IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 1

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1 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 1 InSAR Time-Series Estimation of the Ionospheric Phase Delay: An Extension of the Split Range-Spectrum Technique Heresh Fattahi, Member, IEEE, Mark Simons, and Piyush Agram, Member, IEEE Abstract Repeat pass interferometric synthetic aperture radar (InSAR) observations may be significantly impacted by the propagation delay of the microwave signal through the ionosphere, which is commonly referred to as ionospheric delay. The dispersive character of the ionosphere at microwave frequencies allows one to estimate the ionospheric delay from InSAR data through a split range-spectrum technique. Here, we extend the existing split range-spectrum technique to InSAR time-series. We present an algorithm for estimating a time-series of ionospheric phase delay that is useful for correcting InSAR time-series of ground surface displacement or for evaluating the spatial and temporal variations of the ionosphere s total electron content (TEC). Experimental results from stacks of L-band SAR data acquired by the ALOS-1 Japanese satellite show significant ionospheric phase delay equivalent to m of the temporal variation of InSAR time-series along 445 km in Chile, a region at low latitudes where large TEC variations are common. The observed delay is significantly smaller, with a maximum of 10 cm over 160 km, in California. The estimation and correction of ionospheric delay reduces the temporal variation of the InSAR time-series to centimeter levels in Chile. The ionospheric delay correction of the InSAR time-series reveals earthquake-induced ground displacement, which otherwise could not be detected. A comparison with independent GPS time-series demonstrates an order of magnitude reduction in the root mean square difference between GPS and InSAR after correcting for ionospheric delay. The results show that the presented algorithm significantly improves the accuracy of InSAR time-series and should become a routine component of InSAR time-series analysis. Index Terms Interferometric synthetic aperture radar (InSAR), ionospheric phase delay, split range-spectrum. I. INTRODUCTION INTERFEROMETRIC synthetic aperture radar (InSAR) has been shown to be an effective technique for geodetic imaging of surface ground displacement caused by natural and anthropogenic processes, such as earthquakes, tectonics, volcanic unrest, shallow hydrological processes, landslides, and glacier flow. The accuracy of repeat-pass InSAR measurements of ground displacement can be significantly affected Manuscript received February 0, 017; revised May, 017; accepted June 13, 017. The work of H. Fattahi and M. Simons was supported by the National Aeronautics and Space Administration under Grant NNX14AH80G and Grant NNX16AK58G. (Corresponding author: Heresh Fattahi.) H. Fattahi and P. Agram are with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA USA ( heresh.fattahi@jpl.nasa.gov). M. Simons is with the California Institute of Technology, Pasadena, CA 9115 USA. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TGRS by the extra path delay of the microwave signal passing through a planet s atmosphere. For SAR satellites orbiting around earth, the atmospheric delay is largely caused by the spatial and temporal variations of water vapor in the troposphere [1] [4] and of the number of free electrons in the ionosphere [5], [6], [10] [1]. In the ionosphere, highly energetic solar radiations such as extreme ultraviolet and X-ray radiation partially ionize the atmosphere s neutral atoms and molecules forming a mixture of free electrons, ions, and neutral gas molecules around earth [13] [15]. The ionosphere extends approximately from altitudes of 60 to 1000 km, with a maximum electron density at around 300 km [16]. In addition to altitude-dependent variations, the ionosphere s total electron content (TEC) varies with geographic location, time of day, season, and levels of geomagnetic and solar activities [14]. The ionosphere shows a latitudinal variation, with distinct behaviors in three main regions including low (or equatorial), middle, and high (auroral) latitude regions [14]. In equatorial regions, the ionosphere is dominated by the equatorial (Appleton) anomaly [17] due to a fountain effect whereby ionosphere electron density is higher on both sides of equator rather than at the equator itself. The equatorial anomaly is often asymmetric with a higher density on the winter hemisphere relative to the summer hemisphere. The anomaly may also be disturbed by geomagnetic storms. Irregularities in ionospheric TEC variation that causes sudden change to the phase or amplitude of the microwave signals, known as the scintillation effect, are also maximum at the low latitudes. In contrast to the equatorial region, ionosphere in the mid-latitude regions is less intense and least variable. The TEC variation at mid-latitudes is small, around 0 30% of the average value, and can be accurately predicted by different models [13]. At high latitudes, the irregular variation of TEC is larger than at mid-latitudes and still smaller than at equatorial regions. At polar regions, photoionization and high energy particles are the main sources of ionization. Energetic particles from the magnetosphere are guided by the geomagnetic field lines through atmosphere where they collide with neutral gas atoms and molecules resulting in intense electromagnetic waves. Precipitating particles excite atmospheric elements to higher energy levels, which result in emission of visible lights known as auroral lights [13]. The ionosphere is a dispersive medium with respect to the microwave frequencies. In such a dispersive medium, the delay in the microwave signal is inversely proportional to the signal IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING frequency such that low-frequency signals experience a larger delay than higher frequency signals. The dispersive medium of ionosphere for microwave signals allows space geodetic systems operating in two different microwave frequencies, such as the Global Navigation Satellite Systems, to directly estimate the ionospheric delay [13]. Moreover, ionosphere becomes birefringent media in the geomagnetic field, such that the microwave signals propagating through the ionosphere are decomposed into two circularly propagating waves with slightly different propagating velocities, which depend on the polarization and orientation of the propagation direction with respect to the geomagnetic field [19]. Birefringent ionosphere