Impact of receiver errors on the radiometric resolution of large

Transcription

1 Radio Science, Volume 32, Number 2, Pages , March-April 1997 Impact of receiver errors on the radiometric resolution of large two-dimensional aperture synthesis radiometers F. Torres, A. Camps, J. Barfi, and I. Corbella Department of Signal Theory and Communications, Universitat Politi cnica de Catalunya, Barcelona, Spain Abstract. The specifications of the subsystems that compound a radiometer interferometer devoted to Earth observation are of main concern because they set the viability and final performance of such an instrument. The importance of these errors is related to the exact way they are generated, since this determines if a particular calibration procedure is capable of removing them or if they remain as residual errors. This paper presents a general method to analyze system errors. This method is used to analyze in detail the amplitude and phase errors of the receivers and their impact on the radiometric resolution. Special attention has been paid to nonseparablerrors, since foreseen calibration procedures can only deal with separable phase and amplitude terms. Finally, the results have been used to set the receiver requirements of the instrument called MIRAS (microwave imaging radiometer by aperture synthesis), which is currently being developed by the European Space Agency (ESA). 1. Introduction Aperture synthesis radiometers are of increasing interest in onboard Earth observation applications due to their reduced mass and volume requirements in relation to traditional total-power radiometers. Some works have already been performed to study fundamentals and practical limitations of these instruments [Ruf et al., 1988; Le Vine, 1990; Tanner and Swift, 1993]. Recently, the SMOS (soil moisture and ocean salinity measurements and radiometer techniques consultative meeting) [European Space Agency, 1995] established the interest of these instruments operating at 1.4 GHz in the measurement of soil moisture and ocean salinity, in order to develop global hydrological cycle and climate models. The SMOS [European Space Agency, 1995] has also concluded that at present, "a single frequency, dual polarization instrument in a sun synchronous dawndusk orbit giving 0.5 K radiometric resolution, 10 km spatial resolution and 1-3 days revisit time would satisfy most user requirements" (p.6). Within this Copyright 1997 by the American Geophysical Union. Paper number 96RS /97/96RS scope, the European Space Agency is currently developing an L-band bidimensional interferometer radiometer called MIRAS (microwave imaging radiometer by aperture synthesis) [Martœn-Neira et al., 1994]. This space-borne instrument is configured as a Y-shaped array with 43 antennas per arm, spaced at 0.89, plus the central antenna. In a real system, the recovered image is corrupted by the nonidealities of the subsystems that compound the interferometer. The impact of such system errors on overall system performance is very dependent on the application (Earth observation or radio astronomy) and the particular calibration procedure being implemented. Ruf [1991] analyzes errors in the linear array called ESTAR (electronically steered thinned array radiometer), which makes use of the G matrix method to recover the image. Thompson and D tddado [1982] analyze receiver errors in radio astronomy arrays in terms of loss of sensitivity and their impact on the calibration procedure relating to the National Radio Astronomy Observatory (NRAO) very large array (VLA). A general strategy to address the design of two-dimensional (2-D) arrays devoted to Earth observation is given by Martin-Neira et al. [1996], who summarize the present status of MIRAS, relating system and subsystem conception, system errors, calibration procedures, and inversion

2 630 TORRES ET AL.: RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS algorithms. System errors are classified into (1) antenna errors, (2) receiver channel errors, and (3) receiver baseline errors. In this paper we present a general method to analyze the impact of system errors on the radiometric resolution of a large 2-D radiometer interferometer devoted to Earth observation. Receiver channel errors and receiver baseline egrors are analyzed in depth for a general case, highlighting the need to classify such errors into separable and nonseparablerrors. Finally, the study is used to derive hardware requirements for the MIRAS instrument. 