FFT size decimation = 100 khz 3 meters m/s Fortunately, it is always possible to reprocess passive radar data, reducing its variance at the expense of lower resolution in any of the dimensions of time, range, or velocity 9. This result applies both to individual power spectra from single antennas and cross-spectra between two antennas (for interferometry). In the latter case, rather than squaring the magnitude of the DFT to obtain a power spectrum estimate, we multiply the DFT of the time series on one antenna by the complex conjugate of the DFT from the other antenna. This is the same operation in the frequency domain as the interferometer estimator given in equation Periodogram Variance We have been talking about variance in the radar power spectrum estimates; now we cite some theoretical results describing second-moment properties of the periodogram. Percival and Walden [23] show that, for a zero mean white Gaussian noise process, the periodogram samples (with the exception of those at zero or Nyquist frequency) can be described in distribution by a chi-square random variable with 2 degrees of freedom (i.e., they are the sum of the square of two Gaussian random variables), times one half the variance of the original process. They further show that S 2 [k], 0 < k < N/2 Var {Ŝ(p) [k]} = 2S 2 [k], k = 0 or N/2 (4.57) for a white Gaussian process, where S[k] is the true spectrum of the time series, and for any stationary process, asymptotically as the sample size N gets very large. Thus, the periodogram is an inconsistent estimator: its variance does not decrease as the sample 9 This also gives us the interesting option of mimicking the resolutions of other radars.

99 77 size grows. Also, we see that the variance of the spectrum estimate scales as the square of the value of the true power spectrum itself. This result is similar to the one we show above in which the variance of the target auto- or cross-correlation estimator is strongly dependent on the correlation functions of the transmitter waveform and the targets themselves. It would seem that stronger targets produce spectral estimates which are more difficult to interpret, a counter-intuitive conclusion. Fortunately, we find that the variance of the phase spectrum estimates decreases with increasing SNR, and this is the quantity we are most interested in, at least for the purpose of determining an appropriate transverse resolution for our interferometer. In the next chapter, we describe this result and apply it to the resolution problem.

100 78 Chapter 5 SIMULATIONS In God we trust everyone else has to bring data. Dr. Jesse Poore We have discussed causes of bias and variance in our interferometer data products and derived analytic expressions for these for several cases. Now we turn to numeric simulations to get an empirical estimate of the quality of our data. Simulations are useful tools because they allow us to completely control the environment in which the radar is operating, and we have perfect knowledge of each signal involved. Also, true ensemble averages can be calculated, since we are able to use the same illuminating and target signals over and over again, only changing the noise in the system. However, there are many assumptions and approximations involved with simulating real world processes; we describe in detail our techniques for simulating passive radar data in appendix B. 5.1 A Simulated Target For all of our simulations, we use MATLAB to generate reference and scatter data files, then use our regular radar signal processors (in the C/C++ language) to do the detection and spectrum estimation. An example of simulated interferometer data is shown in figure 5.1. The target present at the displayed range was created with a mean Doppler shift of 100 m/s and a Doppler width of 50 m/s; the echo originates from a single azimuth angle of 30, corresponding to an interferometer phase of -0.9

101 79 Cross Spectrum Phase and Magnitude for a Simulated Target Phase Normalized Magnitude Self Spectra (db) antenna 1 antenna Doppler (m/s) Figure 5.1: A simulated target, with -13 db SNR. radians on an antenna baseline of 16λ; and the SNR of the target signal with respect to the simulated noise is -13 db. However, in the bottom panel of figure 5.1, where the individual power spectra on the two antennas are shown, we see that the peak of the target rises above the noise floor by approximately 6 db. This increase in SNR is due to signal processing gain: a matched filter is used on the target echo, and we also used a coherent integration of 200 samples in this case, which provides a boost of about 23 db to the target signal, which remains correlated over the interval. We are also able to use 19 incoherent averages, with an FFT length of 256 ( ), and this gives us a small processing gain of 10 log db. We have purposefully simulated targets which only exist at a single azimuth angle