rotates the polarization plane of the microwave signals by an angle called Faraday rotation, which can be estimated from full-polarization SAR data [9]. Propagation of microwave signals through the ionosphere causes distortions in the InSAR data, including defocusing of SAR images, Faraday rotation, phase delay, and an extra shift between SAR images in the satellite along-track (azimuth) direction. When the ionospheric variation in the satellite alongtrack direction has a correlation length shorter than the length of the synthetic aperture, the resulting image will be defocused in the azimuth direction [18]. For polarized microwave signals passing through ionosphere, the polarization direction will rotate relative to the original polarization direction by Faraday rotation [16]. Ionospheric delay also introduces an extra phase component to the SAR interferograms, which if not compensated, decreases the accuracy of InSAR measurements of ground displacement [5], [6]. Along-track TEC gradients cause a phase gradient equivalent to an extra Doppler shift, which translates to a time shift in the azimuth direction [6]. In other words, the ionospheric phase gradient introduces extra azimuth offsets between two SAR images that cannot be predicted with geometrical coregistration techniques. Several approaches exist to estimate the ionospheric phase delay from InSAR data. One approach uses Faraday rotation, whereby full-polarization SAR data are used to estimate the Faraday rotation, from which the ionosphere s TEC can be estimated [7] [10], [16] and used to predict the expected phase delay. However, the ionospheric phase delay estimation using Faraday rotation is limited by the availability of fullpolarization SAR data and to high-latitude regions where the geomagnetic field and the radar line of sight (LOS) are largely parallel. Nevertheless, when Faraday rotation is applicable, absolute TEC at each SAR acquisition can be estimated whereas other approaches, which are explained in the following, estimate only a differential TEC between SAR acquisitions. A second approach relies on estimates of the azimuth offsets. These offsets can be integrated along the azimuth direction to estimate the ionospheric phase delay [0]. The azimuth offsets may be estimated from coherent or incoherent cross correlation of SAR images or with spectral diversity from multiaperture interferometry [1]. Offset-based techniques are limited by the direction and spatial wavelength of the ionospheric phase delay such that they are only sensitive to the along-track ionosphere variation with short spatial wavelength. Moreover, the azimuth offsets induced by ionosphere cannot be distinguished from offsets caused by large ground displacement. A third approach uses the dispersive characteristic of the ionosphere for microwave signals and divides the spectrum of the radar signal in range direction into two sub-bands from which two lower resolution SAR images at different center frequencies are formed. In this technique, known as the split range-spectrum technique (also referred to as range split-spectrum in previous studies [1], []), subband images from two acquisitions are combined to form two sub-band interferograms and to estimate the dispersive and nondispersive components of the interferometric phase. The dispersive component represents the ionospheric phase delay. An algorithm for the split range-spectrum technique has been detailed in [3] and has been demonstrated for Sentinel-1 and ALOS- acquisitions [4]. The split range-spectrum technique has already been evaluated for a limited number of InSAR pairs. Here, we extend the split range-spectrum approach in order to estimate a time-series of ionospheric phase delay using a stack of SAR acquisitions. The timeseries of ionospheric phase delay can be used to compensate the InSAR displacement time-series or to evaluate the spatiotemporal variation of the TEC. In Section II, we review the ionospheric phase delay in InSAR data. We discuss the basic concept of estimating ionospheric phase delay from multifrequency SAR acquisitions and using the split range-spectrum technique for narrowband SAR data. We then extend the split range-spectrum technique to estimate the time-series of ionospheric phase delay. In Section III, we discuss the details of the proposed workflow for time-series estimation of ionospheric phase delay from a stack of stripmap SAR acquisitions. In Section IV, we present experimental results; in Section V, we discuss the presented algorithm and the results. II. IONOSPHERIC PHASE DELAY The interferometric phase of an InSAR interferogram formed from two SAR acquisitions at t i and t j contains different components as ( r ij d ij ij + rgeom + rtrop + r ij ) iono φ = 4π (1) λ where λ represents the carrier wavelength of the radar, r ij d is the ground displacement in radar LOS direction, rgeom ij represents the geometrical range difference from radar to the target caused by a nonzero spatial baseline between the two orbits, and r ij ij trop and riono are tropospheric and ionospheric delay, respectively. The ionospheric delay at a given acquisition is the difference between the length of the actual path traveled by the microwave signal through ionosphere and the geometric range from radar to the target, given as [13] riono i = nds ds 0 () where n is the ionosphere refractive index, s is the actual path of the signal, and s 0 is the geometrical path of the signal. The first term on the right-hand side of () denotes the traveled path by the microwave signal and the second term denotes the geometric path. The delay between two acquisitions is the difference of the delay at each of the acquisitions as