2. Miras Basic Description While total-power radiometers measure the power collected by a highly directive antenna, which directly gives the brightness temperature in the main beam direction, interferometer radiometers devoted to Earth observation measure the correlation between the signals given by pairs of nondirective antennas, yielding samples of the coherence function, also called visibility. If Sa (O and S,a(O are the analytic signals of the band-pass noise voltage at the output of a pair of antennas "1" and "2" observing the Earth (Figure 1), the visibility sample V (u,v) is defined as v,2(u,v) lr[s, which is related to the Earth brightness temperature as (1) where Via (u,v) =ff z+nz,?(,q) la(-a:) e -J: C + ) dl[ (2) dq (u,v) spacing between antennas 1 and 2, in wavelengths; T (,q) brighiness temperature; T (,q) modified brightness temperature, K; 1/ /(1- Lq:) obliquity factor; F (,q), F :(,q)normalized antenna voltage patterns; =sin0cos, q=sin0sin directing cosines (see Figure 1); x=(u +v )/fo spatial delay. The term 12( )=rll( )e -2J'f" is called the fringe- wash function. This function takes into account decorrelation effects and depends on the receivers normalized frequency responses through r, 0:): f o* Hn, (f) Hn='(f) e J 2,.e, df (4) where Hni (f) is the receiver normalized band-pass voltage response. Note that as x (the spatial delay) is coordinate dependent, it cannot be compensated as done in radiometry when observing quasi-punctual radio sources. However, when observing the Earth, antennas are closely spaced, and this produces only a slight blur of the image at the swath edges, which can be neglected. It is well known that for ideal receivers, equal antenna patterns, and negligible decorrelation effects, the fringe-wash term can be approximated to (0)-!, and the visibility function becomes the Fourier transform of the modified brightness temperature T(,q): y X % I Figure 1. Interferometer geometry. Antennas 1 and 2 are placed at the observation points in the plane XY. Separation between both antennas in terms of, sets a "baseline." V(u,v) - f fg,+n,s?(,q)e-je' ½u +vn d dq (5) In Figure 2 we have the simplified block diagram of a receiver relating the measurement of one baseline. The real band-passignals out of antennas 1 and 2, bl(t) and b2(t), are demodulated by a pair of coherent in-phase/quadrature (I/Q) demodulators. The output of the receiver is the complex normalized visibility, given by

3 TORRES ET AL.' RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS 631 (t). sin{ /2 z } windowing). Since each pixel remains about 11 s in the field of view (at each polarization), pixel averaging reduces radiometric resolution to 0.9 K. 3. Modeling Amplitude and Phase Errors Amplitude and phase errors can be divided into (1) receiver channel errors and (2) baseline errors. Figure 2. Simplified block diagram of a single baseline Receiver channel errors are those which appear in the MIRAS receiver. measured visibility samples as isolated amplitude factors or phase summands, each one depending on the parameters of a single channel. Baseline errors are r2(u,v) r2(u,v) those errors that have a nonlinear dependence on the parameters of both channels. In the following = v=(o,o) v(o,o) (6) paragraphs, separability will be addressed in more where V:2(u,v ) is the visibility sample V(u,v) detail. Separable errors can be easily accounted for. measured between antennas 1 and 2. V(0,0) is the Let the signal at the I/Q demodulator input of scene averaged power kt^b, assumed to be equal at receiver 1 and 2 be (Figure 2) the output of both antennas. Since the absolute amplitude of the visibility samples is lost due to the b (O=bl(O +ni(o b " (O=b2(O +n:(o (8) normalization, a total-power radiometer is required to measure the sample V(0,0), which is placed at the where b (t) is the ideal signal and n (t) is the zerocenter of the array. As shown in Figure 2, the real and mean narrow-band Gaussian noise introduced by the imaginary parts of the normalized visibilities are receiver. If we take into account that n (t) and n2(t ) computed by means of base-band real correlators. If are uncorrelated signals of average power given by bl(t)=sl(t)cos{rot+o)l(t)} is the real band-pass voltage ktmb, and ktr2b (TR1 and TR2 are the receiver noise at the output of antenna 1, its in-phase and quadrature temperatures, and T^ the scene average temperature), components are given by i (t)=s (t)cos{{ (t)} and then the measured normalized visibility can be written as q (0=S (t)sin{{ (t)}. Taking into account a similar notation for the signal received by antenna 2, the real and imaginary parts of the normalized visibilities are given by (9) /< i (t) i (t) > < i2(t) i2(t) > /<q (0q (0 > <i2(0i2(0> (7) j,, < q (0/a (0 > (10) ]<q (t)q *(t)> MIRAS makes use of 1 bit, two-level digital correlators because of their low consumption, high (11) speed, and high degree of integration [Martin-Neira et a/., 1994]. The cost is a reduction of the effective integration time (41% at the Nyquist sampling If phase errors are taken into account, the I/Q frequency), and the outputs of the 1 bit, two level demodulated signals can be written as digital correlators must be processed in order to derive the desired normalized visibilities, as stated in Figure 2 [Hagen and Farley, 1973]. MIRAS integration time (t) +Oøl is set to 0.3 s, yielding a snapshot resolution due to _0 l the finite integration time of 5.1 K (Blackmann where i w(t)=sl(t) dpl + O l (12) q '(t) =S (t)s. (t)+0o - -)