102 80 (no transverse width or azimuthal distribution). The reason for this is to prevent the misinterpretation of angular spread as variance in the cross-spectrum. The effect of angular spread on coherence and phase spectra is well understood [5]; we wish to evaluate the variance of our estimator. Also, in the data shown in figure 5.1, as well as in all the simulations we describe below, we have used a set number of extra targets at other ranges. We do not vary these extra targets; this is our method for simulating known, controlled clutter in the radar data. Each of the 3 clutter targets we use has an SNR of 0 db (before signal processing). Their range, Doppler, and azimuth characteristics are unique. From the discussion of interferometer bias in section 4.5.1, we expect the crossspectrum noise floor to be raised uniformly by an additional level equal to the transmitter signal energy squared times the sum of all the target signal energies: R 2 x(0) R s (0) This is the clutter contribution to the bias. Each signal in our simulation has an energy of 1000 (the 2-norm of every signal, including the transmitter signal, noise, and all targets, is approximately equal to 1000). Of course, after the -13 db SNR is applied to the main target, its energy becomes = 50. Next, we boost the transmitter and scattered signal energies in our simulations by a factor of 1000 so that when the simulated samples are truncated into the short data type, their information will not be lost (without the gain of 1000, most samples vary between 0 and 1). This adds an additional factor of to the noise plus clutter floor. Finally, we also take into account the processing gains due to the coherent and incoherent integrations. Then, we expect the noise floor in our simulations with clutter to be

103 81 noise floor = 10 log 10 ( ( )) (clutter contribution) + 10 log 10 ( ) (boost of 1000) + 23 db (coherent processing gain) + 7 db (incoherent processing gain) = db = 185 db From figure 5.1, we see that the power spectrum noise floor is approximately 186 db, which is consistent with this rough analysis. 5.2 Empirical Estimation of Cross Spectrum Variance Now we use our simulator to generate many measurements of the variance in our crossspectrum estimates. For these simulations, as discussed above, we use three targets at other ranges to represent clutter. These clutter targets, as well as the transmitter signal and the target of interest, remain constant throughout all our simulations. The only parameters we vary are the SNR associated with the target of interest, the (complex, white Gaussian) noise itself, and the amount of coherent averaging 1 used in the signal processing before spectrum estimation. Also, to avoid complexity, we do not consider normalization of the cross-spectrum magnitude here (we are interested mainly in the statistical properties of the phase estimates), and we do not perform any bias-removal techniques with our signal processor. 1 Instead of coherent integration, we use coherent averaging, which also divides by the number of samples summed together. As we suggested in section 4.6.2, this acts to decrease the variance in the target time series by a factor of 1/D, where D is the number of samples in the integration.

104 82 M spectral estimates... average together for "final" spectral estimate. variance of this estimate = v/sqrt(m). variance of individual spectrum estimates = v Doppler..... Repeat for many instances to get an estimate of the variance of the spectral estimates... Figure 5.2: An illustration of the variance estimation method used here. Figure 5.2 illustrates our variance estimation method: the simulated reference and scatter time series are both 10 6 points long. Depending on the FFT length and the initial coherent integration used, we obtain a certain number (denoted M in figure 5.2) of spectral estimates (at one particular range containing a simulated target of interest) from each simulation. Then, for each frequency bin, we obtain a variance estimate using all the spectra. (Note that the variance of the final spectral estimates, which are the result of averaging the M spectra together, is smaller than the variance we estimate by an incoherent averaging factor of 1/ M. This is an important distinction during our discussion of resolution based on variance in section 5.3.) We also determine the (post-processing) SNR 2 at each Doppler bin, and organize 2 We first estimate the noise floor by taking the average value of the spectrum magnitude with

105 83 our variance results in this way. We repeat this process many times, and average our variance estimates at each SNR. Figure 5.3 shows the resulting variance versus SNR curves; we display the variance of the cross-spectrum in three different forms: complex ( total ), phase, and magnitude. By total variance, we mean the complex 2 nd central moment, Var Ŝ = ( Ŝ Ŝ ) ( Ŝ Ŝ ) (5.1) = ŜŜ Ŝ Ŝ (5.2) The bottom panel of figure 5.3, called points averaged, displays the number of variance estimates (each computed from 19 spectral estimates) that were averaged together to form the final variance estimate at each SNR. Since there are far more frequencies with low SNR, many points went into the variance curves at low SNR, while the number of points averaged drops off for the variance estimates at high SNR. We also compare the variance in cross-spectra after different amounts of coherent averaging of the target time series. Therefore, even though more spectral estimates are available from the simulations using shorter coherent averages, for a fair comparison we have limited the number of spectra used in each variance estimate to 19 (which is the number available from 10 6 data points using a 256-point FFT and the largest decimation we consider, 200). Thus, for every coherent average case, each variance estimate has the same variance. As the top panel of figure 5.3 shows, the spectrum variance decreases as larger coherent averages are used, which we expect (however, the trade-off for large coherent averages/integrations is that we are unable to detect faster-moving targets without additional signal processing tricks). no target present (although the 3 clutter terms are still included), then estimate the SNR by subtracting the noise floor from each value in our spectra. If this quantity is negative, we declare the SNR to be zero. We note that the noise floor will be different depending on the amount of coherent averaging used.