3 FATTAHI et al.: InSAR TIME-SERIES ESTIMATION OF THE IONOSPHERIC PHASE DELAY 3 r ij = r j r i. Based on the Appleton-Hertree formula for the ionospheric refractive index, which ignores the collision effects of the particles [5], and following [6], () can be written as riono i = K f0 N e ds 0 ± 755c f 0 3 Bcos(θ) N e ds f0 4 Ne ds 0 + β (3) where K = m 3 /s is a constant, f 0 is the carrier frequency of the microwave signal, N e is the electron density, c is the speed of light, B is the magnitude of the earth s magnetic field, θ is the angle between the magnetic field and the propagation direction, and β = ds ds 0 represents the effect of bending of the microwave signal through the ionosphere. Bending of the microwave signal in the ionosphere for elevation angles greater than 5 (i.e., incidence angles less than 85 ) is negligible [7]. Since the incidence angle for current and upcoming SAR missions are below 60, bending is ignored in our discussion in the following. The first three terms on the right-hand side of (3) express the first-, second-, and third-order ionospheric delay, respectively. The ± sign for the second-order term in (3) represents birefringence in the ionosphere [9]. A maximum magnitude of the earth s magnetic field of 0.65 gauss (10 4 tesla) and a TEC variation of 0 TECU results in submillimeter delay. The third-order delay is even smaller. Therefore, we keep only the first-order term in the following. Given TEC = N e ds 0, the first-order ionospheric delay between two SAR acquisitions is obtained as r ij iono = K f0 TEC ij (4) where TEC ij represents TEC variation in the slant range direction between the two acquisitions at times t i and t j. Equation (4) shows that ionospheric delay is inversely proportional to the square of the carrier frequency. By substituting (4) into (1), the interferometric phase is expressed as the sum of the dispersive and nondispersive components φ = 4π f 0 r ij non-disp c 4π K TEC ij (5) cf 0 where the first term represents the nondispersive component, which is the sum of the ground displacement, geometrical phase, and tropospheric delay. The second term represents the dispersive component, which is the ionospheric phase delay. A. Ionospheric Phase Delay Estimation From Multifrequency InSAR Data If SAR images at different carrier frequencies of f 0 and f 1 are available, the interferometric phase at f 1 can be expressed based on the dispersive ( φ disp ) and nondispersive components of the interferometric phase at f 0 as φ f1 = f 1 φ non-disp + f 0 φ disp. (6) f 0 f 1 For additional carrier frequencies of f 1,..., f N, a linear system of equations d = Am can be formed, where d = [ φ f0, φ f1,..., φ fn ] T is the observation vector of interferometric phases at different carrier frequencies, m = [ φ non-disp, φ disp ] T is the unknown vector, which is the dispersive and nondispersive phase components at f 0, and A is an incidence matrix, where A = [[1, 1] T, [( f 1 / f 0 ), ( f 0 / f 1 )] T,...,[( f N / f 0 ), ( f 0 / f N )] T ] T. This linear system can be solved to estimate the nondispersive and dispersive phase components at the f 0 frequency using a weighted least squares as ˆm = (A T C 1 d A) 1 A T C 1 d d,wherec d represents the variance-covariance matrix of d. The variancecovariance matrix of the unknowns is therefore obtained as C m = (A T C 1 d A) 1. Currently, no space-borne radar mission operates in multiple carrier frequencies. The L-band mode of the NASA-ISRO SAR (NISAR) mission will nominally include a 5-MHz sideband separated from a 0- or 40-MHz main band [8], allowing to estimate the ionospheric and nondispersive phase components. Moreover, in selected regions, NISAR will operate in an extra S-band frequency [8], which enables ionospheric phase estimation when combined with the L-band acquisitions. We note that a dispersive phase component estimated from S and L bands may not only represent the propagation delay in ionosphere but also may contain dispersive phase components caused by different scattering properties of S and L bands due to surface moisture content, vegetation density, or surface roughness [9]. Separating the two components of dispersive phase caused by surface scattering and propagation delay in multifrequency SAR data is beyond the scopes of this paper. B. Ionospheric Phase Delay Estimation From Narrow-Band InSAR Data Using Split Range-Spectrum For narrow-band SAR images, the full range-spectrum of a radar image acquired at center frequency of f 0 can be split into two narrower nonoverlapping sub-bands at lower and higher center frequencies of f L and f H. Splitting the range-spectrum can be achieved with a bandpass filter (for more details, see Section III). The SAR images at each sub-band can be used to form sub-band interferograms. Using interferometric phase from the two sub-bands, φ L and φ H, A then simplifies to A = [[( f L / f 0 ), ( f 0 / f L )] T, [( f H / f 0 ), ( f 0 / f H )] T ] T, and m =[ φ non-disp, φ disp ] T, which represents the nondispersive and dispersive interferometric phase components at f 0.Due to small separation between f L and f H (usually less than tens of megahertz), the dispersive component from surface scattering is negligible. Assuming that the dispersive component represents only the ionospheric delay, the ionospheric and nondispersive phase components are given by f L f H φ iono = )( φ L f H φ H f L ) (7) φ non-disp = ( f 0 f H fl f 0 ( f H fl )( φ H f H φ L f L ). (8) Assuming the same phase standard deviation for the lowband and high-band interferometric phases of σ φl,h,thevariance of the ionospheric and nondispersive phase components and the covariance between the two components are given as σ iono = fl f H fl ( f H fl ) + f H f0 σ φ L,H (9)

4 4 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING σ non-disp = η iono,non dis = fl f H ( 1 ( f H fl ) f0 fl + 1 ) f H σ φ L,H (10) f L f H ( f H f L ) σ φ L,H. (11) The accuracy of the estimated ionospheric and nondispersive phase components is therefore a function of the subband interferometric phase noise and the central frequencies. Assuming an L-band central frequency of 1.3 GHz and sub-band frequencies with ±10 MHz from the carrier frequency (i.e., f L = 1.9 GHz and f H = 1.31 GHz), based on (9) and (10), phase noise with standard deviation of one radian in the sub-band interferograms amplifies to 46 radians in the estimated ionospheric and nondispersive phase components. Note that the ionospheric and nondispersive phase components are inversely correlated. The covariance of the ionospheric and nondispersive phase components for the numerical example above, equals 11 radians. Therefore, because of the large negative covariance, the variance of the sum of noisy ionospheric and nondispersive components, obtained as σiono+non dis = σ iono + σ non-disp + η iono,non-dis, reduces to around one radian, which is of the order of the sub-band interferogram phase noise. C. Time-Series of the Ionospheric Phase Delay From Split Range-Spectrum To estimate a time-series of the ionospheric phase delay, one could ideally estimate the ionospheric delay for each pair in a single master network of interferograms. However, due to noise amplification caused mainly by the small separation between the low- and high-band frequencies, that is small f H f L, small interferometric phase noise may significantly affect the accuracy of ionospheric phase estimates. Therefore, geometrical and temporal baseline decorrelations limit ionospheric phase delay estimation from a single master network of interferograms. To reduce the impact of