4 632 TORRES ET AL.' RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS in which 0o is the receiver in-phaserror and 0 is its quadrature error. Now if both amplitude and phase errors are considered, the measured normalized visibilities can be expressed as [Torres et al., 1996] (13) =g g: u!s 3.1. Separable and Nonseparable Error Terms Let us have two antennas 1 and 2 and their associate baseline "12," which is measured by means of a nonideal receiver. The measured visibility --12 vraw can be written as _ -- 2 vraw_ V 2, where G 2 is the complex correlator gain. G 2 is referred to as a baseline coefficient (quadrature errors have been neglected to avoid the matrix representation). However, if the correlator complex gain can be factorized into O 2=O Ga =g ejoo,gae (14) 2 where from now on the phase terms stand for residual phase errors and A V,2 takes into account all amplitude where G and G: are the complex gains associated errors. Since it can be considered that after calibration, with each channel, it is said that G : is separable. G these errors are much lower than one, then and G: are channel complex coefficients. In general, A V 2--Ag +Ag2+Ag 2+A V o. the complex gain G 2 can be factorized into both separable and nonseparable coefficients [National Radio Astronomy Observatory, 1989]: (15) 012 =gl g2g12 ej(øø'-øø'- If the normalization term V(0,0) and quadraturerrors are included, we require a matrix relationship of real coefficients to relate the measured visibility to the ideal magnitude: (16) Vj) =g'g:g':v(o'o) u This consideration is of main importance because the so-called channel-based calibration procedures make use of separability to estimate the channel coefficients. Nonseparable terms will contaminate such procedures and remain as residual errors. Hence hardware requirements on subsystems producing separable or nonseparablerrors must be treated accordingly Calibration Considerations Accurate system design and manufacturing cannot reduce amplitude and phase errors to the level which is required. Some kind of calibration procedure (e.g., self-calibration, noise injection, ground test, etc.) must be allocated to estimate such error coefficients. If the estimated amplitude coefficients are written as, 1,2 =gl.2 (1'/ gl.2), 12 = 1 +/ g12 t(o,o)= v(o,o)o+ av' o) (17) and a similar notation is used to write the estimated l,,rraw phase coefficients, a new -n can be obtained by multiplying (16) by the inverse of the matrix of estimated coefficients. This yields (18) 4. Effect of Amplitude and Phase Errors on the Radiometric Resolution In compact notation, equation (18) can be rewritten to relate each visibility sample as Vu = Pu Vu (19) where u is the 2x2 matrix which includes all phase and amplitude error terms relating to each measured baseline (equation (18)). M is the number of visibilities used to measure a scene. Note that the generalization for the whole system produces a sparse system of linear equations, which allows us to generate the measured visibilities V aw out of the ideal samples. The reference scene used in all the

5 TORRES ET AL.' RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS J.-' "... "[."..it c,rcle (-).- "-,../'" I t ",// 'LI I / '"'N'"'... -._ / ' E (rth-sky bo er (4'"'"..., / ',,/-.. ' /., -,./ \ / ""'"'" I '"'"'., ' for a Y-shaped array or d=x/2 for a T-shaped array -0.8./' /..,.,.' /'%.. '.,. [Mersereau, 1979; Camps et al., 1995a]. However, -. since MIRAS baseline length has been set to 0.89X, to reduce the number of antennas, the recovered scene is overlapped with its aliases. Hence the alias-free Figure 3. Earth and sky regions in the directing cosines region is reduced to the area shown in Figure 3, in domain from a platform of 800 km height, tilted 31.2 ø. which the Earth and sky alias have also been Earth and sky aliases, the field of view (FOV) (trapezoid), represented. and region where the errors are computed (inner circle) are Simulations have shown that phase and amplitude shown. errors are multiplicative errors which do not depend on the number of elements per arm (NEu>10). Hence, simulations consists of a modified brightness in order to increase simulation speed and reduce temperature distribution inside the Earth-sky border of memory requirements, the number of antennas per arm 200 K (Figure 3): has been set to NEu=15. Since it is not clear that pixel averaging reduce system errors, phase and amplitude errors have been constrained to