106 84 Complex Cross Spectrum Variance vs. SNR Coherent Average Length Phase Magnitude Points Averaged SNR (db) Figure 5.3: The top three panels show different representations of cross-spectrum variance versus SNR. The bottom panel shows the number of variance estimates that were averaged together to form the final value.

107 85 Another observation about the total cross-spectrum variance in figure 5.3 is that it tends to increase with SNR. We expect this increase in variance with SNR, because when the SNR increases, the spectral values themselves become larger (by a factor of a, for example), which will increase the variance by a factor of a 2 : Var aŝ = ( aŝ ( aŝ ) aŝ aŝ ) = ( aŝ a Ŝ ) ( aŝ a Ŝ ) = a ( 2 Ŝ Ŝ ) ( Ŝ Ŝ ) = a 2 Var Ŝ (5.3) Indeed, as we describe in section 4.8.3, the variance of the periodogram estimator is proportional to the square of the value of the true spectrum. We find this approximately to be the case in the top panel of figure 5.3: the relationships between variance and SNR are all roughly linear with a slope of 2 db per db of SNR. A linear increase on a log-log plot indicates a power-law relationship, with the power equal to the slope of the line. Thus, the variance is proportional to the SNR raised to the 2 nd power, and our estimator is performing as we expect. Next, we compare the simulated variance to that predicted by the (first-order approximation) analytic expression from equation 4.54, Var ˆK R 4 x(0)c 2 (0) (5.4) Comparison with a more detailed expression for the variance (such as equation 4.50) is difficult due to the many specific clutter properties required in the formula. The expression for total variance in equation 5.4 was derived for a single crosscorrelation estimate (i. e., no coherent averaging), so we expect it to agree roughly with the dotted line (corresponding to a coherent average length of 1) in the top panel of figure 5.3.

108 86 Proceeding in the same manner as in section 5.1, we evaluate the variance formula with the 4 targets and transmitter power: variance 4 10 log 10 (1000) (transmitter power) log 10 (1000) (boost of 1000) log 10 ( ) (clutter) db 310 db The value of is in the appropriate area of the top panel of figure 5.3, which indicates that our very rough analysis is consistent with simulated data. We are most interested, however, in the quality of the cross-spectrum phase and magnitude estimates separately (especially the phase, since it determines our interferometer s transverse resolution). The first thing we note from figure 5.3 about the phase and magnitude variances is that they don t seem to be affected by coherent averaging (a counter-intuitive result). The magnitude variance still increases with SNR, which is not surprising given the discussion above. The phase variance, though, becomes smaller as the spectrum SNR grows larger. This is an excellent and intuitive result for us, since it indicates that we will be able to use finer transverse resolution with stronger targets. As a target, which exists in one particular region in the sky, returns more power to the scatter receiver, the cross-spectrum phase becomes more likely to take on the value corresponding to that particular direction than any random fluctuation. The range of possible phase values effectively shrinks. In the case of the magnitude, when a target is present, the mean value of the magnitude shifts upwards, but the range of possible values also grows, and no rule exists to constrain those values.

109 87 3 Variance of Cross Spectrum Phase Estimates vs. SNR Phase Variance (radians) Signal to Noise Ratio (db) Figure 5.4: Cross-spectrum phase variance plotted as the spectrum SNR increases. 5.3 Evaluation of Interferometer Resolution In figure 5.4 we show again a plot of variance in cross-spectrum phase estimates versus SNR. To create this plot, we have used an initial coherent average length of 50, and included the full 78 resulting spectra in each estimate of the variance. We ran the simulation 30 times, varying the (pre-processing) target SNR between -5 and -15 db. The resulting plot is somewhat smoother than the one in figure 5.3, and we use it in determining transverse resolutions for the interferometer. logic: We relate variance in the phase estimates to transverse resolution by the following