geometrical and temporal baseline decorrelations, we use a network of coherent small baseline interferograms to estimate the network of ionospheric phase delay. Given a connected network of ionospheric phase delays, the vector of ionospheric pairs at each pixel φ i, j iono is related to a vector of the sequential phase differences φ k,k+1 iono, hereafter called differential time-series, as φ i, j iono = G φk,k+1 iono, where G is a design matrix [30]. Assuming N SAR acquisitions used to estimate the network of ionospheric phase delay, i, k [1, N) and j [, N]. For a connected network of small baseline pairs of ionospheric phase delay, G is full rank and the system of equations is overdetermined. We notice that disconnected networks with small subsets, such as those networks commonly used for displacement time-series analysis [30], would bias the estimated differential time-series of ionospheric phase delay and should not be used. We estimate the differential time-series of the ionospheric phase delay using a weighted least-squares as ˆφ k,k+1 iono = (G T C 1 G) 1 G T C 1 φ i, j φ i, j φ i, j iono, where C φ i, j is the variance-covariance matrix of the network of ionospheric phase delay estimated from split range-spectrum. The variance-covariance matrix of the estimated differential time-series is obtained as C ˆφ k,k+1 = (GT C 1 G) 1. φ i, j Since each element of ˆφ k,k+1 iono is differential between two consequent acquisitions, a time-series of ionospheric phase delay relative to a reference acquisition is obtained by temporal integration of ˆφ k,k+1 iono over its elements. The square root of the diagonal components of C ˆφ k,k+1 represents the uncertainty of differential time-series of ionospheric phase delay. The covariance between the elements of ˆφ k,k+1 should be taken into account to express the uncertainty of the ionospheric delay time-series at each acquisition date relative to a reference date. For example, given ˆφ 1, and ˆφ,3 as the first and second elements of the differential time-series, the uncertainty of ˆφ 1,3 in a time-series relative to the first acquisition is given as σ ˆφ 1,3 = σ ˆφ 1, + σ ˆφ,3 + η ˆφ 1,, ˆφ,3 (1) where η ˆφ 1,, ˆφ,3 is the covariance between ˆφ 1, and ˆφ,3. III. PROCESSING WORKFLOW FOR TIME-SERIES ESTIMATION OF THE IONOSPHERIC PHASE DELAY The processing workflow for the time-series estimation of the ionospheric phase delay from a stack of SAR data is summarized in Fig. 1. The workflow can be divided into three main blocks that include: 1) estimating accurate timeseries of offsets to coregister the stack of single-look complex image (SLC) images to a common coordinate system; ) splitting the range-spectrum and resampling the sub-band SLCs using the precise offsets from the previous step; and 3) forming networks of low-band and high-band interferograms to form a network of ionospheric phase delay and finally estimating the ionospheric phase time-series. We explain the workflow in more detail in the following sections. A. Coarse Coregistration Using Geometrical Offsets For estimating a time-series of the ionospheric phase delay, stacks of full-band and sub-band SLCs should be coregistered to a common master coordinate system, here after called the stack master. We choose an arbitrary full-band SLC image as the stack master and compute the geometrical offsets between all slave full-band SLCs and the stack master. The geometrical offsets are obtained using a digital elevation model and precise orbits of the SAR satellite [31]. Using the geometrical offsets, the stack of full-band SLCs is coregistered to the stack master. This process is the same as the first stage of geometric coregistration in the network-based enhanced spectral diversity approach, which was developed to coregister a stack of SAR acquisitions acquired with Terrain Observation with Progressive Scan [3]. The geometrical offsets in the azimuth direction usually require refinement to account for possible timing-errors, along-track orbital errors or ionosphere-induced azimuth offsets [33]. B. Time-Series of Azimuth Offsets In order to refine the azimuth offsets, we estimate a time-series of azimuth misregistration with respect to the

5 FATTAHI et al.: InSAR TIME-SERIES ESTIMATION OF THE IONOSPHERIC PHASE DELAY 5 to form the sub-band SLCs at frequencies f L and f H.Given an SAR SLC focused to the carrier frequency with the range bandwidth of B, the spectrum of the complex signal in range direction for each line of the image can be split into sub-bands using bandpass filtering. After bandpass filtering of the rangespectrum, each sub-band SLC is obtained by computing the inverse Fourier transforms of the sub-band spectrum. Finally, to center the low-band and high-band spectra at f L and f H, respectively, we demodulate the sub-band SLCs as I L = I L e jπ f Lt, t [0,w/RSR] (13) I H = I H e jπ f Ht, t [0,w/RSR] (14) where I L and I H are the low-band and high-band SLCs, respectively, t is the range time, w is the number of samples in range direction, and RSR is the range sampling rate. Fig. 1. InSAR processing workflow for time-series estimation of ionospheric phase delay. The colors indicate different processing blocks as orange: estimating time-series of offsets to coregister the stack of SLCs, blue: splitting range-spectrum for the stack and using the time-series of offsets to coregister the stacks to a common coordinate system, and green: processing block for estimating the time-series of ionospheric phase. For the sake of simplicity, different acronyms have been used as follows. geom: geometrical. coreg: coregistered. Amp: amplitude. corr: correlation. Net: network. SB: small baseline and refers to pairs of SAR acquisitions with small spatial and temporal baselines. Az: azimuth. Tseries: time-series. LB: low-band. HB: high-band. igrams: interferograms. iono: ionospheric. All SB networks are connected networks. stack master. For a stack of stripmap acquisitions, the timeseries of azimuth misregistration can be obtained by amplitude cross correlation of the stack master with the coarse coregistered full-band SLCs. However, the accuracy of the azimuth misregistration decreases with increased timedifference or spatial baseline from the stack master acquisition. To reduce the impact of the temporal and spatial baseline decorrelations on the azimuth offsets, we use pixel-offset small baseline technique in which we compute the pixel-offsets for a network of small baseline pairs of full-band SLCs and then invert the network of the azimuth offsets to estimate the timeseries of azimuth offsets with respect to the stack master [34]. A connected network