give a radiometric = -200 K; (,q) ceat (,q) sensitivity of AT=0.5 K. This goal is below the absolute MIRAS noise floor given by pixel averaging: (20) AT=0.9 K. Of course, since phase and amplitude errors will be constrained to give radiometric errors where (, 1) are the directing cosines ( 2+ 12<1). The below this noise floor, the effect of noise is neglected sky temperature has been set to 0 K. The recovered in all the simulations. This is equivalento considering distribution is then given by an infinite integration time for the correlators. It must be pointed out that the radiometric resolution that is (21) obtained for a null phase and amplitude error is not zero because of the discretization and finite coverage where W(u,v) is the 2-D blackmann window. The of the (u,v) plane. It has been found that for a number inverse Fourier transform is performed over the (u,v) of antennas per arm NEu--43 (MIRAS case), this error hexagonal grid and its reciprocal (, 1) basis [Camps is AT=0.03 K, and for NEL=15, AT K, both et al. 1995a, b]. The error is computed as the below the goal of AT=0.5 K that has been established root-mean-square value of the difference between the to constrain the effect of phase and amplitude errors. real temperature and the recovered value: M AT= M- 1 /=1 (22) I E ( '( "q') - T( "q')) where M is the number of pixels in the area where the error is computed. The radiometric resolution AT given in all the tables and figures is the average error given by (22) out of 50 realizations. The computation of the error is restricted to the inner circle in Figure 3 (radius 0.16 in the (, 1) plane), which is placed at the center of the field of view (FOV). When sampling the frequency spectrum, its inverse Fourier transform repeats periodically and there is some danger of aliasing. To avoid this, the maximum sampling period is given by the antenna separation. For a 2-D signal supported by the unit circle, the maximum antenna spacing that satisfies the Nyquist criterion is d=xi ] Effect of Separable Amplitude Errors Initially, the measured visibilities have only been contaminated by separable amplitude errors, other errors in (18) having been set to zero. That is, the amplitude error in the visibility sample measured by means of antennas "rn" and "n," after calibration, is given by

6 634 TORRES ET AL.: RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS Table 1. MIRAS Radiometric Resolution AT versus Normalized Amplitud Error O, v RADIOMETRIC RESOLUTION (BLACKMANN, NEL=15) Oav AT, K (o) AT, K 10 '2 3,96 0, _.1.2 3,2 10 '3 1,25 0, l0 '3 0,39 0, ,2 l0-4 0,12 0, ATR is the error in the estimated receiver noise temperature that gives Oav 0.2 o 20 2 ' 24 ' I I I ß S/N (db) V'S "= V,,,,,(1 + A V,,,n) =V,,,n (1 + AS., + AS,,) (23) where AVmn is a zero-mean error of deviation Oav, which accounts for all amplitude errors. In terms of the signal-to-noise (S/N) ratio (Vm u over Vnoise ), we can write ( )=-10 1ogoav=10 1og( / oas,, ) (24) Inversion of the measured samples by means of the inverse fast Fourier transform gave the estimated temperature distribution. Only half of the visibilities have been corrupted with amplitude errors. The other half have been calculated by hermiticity in order to simulate MIRAS operation mode. The radiometric error, computed at the center of the scene is shown in Table 1. The solid line, the asterisk, and the cross in Figure 4 represent the same process: uncorrelated amplitude errors in each visibility sample, computed by different methods, which do not correspond to a real-life instrument. However, this case is much easier either to simulate or to derive an analytic expression. It is useful to compare and check the results of the real MIRAS case. The solid line represents the expression developed in the appendix relating the case of uncorrelated amplitude errors Agmn. The asterisk in Figure 4 is a simulation performed by adding an uncorrelated amplitude error A Vmn to each visibility sample in (23). The cross in Figure 4 is computed by adding an uncorrelated channel amplituderror ag m in (23) but recalculating Ag m to generateach Vm w (this causes uncorrelated amplitude error in V raw ). Note Figure 4. Impact of amplitude errors on MIRAS radiometric resolution. Uncorrelated amplitude errors in the visibilities: theoretic expression (solid line) and two type of simulations (asterisk and cross). Uncorrelated amplitude errors in the channels, MIRAS case, are shown by open circles. that since AVmn= Agm+ n, then oav = /2o Agm' Note that these three plots must give the same results since they representhe same process. Finally, the open circle represents the MIRAS case: Uncorrelated amplituderrors ag m have been added to each channel in (23) to calculate all the --mtl V ra ø Note that since the number of visibilities Vm ] (5676) is much higher than the number of channel errors ag m (130), the amplitude errors A Vmn = Z gm+ z g n are highly correlated. Note in Figure 4 that all results are very similar when high S/N ratios are used. Correlation effects are only notable at low S/N ratios. As a conclusion, to achieve Table 2. MIRAS Radiometric Resolution AT versus Error in the Phase of the Visibility Samples OOoi, deg (o) AT, K 0,6 0,33 0,8 0,4 1,0 0,43 1,2 0,63 II