110 88 Variance = (Standard Deviation) 2 2 Standard Deviation = Resolution, in radians 2π/Resolution = Number of Phase Bins over 1 Interferometer Lobe We are defining the transverse resolution (in radians) to be that phase range which includes a phase estimate ± its standard deviation; this is a common way to display error bars on data. To determine the variance implied by the use of, say, 100 phase bins, we write 2π/100 = 2 std = var = std 2 = SNR > 35 db We see immediately that 100 phase bins is an unreasonably fine grid. Indeed, even a very grainy image with only 5 bins across the 2π radians of interferometer phase requires an SNR of 15 db or greater, judging by our results here. It appears that our interferometer has almost no resolution at all. However, we have not yet considered the effect of incoherent averaging of the spectral estimates, and this reduces the variance from that shown on the curve in figure 5.4 significantly. As we described in section 4.8.2, our final spectrum estimate is actually the average of several spectra computed independently from the available data (we assume the target is statistically stationary over the time period of observation). Therefore, the final cross-spectrum phase exhibits a variance reduced additionally by a factor of 1/ N, where N is typically 78, for an initial decimation of 50 (in other words, the number of points that went into our variance estimates shown above).

111 89 Table 5.1: Interferometer transverse resolutions appropriate for a given SNR. SNR (db) Variance Resolution Resolution # Phase Bins (radians) (radians) 1000 km) π/ π/ π/5 10 any value π/5 20 We can now state appropriate transverse resolutions for our interferometer data, given the signal-to-noise ratio of the target, in the particular case 3 with a coherent integration of 50 points and an FFT of length 256. These are summarized in table 5.1. These values can be considered at best rules of thumb; we have made many assumptions in coming to this conclusion and the simulations themselves are only approximations. However, it is useful to have a general idea of the granularity of data to expect out of our instrument. The method we recommend for determining an appropriate transverse resolution is (unfortunately) on a case-by-case basis. The post-processing signal-to-noise ratio of the corresponding power spectrum should be estimated, and a decision about resolution made using that information. In figure 5.5, we show as an example an auroral echo with 4 different transverse resolutions (the range resolution remains 1.5 km throughout the figure). The leftmost panel, with 2 km transverse resolution, corresponds to an echo with SNR 26 db, according to our rule of thumb. The next two panels have resolutions accommodating SNR s greater 3 The resolution limitations only relax for larger coherent integrations, so the values we state here are lower (i.e., worst) bounds, in the sense that we rarely would want to use a decimation less than 50.

112 90 50 bins / 2 km 20 bins / 5 km 10 bins / 10 km 5 bins / 20 km 1100 Range (km) Transverse Position (km) Figure 5.5: A comparison of transverse resolutions in interferometer data. From left to right, the phase is separated into consecutively fewer bins, resulting in degrading transverse resolution. The range resolution in each figure is 1.5 km. than 17 db and 11 db, respectively. Finally, the rightmost panel, with a transverse resolution of 20 km (at a range of 1000 km), can be considered accurate at any SNR. In the case of the echo shown, which occurred at UT 05:11 on February 2, 2002, we suggest that a resolution between 5 and 10 km may safely be used, according to the power spectra from the same echo shown in figure 5.6. Determining the resolution is a difficult problem, though, since the SNR varies with range (and Doppler velocity, for that matter). We suggest a strategy of using finer resolutions when possible, while keeping in mind the limitations of the instrument. The risk in using too fine a resolution is that it becomes possible to misinterpret random fluctuations for true

113 km Power Spectra at Different Ranges for an Auroral Echo Max SNR: 18 db 989 km 1005 km Max SNR: 13 db Max SNR: 6 db Doppler Velocity (m/s) Figure 5.6: Power spectra at 3 different ranges for the auroral echo we observed on February 2, 2002, at UT 05:11. transverse structure in the lower SNR areas. This uncertainty associated with whether the data is showing structure or coincidental patterns in noise is why we have spent so much effort analyzing the variability of our data products. 5.4 Target Detection by Phase Compactness As a final note, we describe another useful consequence of the decrease in phase variance with increasing SNR. As we have seen, the presence of a target causes a certain degree of organizedness in the cross-spectrum phase; the decrease in variance is directly related to this phenomenon. Figure 5.7 shows a spectrum containing a target in