of small baseline pixel-offsets should be used to ensure unbiased estimation of the time-series of the azimuth offsets. C. Splitting the Range-Spectrum Given a stack of SAR acquisitions, we focus the data to form full-band SLCs at the carrier frequency f 0 and sub-band SLCs at low-band and high-band frequencies of f L and f H. Alternatively, one may first focus the stack to the carrier frequency and then split the range-spectrum for each image D. Fine Coregistration and Interferogram Generation Using the geometrical offsets refined with the time-series of azimuth offsets, the stacks of full-band and sub-band SLCs should be coregistered and resampled to the stack master. Afterward, a connected network of small baseline full-band and sub-band interferograms can be computed by cross multiplying coregistered SLCs at each sub-band. We note that the sub-band frequencies should be taken into account for resampling the sub-band SLCs and for flattening the sub-band interferograms. E. Ionospheric Phase Delay Estimation In order to estimate the dispersive and nondispersive phase components for each small baseline pair, the sub-band interferograms need to be multilooked, filtered, and unwrapped. Due to the amplification of noise, the estimated dispersive and nondispersive components need to be low-pass filtered. To mitigate possible artifacts caused by low-pass filtering, we use an iterative masking-interpolation filtering approach. In this approach, we first create a mask for invalid regions, including areas covered by water (e.g., lakes, ocean), shadow areas, and regions with low coherence. We mask out invalid regions of the dispersive and nondispersive phase components, fill in the masked regions using a nearest neighbor interpolation and apply the low-pass filter to the masked and interpolated data. In the second iteration, we reset areas with good data to their original unfiltered values and fill in the masked region by interpolating the filtered data. The masked and interpolated data are filtered again. This process is repeated until convergence. We normally use 5 6 iterations. The filtered dispersive and nondispersive components along with the sub-band interferograms are used to estimate possible phase unwrapping errors in the sub-band interferograms [3]. Afterward, the sub-band interferograms corrected for unwrapping errors are used to re-estimate the dispersive and nondispersive pairs. Then, the iterative filtering should be applied to reduce noise in the two components. The resulting network of pairs of ionospheric delay is inverted to estimate the timeseries of the delay, which can be used to compensate the fullband interferograms or InSAR time-series. The time-series of delay can also be used to produce TEC time-series with respect to a reference acquisition.

6 6 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING Fig.. Location of ALOS-1 acquisitions in Chile and California. IV. EXPERIMENTAL RESULTS In order to evaluate the proposed algorithm to estimate time-series of ionospheric phase delay, we use three stacks of L-band acquisitions in Chile and California (Fig. ), acquired by ALOS-1 from 007 to 011. ALOS-1 stacks were acquired in two modes, fine beam dual polarization (FBD) and fine beam single polarization (FBS) with bandwidths of 14 and 8 MHz, respectively. In order to evaluate the performance of the algorithm with a smaller range bandwidth, we downsample the raw FBS acquisitions of each stack to 14-MHz bandwidth. We then split the range-spectrum of the SLCs to form the sub-band SLCs at ±3.5 MHz from the carrier frequency with the sub-band bandwidth of 6 MHz. We use the shuttle radar topography mission digital elevation model (DEM) and precise orbits to compute the geometrical offsets for each stack and refine the azimuth offsets by amplitude cross correlations using a pixel window size. After inverting the azimuth offsets, we compute temporal coherence [35] to evaluate the inversion and mask out the estimated azimuth offset time-series using the temporal coherence with a threshold of 0.8. We then use a nearest neighbor interpolation approach to fill the masked regions of the azimuth offsets and oversample the offsets to the same grid size of the geometrical azimuth offsets. We multilook each interferogram by 4 and 14 looks in range and azimuth directions, respectively, filter using a Goldstein filter, and unwrap using statistical-cost network-flow algorithm for phase unwrapping [36]. After estimating the dispersive and nondispersive components, we use a -D Gaussian filter with a kernel size of 100 pixels in both range and azimuth directions to low-pass filter the estimated ionospheric and nondispersive phase components using the iterative filtering-interpolation approach explained in Section III. For each stack, we also estimate a raw InSAR time-series, in which a network of full-band interferograms (14-MHz interferograms) is inverted to estimate the InSAR time-series. We call this time-series, a raw InSAR time-series because the interferograms contain ground displacement, tropospheric and ionospheric delays, and possible geometrical residuals (e.g., caused by DEM errors). Fig. 3 shows the raw InSAR differential time-series, the estimated differential time-series of ionospheric phase delay, the differential InSAR time-series after compensation for the ionospheric delay, uncertainty of the estimated ionospheric phase delay, and the differential time-series of the azimuth offsets for track 104 in Chile. Note that the estimated differential phase between and contains the coseismic ground displacement caused by the Mw Tocopilla earthquake. Fig. 3 indicates a maximum relative ionospheric delay of 1.3 m across the 65-km-long swath. The differential time-series of azimuth offsets indicates a maximum 1.4-m azimuth offsets caused by an ionospheric phase gradient in the azimuth direction. The differential InSAR time-series compensated for ionospheric delay reveals coseismic ground displacement due to the Tocopilla earthquake between and acquisitions. Differential time-series of ionospheric delay for track 103 in Chile (Fig. 4) shows that the delay varies more than m between the acquisition dates along the 445-km azimuth extent of this track. The differential time-series of the azimuth offsets also indicates maximum azimuth offsets of 5 m caused by the ionospheric phase gradient in the azimuth direction. Parts of the coseismic ground displacements caused by the 007 Tocopilla earthquake are evident in the differential phase between and acquisitions in the differential InSAR time-series compensated for the ionospheric delay. Comparing the