7 TORRES ET AL.' RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS 635 a radiometric resolution of AT=0,5 K over a constant calculate all the V raw Note that since the number of --mtl ø distribution of T=200 K, it is required that visibilities Vm n aw (5676) is much higher than the oar< 7.3x10-3(a S/N ratio of about 21 db). number of channel errors 0om (130), the baseline phase errors Oom n ---- Oom - Oon are highly correlated. As 4.2. Effect of In-Phase and Quadmasre ElmeS seen in Figure 5, correlation reduces the effect of In this case we proceed as we did when dealing phase error in relation to the uncorrelated case. It can with amplitude errors. The radiometric error due to be demonstrated that correlation of the phase errors phase errors is presented in Table 2. The solid line in reduces their effect when observing an even Figure 5, the asterisk, and the cross representhe same temperature distribution. Note that the Earth seen process: Uncorrelated phase errors in each visibility from space has a dominant even temperature sample, although they are computed by different distribution and correlated channel phase errors methods. The solid line plots the equation developed partially cancel. Similar simulations proved that both in the appendix, an approximate expression of the quadrature and in-phaserrors impact in the same way radiometric resolution due to visibility uncorrelated on radiometric resolution. Since in-phase and phase errors. The asterisk in Figure 5 is a simulation quadratur errors are uncorrelated, the variance of performed by adding an uncorrelated phase error Oom n phaserrors is 2 e = 2 o + O:q0. to each visibility sample in (18); other phase terms are If snapshot radiometric resolution is constrained to set to zero. The cross in Figure 5 is computed by AT=0.5 K over a constantemperature distribution at adding an uncorrelated channel phase error 0om in Ta=200 K, then oe < 1.1 ø. If this phase error is (18), but recalculating this 0om to generate each Vm n aw equally split into residual in-phase and quadrature (this causes an uncorrelated phase error in Vm n aw ). Note errors, then o < 0.8 ø and O0q< 0.8 ø. that since Oom n = Oom - Oon, then OOomn= 200om. Note that these three plots must give the same results since 5. Receiver Requirements they represent the same process. Finally, the open circle represents the MIRAS case: Uncorrelated phase As shown in section 4, to constrain each error errors 0om have been added to each channel in (18) to contribution to a radiometric resolution of AT=0.5 K, the receiver must accomplish Oar< 7.3x10-3, o<0.8 ø, and O0q<0.8 ø. Hereafter we will set receiver requirements to achieve these specifications. RADIOMETRIC RESOLUTION (BLACKMANN, N EL=I 5) -f- 2.5 o o 0.5 o o o I I theta (deg) Figure 5. Impact of phase errors on MIRAS radiometricresolution. Uncorrelated phase errors in the visibilities: theoretic expression (solid line) and two type of simulations (asterisk and cross). Uncorrelated phase errors in the channels, MIRAS case, are shown by open circles Separable Phase Ermls In a real-life instrument, in-phase errors are large and they are not expected to remain constant due to aging, temperature drifts, etc. Hence a periodic calibration procedure is mandatory to reduce in-phase errors to about 0.8 ø. MIRAS performs an analog I/Q demodulation at 1.4 GHz to ease signal distribution to the correlator unit. This can give quadrature errors of about 5 ø [Scientific Components, 1994], which must be either ground tested (and drift minimized) or included in the calibration algorithm to achieve about 0.8 ø of residual error Separable Amplitude Errors Once phase errors are calibrated, amplitude errors can be expressed as separable (25)