114 92 the top panel; in the bottom panel, the standard deviations of the phase estimates are plotted versus Doppler velocity. We claim that it is possible to automatically detect targets by setting a phase variance threshold and testing spectra for consecutive samples that fall below that threshold. This method would be no more computationally intensive than search or test methods based on detecting peaks in spectrum magnitude: each variance (or standard deviation) estimate can be computed directly from the cross-spectral estimates already available from the radar signal processing. We have not implemented this technique, but we have noted on many occasions that while magnitude spectra can be unreliable, the phase becomes compact quite predictably when a target is present. One caveat to note when applying this method is that one must consider the unnatural discontinuity introduced into the phase by unwrapping the circular spectrum in the complex plane and plotting it as shown in figure 5.7. If the target phase exists near π or +π, small deviations in the phase estimates may result in much larger standard deviation estimates, since the values may roll over (essentially, cross a branch cut). There are many possible ways to deal with this problem; we do not discuss them here.

115 93 Cross Spectrum Magnitude Detection of Target by Variability in Phase Estimates std of Phase Estimates Doppler Velocity (m/s) Figure 5.7: An example of cross-spectrum phase compactness where a (simulated) target exists. The standard deviation of phase estimates can be used to detect targets, in some cases more reliably than power spectrum magnitude.

116 94 Chapter 6 ANALYSIS OF IONOSPHERIC EVENTS WITH INTERFEROMETRY Hey, that one s cool! It looks like a jester s hat! That s a bat. Explosion. Explosion. Funnel-cone ice cream thing. This is your brain. This is your brain on drugs. Dr. Mark Oskin In this chapter we analyze several E-region field-aligned irregularities with the interferometric techniques discussed earlier, and show what additional information can be learned given knowledge of their transverse structure. We demonstrate that our passive radar interferometer data products are consistent with what we might expect from an active radar interferometer, such as CUPRI, and that we are able to make the same types of measurements that have been done before with other active radar interferometers [25, 26]. We further claim that the very fine resolution of the MRR interferometer 1 allows us to see for the first time that significant structure exists in the irregularities at very fine scales. Finally, we note that the E-region irregularities we detect must be inherently small, with scattering areas on the order of km in the transverse dimension, for our finely-spaced interferometer lobes to detect them at all. Such narrow structure has hitherto been unconfirmed, to the author s knowledge. 1 Typically, km in the plane to B. This is an extremely fine resolution relative to most other ionospheric coherent scatter radars (cf. SHERPA, km; STARE, km).

117 Description of Irregularity Backscatter As we discussed in section 3.3, MRR detects ionospheric phenomena via the Bragg scattering mechanism, and thus it is sensitive to structures with scale sizes near 1.5 meters. We often find irregularity velocities to be near c s, the ion acoustic speed, and a study of the Doppler statistics of echoes observed by the Manastash Ridge Radar over a one-year period 2 indicates that the majority of irregularities detected by MRR are excited by the modified two-stream instability (otherwise known as the Farley-Buneman instability [17]). These echoes might be classified as type 1, or fast moving type 2, due to their large Doppler shifts and narrow- to moderate- spectral widths. Since the plasma wave modes believed to be causing the coherent backscatter are very strongly damped in directions parallel to the magnetic field [28, 20], we can expect scatterers detected by MRR to be localized to regions where the radar line of sight is within 1 2 of perpendicular to B. Figure 6.1 shows the radar field of view, over southwestern Canada, along with contours of constant bistatic range and aspect angle. Even without direction-of-arrival information from the interferometer, we can roughly determine the location of scatterers using the intersection(s) of the appropriate range contour and the 90 aspect angle contour. Furthermore, the altitude from which irregularity scatter originates is limited to E-region heights, approximately km. 2 In work not yet published, but presented at the IUGG General Assembly in Sapporo, Japan, July 2003: Meyer, M. G., J. D. Sahr, D. M. Gidner, and C. Zhou, A Year in Review: High Latitude Ionospheric Irregularity Observations with Passive VHF Radar.