differential time-series of the azimuth offsets with the differential InSAR timeseries compensated for ionospheric delay, reveals that after ionosphere compensation, differential phase maps with larger residuals also show significant azimuth offsets, implying that the residuals are mainly caused by high-gradient ionospheric phase delay (scintillation) originating from ionospheric irregularities. Two geocoded interferograms from tracks 103 and 104 (Fig. 5), containing the coseismic ground displacement, are more consistent in their overlap region, after ionospheric delay correction. The remaining inconsistency in the southern part of the overlap of the two tracks after the correction is most likely due to tropospheric delay. The differential time-series for a stack of 3 ALOS-1 acquisitions in California (Fig. 6) shows a maximum ionospheric delay of less than one phase cycle (11.8 cm) across 160-km ALOS-1 azimuth extent. The smaller ionospheric delay in California relative to Chile is likely due to the fact that California is located at mid-latitudes with less intense and smaller TEC variations as compared to Chile. A. Comparing InSAR Time-Series With GPS Time-Series In order to validate the InSAR time-series after ionospheric delay correction, we use several GPS time-series, whose locations are plotted in Fig. 5(b). Fig. 7 compares time-series of pairwise GPS positions with InSAR time-series before

7 FATTAHI et al.: InSAR TIME-SERIES ESTIMATION OF THE IONOSPHERIC PHASE DELAY 7 Fig. 3. Differential InSAR time-series for ALOS-1 track 104 in Chile. Each map shows differential quantities between two consequent acquisition dates. First row shows differential raw InSAR time-series, second row shows differential time-series of ionospheric phase delay, third row shows differential InSAR time-series compensated for the ionospheric delay, fourth row shows uncertainty of the differential time-series of ionospheric delay, and fifth row shows the differential time-series of azimuth offsets. and after ionospheric delay correction. Comparing the InSAR and GPS time-series reveals that the scatter of InSAR timeseries significantly reduces after ionospheric delay correction. To quantifying the improvement of the InSAR time-series after ionospheric delay correction, we assume the GPS timeseries as truth and compute the deviation of InSAR timeseries relative to GPS time-series using a root mean square error (RMSE) given as RMSE = N i=1 ( d gps i di InSAR ) N (15) where d gps i and di InSAR represent the GPS and InSAR timeseries, respectively, at the ith InSAR epoch, and N is the number of InSAR acquisitions in the stack. Note that the two time-series are compared only at their common epochs. Before correction, the RMSE between InSAR and GPS varies from 14.0 cm at MCL1-VLZL stations to 48.0 cm at CRSC-SRGD stations (Table I). Correction of ionospheric delay significantly reduces the RMSE, varying between 1.0 cm at CDLC-SRGD to 6.8 cm at CRSC-CTLR (Table I and Fig. 7). V. DISCUSSION A. Noise Amplification in the Estimated Ionospheric Phase Delay The uncertainty of estimated ionospheric phase delay given in (9) is mainly controlled by the separation of sub-band frequencies. Fig. 8 shows the ratio of the standard deviation

8 8 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING Fig. 4. Similar to Fig. 3, but for track 103. of the ionospheric phase to that of the sub-band phase as a function of frequency separation between the two subbands, assuming a full-band central frequency of 1.7 GHz that is within the L-band range of the microwave spectrum. As expected, the ratio is large for small separation between f L and f H implying that noise of the sub-band interferograms

9 FATTAHI et al.: InSAR TIME-SERIES ESTIMATION OF THE IONOSPHERIC PHASE DELAY 9 Fig. 5. ALOS-1 interferograms, containing the ground displacement caused by the Tocopilla earthquake, from adjacent tracks (Left) 103 (acquisitions and ) before ionospheric delay correction and (Right) 104 (acquisitions and ) after ionospheric delay correction. Black diamonds: location of GPS stations whose time-series has been compared with InSAR time-series in Fig. 7. TABLE I RMSE BETWEEN GPS TIME-SERIES AND InSAR TIME-SERIES BEFORE AND AFTER IONOSPHERIC DELAY CORRECTION FOR SEVERAL STATIONS IN CHILE significantly amplifies in the estimated ionospheric phase. The ratio decreases with increased separation between f L and f H. For two sub-bands, 7.0 MHz apart (similar to ALOS-1 FBD acquisitions), σ iono is 131 times larger than σ φl,h. A nominal NISAR L-band mode will include main band and sideband of 0 and 5 MHz, respectively, at the two ends of a total 85-MHz bandwidth. This acquisition mode results in 7.5-MHz separation between the two bands. This large separation will reduce the noise amplification factor to around 13 for NISAR data. In other words, assuming the same scene scattering properties, the estimated ionospheric phase delay from NISAR acquisitions is expected to be an order of magnitude less noisy than the ionospheric phase delay estimated from the ALOS-1 FBD acquisitions. Fig. 9 shows an ALOS-1 FBS interferogram covering the Feb. 7, 010 Mw 8.8 Maule earthquake in Chile and the estimated ionospheric and nondispersive phase components before and after low-pass filtering. The ionospheric and nondispersive phase components [Fig. 9(b) and (e)] are sufficiently noisy, such that the phase fringes are not evident in the re-wrapped phase [Fig. 9(h) and (k)]. Given the nominal L-band NISAR acquisition parameters [8], and assuming the same noise level Fig. 6. Differential InSAR time-series for ALOS-1 track 16 in California. First and second rows: differential raw InSAR time-series. Third and fourth rows: differential time-series of ionospheric phase delay. Fifth and sixth rows: differential InSAR time-series compensated for the ionospheric delay. Seventh and eights rows: uncertainty of differential time-series of ionospheric delay. in the sub-band ALOS-1 FBS interferograms and NISAR sideband and main-band interferograms, the noise amplification in the ionospheric and nondispersive phase components reduces by a factor of five. Fig. 9(d) shows expected ionospheric phase delay before filtering, assuming NISAR L-band acquisition parameters. For the expected NISAR ionospheric phase delay, we estimate noise as the difference of unfiltered [Fig. 9(b)] and filtered [Fig. 9(c)] ALOS-1 ionospheric phase delays and scaled the noise by 1/5 to simulate the expected noise in NISAR ionospheric phase. The expected NISAR ionospheric phase [Fig. 9(d)] is the sum of the scaled noise and the filtered ionospheric delay [Fig. 9(c)]. Fig. 9(j) suggests that with nominal NISAR parameters, the ionospheric phase fringes are clearly visible even before low-pass filtering. B. Residual Ionospheric Delay Low-pass filtering of the ionospheric phase delay estimated from the split range-spectrum technique reduces noise at the expense of spatial resolution. In other words, with the split range-spectrum technique, only the long spatial wavelength ionospheric phase delay can be estimated. The shortest spatial wavelength of the estimated ionospheric delay depends on the size of the low-pass filter. This mainly depends on the separation of sub-band center frequencies such that larger filter sizes are required for a smaller separation of sub-band frequencies and smaller filter sizes for a larger sub-band separation. Gaussian low-pass filters with larger kernel sizes