8 636 TORRES ET AL.: RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS where the factor V(0,0) is given by a total-power radiometer, and gig2 is the correlator gain, which is split into each channel factor gm' The error in the normalized visibility module oa, is given by m =gin (1 + Ag ) aa =v aag. (26) Denormalization of the visibility samples also gives an error due to the total power uncertainty: (31) where i is a gain factor, fi is the filter's central frequency (around fo), B is the filter's noise bandwidth, x is the group delay, and { is the filters' phase (including the contribution of the local oscillator). Now, the tinge-wash function can be computed as Hence avo r (27) AT oar (28) - O a - r R120:) =F-I[Hi(f) H (f)]=/ 120: ) ej2¾o ß (32) being the spatial delay. Under these assumptions and by normalizing the gain, Uncertainty of a total-power can be as low as Oar=0.015 K [Thompson eta/., 1986], which gives an uncertainty of the visibility OaVo=7.5xlO ' (S/N ratio of 41 db). Even an uncertainty of oar=0.1 K would give a S/N ratio of 33 db. Since these values are much lower than required, the amplitude error can be assigned to the normalized visibility. This is (29) The amplitude error in the normalized visibility oa, is directly related to the uncertainty in the receiver noise temperature ATa [Torres et al., 1996]: (30) Now taking into account MIRAS receiver specifications, Ta=80 K and T^=200 K, error in the receiver noise temperatures is related to the radiometric resolution AT in Table 2. Then, if radiometric resolution is constrained to AT=0.5 K, receiver noise temperatures must be estimated within ATR< 2.9 K Nonseparable Amplitude and Phase Ermls In this paragraph we will study the effect of slight differences among the transmission channels from the antennas to the correlators' inputs. Assuming Gaussian filters, the equivalent RF band-pass voltage transfer function of the "i" receiver takes the form (33) wheref 2 is the equivalent central frequency deviation, given by 2 2 Bt +B2 (34) in whichfi=fo+afi. Taking into account (33) the fringe- wash term can be rewritten as 12(_g) = 12 œ -2zBt22( -'-,t+,2)2e -J2zaft 'œj t2 (35) where B 2 is the equivalent bandwidth I 2 Bt +B BI2 = 2 2 ~B(1 + est2) (36) and A 2 is the fringe-wash gain 12= 2 2 (37) To express the phase term { 2, each receiver absolute group delay can be decomposed into x =Xo+AXi, where

9 TORRES ET AL.: RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS 637 'c o is the average delay. of all channels. Then B12,1;i +B 2 2 (38) The spatial delay 'c in (33) depends on the spatial coordinates (, 1). This causes a phase and amplitude aberration of the image called fringe-wash the effect of which is negligible inside the field of view: + ( x2 A- x f9 n(- )=e - 1 ASn(- :)=-2 Af 2 : - 0 (42) Note that ( t2 is decomposed into a separable part and a nonseparable part A 2. Now we can estimate the magnitudes involved. From a commercial catalogue [Scientific Components, 1994], with standard, nonmatched components (a really worst case for satellite payload), we can estimate the following error bounds for the magnitudes involved: f,,=l.4 GHz, B = 25 MHz, Xo = 80 ns, I Af,[ < 0.5 MHz, lab i [ < 1MHz, and I Axil < 2 ns. This yields an amplitude error 1^21< 1.2x10-3(S/N ratio > 29dB), which gives a radiometric error of AT< 0.15 K (Figure 4). Phase errors can in principle be eliminated from the phase closure condition. However, the non- separable part of the phase error A 2 contaminates the closure condition as A r=a n + A 2 a + A a -<1.08 * (39) 6. Offset Resid. n! Errors The most direct cause of offset errors is the presence of correlated noise in the receivers coming from the local oscillator (Figure 6). The local oscillator signal can be expressed as xa(0=a(1 +mn(t))coa( t + n(t)) +n(t ) (43) where m,(t) is the local oscillator amplitude noise, O)n(O is its phase noise, and n(t) is the wideband thermal noise. The AM noise modulates the mixer RF conversion loss, but if the mixer is driven at saturation level, this effect is minimized: Lc(mn(t))=L c. Phase noise does not contribute to the offset since the interferometer performs a power detection. Note in (1) that a common phase term, random or not, is canceled due to the conjugate term. However, thermal noise generated by the local oscillator (LO) at the RF and The equivalent channel phase error is about image bands is downconverted to IF by the mixer. It 0 q< 0.25 ø, which gives a radiometric error can be accounted as an equivalent offset temperature AT< 0.2 K (Figure 4). Differential delays introduced at the receiver input, given by by filter responses and transmission lines cause a decorrelation of the signals and hence an amplitude error. When only small differential delays are taken into account, it is derived from (35) that L½ (44) where Tph is the physical temperature of the LO, G is (40) the RF preamplifier gain, and L o is the conversion Then, the amplitude error is given by (41) b.(t) b n (t) If this error is to be negligible, then to achieve a radiometric resolution of AT=0.1 K, AV=3.3x10 '4 (t) and Ax 0.5 ns. This gives a differential path Figure6. Block diagram of a front-end to show mismatch of about 15 cm. Hence differential delays noisecontribution of the local oscillator: preamplifier gain are mainly contributed by filter differential group G, mixer conversion loss Lc and conversion loss of the RF delay. band at the local oscillator port Loa.