118 96 Figure 6.1: The MRR field of view, over southwestern Canada, shown with contours of constant range and aspect angle. The edges of the figure are labeled with geographic latitude (N) and longitude (E). Credit for creating the figure goes to Dr. Frank Lind. 6.2 February 2, 2002 An example of an auroral irregularity observation from February 2, 2002 is shown in figure 6.2. The range-doppler diagram shows a scattering volume that extends from approximately 970 to 1070 km; an area of high intensity exists which has a large Doppler shift near 670 m/s, and a diffuse area with lower intensity and large Doppler width exists alongside the strong scatterer. Investigation of the interferometer data reveals that this auroral echo is most likely

119 February 2, 2002 UT 05:11 db Velocity (m/s) Range (km) Figure 6.2: A range-doppler display from MRR showing an auroral echo near 1000 km; an area of high intensity and large Doppler shift is flanked by a larger, diffuse type 2 echo, which we show to be spread across the transverse dimension, indicating a shear. limited to one interferometer lobe. The spectra shown in figure 6.3 were computed at the range of 980 km, which includes part of both the bright spot and diffuse scattering area. The coherence is large where the cross-spectrum phase is highly organized, indicating a scatterer contained wholly within one lobe (approximately 100 km wide at this range). Again, we note that the self-spectra on both antennas are consistent with each other and with the coherence. Using the same methods for interpreting the phase information as we did in chapter 3, we estimate that the irregularity scattering volume is 30 km wide (at the particular range of 980 km), and its longitudinal length is estimated to be 100 km from the

120 98 Cross Spectrum Phase and Magnitude for Range 980 km Phase Normalized Magnitude Self Spectra (db) Doppler (m/s) Log Periodic Yagi Figure 6.3: The cross-spectrum and self-spectra for one range in the auroral echo from figure 6.2. Due to the highly organized phase, we observe that the echo seems limited to one interferometer lobe; the phase width implies a scattering volume extent of approximately 30 km. The 95% significance level for the coherence is also shown in the middle panel. range information. Another feature to note in figure 6.3 is the upward trend in interferometer phase across a Doppler width of about 500 m/s. The steady increase in phase over these likewise increasing Doppler bins suggests a velocity shear across the scattering volume in the transverse direction. In general, we expect some sort of shear when large field-aligned currents are present, so this is consistent with the disturbed conditions under which MRR detects irregularity scatter.

121 99 Cross Spectrum Phase and Magnitude for Range 984 km Phase Normalized Magnitude Self Spectra (db) Doppler (m/s) Log Periodic Yagi Figure 6.4: Cross spectrum at range 984 km for the auroral echo on February 2, 2002, UT 05: Natural Progression to Two-Dimensional Images Examining the cross spectra from this echo for the next few ranges reveals the same linear upward trend in phase across Doppler, but at more distant ranges, a discontinuity appears and the phase steps from one level to the next, indicating the separation of the slower- and faster-moving parts of the scatterer. For example, the step can be seen beginning to form in the interferometer phase at a range of 984 km (figure 6.4); at ranges 989 km (figure 6.5) and 992 km (figure 6.6), the two distinct levels of phase (transverse position of scattering volumes) are clear; and by range 1005 km (figure 6.7), the bright, narrow-doppler feature is gone, and the type 2 feature exists primarily at a single level of interferometer phase. We find it useful to consider interferometer data such as this over the entire set of ranges which contain irregularity scatter. For example, figure 6.8 shows magnitude and phase-difference spectra from the same auroral echo, stacked up over several

122 100 Cross Spectrum Phase and Magnitude for Range 989 km Phase Normalized Magnitude Self Spectra (db) Log Periodic Yagi Doppler (m/s) Figure 6.5: Cross spectrum at range 989 km for the auroral echo on February 2, 2002, UT 05:11. Cross Spectrum Phase and Magnitude for Range 992 km Phase Normalized Magnitude Self Spectra (db) Log Periodic Yagi Doppler (m/s) Figure 6.6: Cross spectrum at range 992 km for the auroral echo on February 2, 2002, UT 05:11.

123 101 Cross Spectrum Phase and Magnitude for Range 1005 km Phase Normalized Magnitude Self Spectra (db) Log Periodic Yagi Doppler (m/s) Figure 6.7: Cross spectrum at range 1005 km for the auroral echo on February 2, 2002, UT 05:11. adjacent ranges. A footprint of the echo can be seen in the stacks, both as a bump in the magnitude spectra (self-power spectra from a single antenna) of the left panel and as a compact area of phase in the right panel. This naturally led us to create twodimensional interferometer images (as discussed in section 3.4.1) by forming phase histograms plotted versus range, expecting the footprint of organized phase to show up as a bright area in two-dimensional colorscale plots. Additionally, we weighted the phase histograms with the cross-spectrum magnitude by summing not only the occurrence, but the corresponding magnitude, of each data point as it went into the appropriate histogram bin. This method performed to our satisfaction 3, in that it is a convenient and straightforward way to visualize the interferometer data. The resulting two-dimensional interferometer image for the echo discussed above is shown 3 There are other methods for generating 2D interferometer images that we wish to try in the future. For example, one which makes use of the normalized cross-spectrum magnitude estimate of transverse width proposed by Farley et al. [5].