10 10 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING Fig. 8. Ratio of the ionospheric phase standard deviation to the sub-band phase standard deviation as a function of the separation of sub-bands center frequencies. For smaller values of ( f H f L ), noise of the sub-band interferograms significantly amplifies in the estimated ionospheric phase. The noise amplification reduces with the increase in ( f H f L ).Note the logarithmic scale of both the horizontal and vertical axes. Fig. 7. (Left) Differential GPS time-series (block circles) compared with InSAR time-series before (yellow squares) and after (green circles) ionospheric delay correction. Both GPS and InSAR time-series are differential between the GPS sites. InSAR time-series were obtained by temporal integration of the differential time-series and are relative to a reference date, which is for track 104 and for track 103. All InSAR timeseries are from track 103 except for MCL1-VLZL, which is from track 104. GPS time-series are referenced to the same reference date as InSAR timeseries. For each GPS time-series, all three components in east, north, and vertical directions were used to project to the InSAR LOS direction. Orange vertical dashed line: 007 Mw 7.7 Tocopilla earthquake. (Right) Residual time-series that is the difference of GPS and InSAR after ionospheric delay correction. Each residual time-series is shifted to its mean. Shaded gray bars in the residual plots for CRSC-CTLR, CDLC-CTLR, and CTLR-SRGD indicate acquisition dates with large azimuth offsets at station CTLR (see Fig. 10 for a time-series of azimuth offsets at stations CTLR and CDLC). remove a larger spectrum of the ionospheric delay through noise reduction process. The time-series of residuals between GPS and InSAR after ionospheric delay correction at different GPS station pairs (Fig. 7) indicates a higher RMSE for CRSC-CTLR, CDLC-CTLR, and CTLR-SRGD pairs, which all share the CTLR station and higher residuals at the first and 8th epochs of the time-series. These two epochs also show significant azimuth offsets at station CTRL (Fig. 10). Azimuth offsets are caused by high-gradients in the ionospheric phase delay in the azimuth direction. Therefore, the correlation between epochs with large residuals (Fig. 7) and large azimuth offsets (Fig. 10) imply that the residual of InSAR and GPS time-series in Fig. 7 are most likely dominated by the highgradient ionospheric delay that is lost due to the necessary low-pass filtering. Fig. 9. Noise amplification and the need for low-pass filtering of the ionospheric and nondispersive phase components estimated with the split range-spectrum technique. (a) Full-band ALOS-1 interferogram formed from two FBS (8 MHz) acquisitions, acquired on and from track 116 (frames 6410 to 6460) covering the coseismic displacement caused by the Mw Maule earthquake. (b) Estimated ionospheric phase delay before filtering. (c) Similar to (b), but after filtering using a -D Gaussian low-pass filter with a kernel size of 100 pixels in both the range and azimuth directions. (d) Expected ionospheric phase delay before filtering if L-band NISAR acquisition strategy is assumed with two 5- and 0-MHz bands, 7.5 MHz apart from each other. The expected NISAR-like ionospheric delay was obtained by scaling noise [difference of (b) and (c)] to one-fifth and adding it back to (c). (e) Estimated nondispersive phase before filtering. (f) Similar to (e) but after low-pass filtering. (g) (l) Same as top row but rewrapped to ALOS-1 L-band natural fringe rate. The range-spectrum of each full-band ALOS-1 FBS SLC was split into two sub-bands with 14-MHz difference between f H and f L and with a sub-band bandwidth of 1 MHz. The high-gradient ionospheric phase variations in azimuth direction can be potentially estimated by integrating azimuth offsets [19], [0]. However, this approach cannot distinguish between ionosphere-induced azimuth offsets and those caused by large ground displacements. The expected high frequency ionospheric phase delay should also be reduced with dusk-

11 FATTAHI et al.: InSAR TIME-SERIES ESTIMATION OF THE IONOSPHERIC PHASE DELAY 11 Fig. 10. Time-series of azimuth offsets at stations CTLR and CDLC from track 103. The vertical gray bars indicate epochs with significant azimuth offsets compared to the rest of the time-series at station CTLR. Fig. 11. Ionospheric delay at different frequencies shown as wrapped phase for the corresponding wavelength. (a) Observed ionospheric phase delay at L-band [the same as Fig. 9(i)] and predicted delay for the same TEC variation at different frequencies (b) S-band, (c) C-band, and (d) X-band. If shown as unwrapped phase, the difference between L- and X-bands would be about a factor of 60. down acquisition strategy, when ionosphere scintillation is at a minimum [37]. Less noisy ionospheric delay from NISAR will require lighter low-pass filters and, therefore, a higher frequency ionospheric delay will be preserved. C. Expected Ionospheric Delay at Different Carrier Frequencies The first-order ionospheric delay given in (4) is inversely proportional to the square of carrier frequency of microwave signals. Given the same variation of ionosphere TEC and a similar imaging geometry, ionospheric delay at L-band frequency (1.7 GHz) is 3.9, 18.1, and 57.7 times larger than S-band (.5 GHz), C-band (5.405 GHz), and X-band (9.65 GHz) frequencies, respectively. Fig. 11 shows estimated ionospheric delay in Chile at L-band frequency compared with expected delay at S-, C-, and X-band frequencies assuming the same TEC variation observed with ALOS-1 and same Fig. 1. Impact of the ionosphere-induced azimuth offsets on the interferometric phase and coherence for a pair of ALOS-1 acquisitions, acquired on and from track 103 over Chile. (a) Interferometric