124 102 in figure 6.9; the larger-scale structure of the echo is now easily visible. We see that the high-intensity, narrow-doppler feature splits apart from the lower-amplitude feature at a range of about 1000 km, as we predicted by examining the cross-spectra at individual ranges. Due to the aliasing issue, it is unclear whether these two features are part of the same scattering volume, or whether they exist in different parts of the sky. Also, the overall location (of either or both echoes) in the sky is unknown. However, recognizing the highly field-aligned nature of these phenomena, it is reasonable to speculate that the observed scatter exists in a single region of the sky. At a range of 1000 km, 2 of aspect angle (90 ±1 ) spans about 200 km. At this same range, our interferometer beams are approximately 100 km wide, indicating that any irregularity scatter we see will most likely be contained within two lobes. 6.3 March 24, 2002 On March 24, 2002, MRR detected irregularities over a period of roughly 10 hours. This extensive amount of data allowed us to make many successive interferometer measurements, spaced at intervals of 4 minutes. Figure 6.10 shows 3 images from adjacent time intervals during the 8 minute period from UT 05:07 to 05: Velocity Vector Measurements By determining the phase progression of the auroral echo from one panel to the next, we can estimate the transverse drift velocity of the irregularity. For the timestep from UT 05:07 to 05:11, at 5 different ranges (990, 1005, 1020, 1035, and 1050 km), we find transverse velocities with both positive and negative values (both eastward and westward drifts) on the order of 15 m/s (which accompany Doppler shifts of m/s). For the timestep from UT 05:11 to 05:15, the transverse velocity at each of the

125 103 Power Spectra Interferometer Phase Increasing Range Increasing Range Doppler Velocity (m/s) Doppler Velocity (m/s) Figure 6.8: Power spectra (from one antenna) and interferometer phase vs. range for the auroral echo on February 2, 2002, UT 05:11.

126 104 Auroral Echo: 2 February 2002, UT 05: Range (km) Transverse Position, Measured from Lobe Center (km) Figure 6.9: The February 2, 2002 auroral echo, represented in a 2-D image of range vs. transverse width. The resolution in this plot is 3 5 km (range transverse dimension). 5 ranges is eastward, and slightly larger (30-60 m/s). The Doppler shifts in this case are smaller, between 700 and 800 m/s. With an unaliased interferometer, it is possible to determine whether a scatterer lies to the east or the west of the radar receiver simply by observing the sign of the phase difference between the east and west antennas. While the many interferometer lobes in our system prevent us from determining a unique bearing to targets from our scatter receivers, it is still possible to determine, by examining the phase progression in time on each antenna, whether a moving target s motion is in an eastward or westward direction. Given this information along with the transverse drift speed

127 105 UT 05:07 UT 05:11 UT 05: Range (km) Transverse Position (km) Figure 6.10: 2D interferometer images from March 24, 2002, UT 05:07 05:15. and Doppler shift, we know two perpendicular components of the irregularity drift velocity: the Doppler shift, measured along the radar line of sight, and the transverse velocity, in a direction normal to the radar line of sight. Assuming the radar is looking perpendicular to B, and that all irregularity activity exists in the plane perpendicular to B, this means we know the total velocity vector describing the irregularity motion. We have analyzed the echo shown in figure 6.10 in this way. Figure 6.11 shows an illustration of the velocity vectors for the two timesteps at 5 different ranges (15 km apart). We see that the echo remains highly blueshifted over the entire time interval, with a net drift toward the radar receiver. Also worth noting is the small flow angle.

128 March 2002: Velocity Vectors vs. Range and Time Range: km Westward Eastward Toward Radar Radar Line of Sight UT 05:07 05:11 UT 05:11 05:15 Time Figure 6.11: Measured velocity vectors for the drifting irregularity shown above in figure 6.10.

129 107 We often find it to be the case that the Doppler shifts of irregularity echoes are much larger than their transverse drifts. It is possible that the reason behind the small transverse drifts is actually not small flow angle, but rather aliasing in our measurements of interferometer phase progression. Since the lobes are so finely spaced, a fast moving irregularity could jump from one lobe to the next during the 4 minute time period between observations and falsely appear to have very little transverse velocity. If this were the case, then echoes at a range of 1000 km could be aliased in multiples of roughly 400 m/s. This is a rather large error to incur (although the very large Doppler shifts in this particular example still dominate the measured velocities even if the transverse error is 400 m/s). The long 4-minute interval between observations is unfortunate. Experimentation with splitting the 10-second datasets into multiple parts to get finer time resolution was unsuccessful, as the variance of the phase estimates grew too large (from the necessary decrease in the number of incoherent averages) to distinguish small changes in the interferometer phase. One possible way to alleviate this problem is to take 10-second bursts of data much more often 4, opting to send only a few of the datasets over the internet link to be processed, unless we find that they contain echoes we wish to analyze further. Unfortunately, this technique was not in use during the March 24 event (or for any event thus far). We do have 2-minute-long data bursts from recent irregularity events, which would also serve to eliminate transverse drift aliasing, but these are plagued by as-yet-unsolved interference problems. 4 Once every minute ought to be frequent enough for our purposes, as we do not often see irregularities with drift speeds greater than 1600 m/s.

130 Electric Field Measurements If we assume that the plasma motion is due entirely to E B drift (Hall drift), and that the magnetic field is known and static (Earth s dipole field), then we can use the velocity vectors measured by MRR to roughly estimate the electric field in the vicinity of the scatterer. The direction of E will be perpendicular to both B and the irregularity drift velocity (v), such that v = E B B 2 (6.1) To determine the electric field strength, we take B to be Tesla, a typical value for E-region heights in the MRR field of view. In the ionosphere, the ψ parameter also applies, which is related to the collision and cyclotron frequencies of the ionized particles. A typical value for ψ in the E-region is We use these values with the equation E = (1 + ψ)bv (6.2) to obtain estimates of E. If we only have Doppler shift information (as opposed to a total velocity measurement), we can include the flow angle θ: E = (1 + ψ)bv cos θ (6.3) In the example above, with velocities as shown in figure 6.11, we find the total irregularity drift speeds v to be between 700 and 900 m/s. These speeds correspond to electric field strengths of mv/m, which are relatively large values and indicate the disturbed ionospheric conditions under which they were measured 5. We can determine the direction of E by considering figure 6.11; B in this case is into the page, which means the electric field is directed westward. 5 Kp during these observations was 6.0; an overflight of DMSP satellite F14 at the same time showed disturbed magnetosphere conditions; GPS total electron content data showed depletions in the MRR field of view, indicating low conductivity and a region able to support large electric fields.

131 March 30, 2002 UT 04:59 db Velocity (m/s) Range (km) 220 Figure 6.12: An example of Doppler power spectrum versus range from irregularity scatter obtained on March 30, March 30, 2002 A geomagnetic storm lasting from 21:00 to 23:00 local time on March 30, 2002 caused irregularities over western Canada which we were also able to observe with our interferometer. The range-doppler spectra, one of which is shown in figure 6.12, reveal echoes centered around 1050 km, ranging over approximately 200 km, with mean Doppler shifts near 500 m/s and with Doppler widths from 100 m/s to 300 m/s. All of these echoes appear to be type 2, with power spread uniformly across their Doppler widths. In figure 6.13 we show two-dimensional images from data taken at 4-minute intervals over a time window of 16 minutes. In the first three frames, the scattering volume

132 110 Figure 6.13: A sequence of 5 consecutive interferometer images from the March 30 event, 4 minutes apart in time. The first three frames indicate a mean transverse drift speed of 70 m/s. The final frame clearly shows that the scattering volume has split, a feature unrecognizable from the range-doppler spectra in figure can be seen drifting in the transverse direction at roughly 70 m/s, which is consistent with a possible guiding center E B drift. Later, at UT 04:59, the scattering volume separates into two main pieces (above and below 1050 km), not an apparent feature when only the range-doppler diagram in figure 6.12 is considered. Figure 6.13 also illustrates the phase-wrapping inherent in the aliased interferometer measurements; in the last three frames we can see that the echo is split between the two edges of the frame, crossing an arbitrary phase boundary between interferometer lobes. Using the transverse drift speed (70 m/s) inferred from the interferometer data