phase and (b) coherence between the two SLCs coregistered with only geometrical offsets. (c) and (d) Same as (a) and (b) but with azimuth offsets adjusted with amplitude cross correlation between the two acquisitions. See Fig. 4 for a plot of azimuth offsets between the two acquisitions. imaging geometry as ALOS-1. A TEC variation of around 10 TECU between the two SAR acquisitions ( and ) and along 360-km azimuth extent of the imaged track, results in more than 1 interferometric phase cycles in L-band while it would only generate 10, 5, and less than three phase cycles in S-, C-, and X-band interferograms, respectively. D. Impact of Azimuth Misregistration Ionospheric phase gradient in the azimuth direction may introduce significant azimuth offsets, causing misalignment of SAR images and resulting in noisy interferograms [6], [0]. Fig. 1 shows an interferogram and the associated coherence map obtained with only geometric coregistration compared with the same interferogram and coherence map after accounting for the azimuth offsets induced by the ionospheric phase variation. The azimuth offsets for this pair can be seen in Fig. 4. If not accounted for, the large ionosphereinduced azimuth offsets reduce coherence, as well as introduce noise to and bias the interferometric phase. We note that the azimuth offsets are induced by the gradient of the ionospheric phase delay in the azimuth direction and not by the absolute

12 1 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING magnitude of TEC. For example, the spatially low-frequency large ionospheric delay variation of around 1.5 m (6.1 TECU) between and acquisitions in Fig. 4, does not introduce significant azimuth offsets. In contrast, the spatially high frequency ionospheric phase delay between and acquisitions in Fig. 4 introduce large phase gradients that cause significant azimuth offsets. E. Phase Ramps in InSAR Data Long-wavelength phase ramps in InSAR data have been traditionally attributed to residual geometrical phase mainly caused by inaccuracy of satellite orbits. However, based on reported accuracies of the orbits of SAR satellites, the residual phase caused by orbital errors even for older SAR missions, such as European remote sensing satellite and environmental satellite, is expected to be smaller than a phase cycle over around 100 km [1]. The expected uncertainty of orbital errors in ground displacement velocity fields obtained from InSAR time-series is of the order of 1- mm/yr over 100 km for older satellites and should reach below 1 mm/yr over 100 km for modern satellites with precise orbits [38]. Here we found that long-wavelength phase ramps in L-band SAR data acquired over Chile in the equatorial belt are dominated by ionospheric delay. The evaluated stack in California is affected by smaller ionospheric delay, such that the magnitude and temporal variation of the ionospheric delay is of the order of expected tropospheric delay in this region. After ionospheric delay correction, the residuals are most likely due to tropospheric delay, DEM error, and residual ionospheric delay caused by high-gradient TEC variation. The contribution from orbital errors is most likely negligible. VI. CONCLUSION We presented an algorithm based on the split rangespectrum technique for time-series estimation of ionospheric phase delay from a stack of SAR acquisitions. We applied the algorithm to three 14-MHz ALOS-1 stacks in Chile and California with different ionospheric delay characteristics. Estimated time-series of ionospheric delay shows significant variation of ionospheric delay of up to m along 445 km in Chile with both low and high spatial frequencies. The temporal variation of the delay in California reaches maximum 10 cm over 160 km with only low spatial frequencies. Correction for ionospheric delay in regions with high TEC variation reduces the temporal variation of the InSAR time-series from meter levels before the correction to centimeter levels after correction for ionospheric delay. Comparing independent GPS time-series to InSAR time-series demonstrated significant reduction in deviation of InSAR from GPS with an RMSE of 14.0 to 48.0 cm before correction to 1.0 to 6.8 cm after correction. Uncertainty in estimates of the ionospheric delay time-series is a function of the separation between low-band and highband frequencies and the coherence of the interferograms. For highly coherent stacks in arid areas of Chile, stacks of L-band SAR data with small full-bandwidth of 14 MHz resulted in uncertainties of less than 1 cm for most acquisition dates. For the California stack, with less coherence between the SAR acquisitions, uncertainties were larger (around -3 cm for most acquisitions). Larger range bandwidth and separation between low-band and high-band center frequencies, shorter spatial baselines, and more frequent acquisitions will all reduce the uncertainty in estimates of the ionospheric delay timeseries. 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His research interests include algorithm development for SAR, InSAR, and InSAR Mark Simons received the B.Sc. degree in geophysics and space physics from the University of California, Los Angles, Los Angles, CA, USA, in 1989, and the Ph.D. degree in geophysics from the Massachusetts Institute of Technology, Cambridge, MA, USA, in He has been with the California Institute of Technology, Pasadena, CA, USA, since 1996, where he is currently a Professor of geophysics with the Seismological Laboratory. His research interests include studying processes that deform the solid earth including those associated with the seismic cycle, migration of magma and water in the subsurface, tides, and glacial rebound; tectonics and the relationship between short and long time scale processes; glaciology, particularly basal mechanics and ice rheology; tools and applications using space geodesy, particularly GNSS and SAR; Bayesian methods for large geophysical inverse problems; and application of space geodesy for monitoring and rapid response to natural disasters. Piyush Agram (M 10) received the B.Tech. degree in electrical engineering from IIT Madras, Chennai, India, in 004, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, USA, in 010. He was a Keck Institute of Space Studies Post- Doctoral Scholar with the Caltech s Seismological Laboratory, Pasadena, CA, USA, until 013, and then joined the Radar Algorithms and Processing Group, Jet Propulsion Laboratory, Pasadena, CA, USA. His research interests include algorithm development for SAR focusing, radar interferometry for deformation time-series applications, and geospatial big data analysis.