10 638 TORRES ET AL.: RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS loss of the RF and image bands at the LO input. To assess the order of magnitude of this offset, let us have a cavity oscillator, directly feeding the mixer, at Tphi=298 K, Lc=7 db, Lcol=20 db, and G=20 db. This gives Toff=0.3 K. However, if the LO is buffered via a 20-dB amplifier, the Toff increases to 30 K. As a conclusion, LO noise contribution must be minimized by using a high-gain preamplifier and increasing L ol (LO filter, low-gain buffer, high LO rejection mixer, etc.). Other causes of offset error are offset currents at the analog correlator inputs or voltage offsets at the threshold level of digital correlators [Laursen and Skou, 1994]. In all the cases, once the offset is calibrated by uncorrelated noise injection, the offset can be written as 2(u,v) (45) (u,v) lr(o,o) where the error is split into the real and imaginary part. When the absolute values of the visibilities are recovered, the measured denormalized visibilities can be written as (46) = + (%+jq,pv'(0,0) Each real correlator introduces an offset e relative to P0 max If residual calibration errors be as zero-mean Gaussian noise, we can write are assumed to (47) Since offset errors can be treated as zero-mean Gaussia noise, their impact on radiometric sensitivity can be evaluated from the signal-to-noise ratio. Assuming Blackmann windowing, to achieve a snapshot radiometric resolution of AT=0.25 K over a constant distribution at T=200 K, it is required that the S/N ratio be 45 db, which gives O r, ;i 2.2x10'5 [Bard et al., 1996]. 7. Conclusions This paper has presented a general method to analyze the impact of system errors on the radiometric resolution of large 2-D interferometeradiometers devoted to Earth observation applications. The study uses a simple model of the Earth seen from space in order to determine the alias-free field of view and set a reference temperature distribution. The work has been focused on receiver phase and amplitude errors, although the method can be extended to account for other system errors, such as correlator or antenna errors. More complex models of the Earth can also be easily introduced. Receiver errors have been classified into separable and nonseparable phase and amplitude errors. The importance of this classification has been addressed, since foreseen calibration procedures can only deal with separable coefficients. It has also been shown that receiver errors produce highly correlated baseline errors. This is an important result, since correlation highly reduces the impact of channel phase errors. Note that a simpler simulation that does not consider correlation would predict larger radiometric errors. This is due to the fact that the Earth presents a dominant, even temperature distribution. The effect of correlation on amplitude errors has been found to be of less consideration. Simulations established the upper bound of phase and amplitude errors as a function of MIRAS-required radiometric resolution. Since separable phase and amplitude errors are expected to be partially corrected by a suitable calibration procedure (methods based on noise injection or phase closure relationships), special attention was paid to nonseparablerrors. It has been found that these nonseparable errors are mainly contributed by interchannel mismatches of the receivers' frequency response. A detailed analysis of receiver response, using Gaussian filters, allowed us to develop analytic expressions to evaluate radiometric resolution as a function of filter parameter dispersion: center frequency, noise bandwidth, differential phase, and differential group delay. Appendix: Impact on Radiometric Resolution of Module and Phase Resid._al Errors The appendix is devoted to developing an analytical expression for the radiometric resolution when uncorrelated phase and amplituderrors are present in each sample of the measured visibility function. For an ideal interferometer, the expression of the expected brightness temperature can be written as

11 TORRES ET AL.: RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS 639 q(, )= Vr d2 2 (A1) where rr(k,n)-<'}(k,n)>] a stands for Gibbs phenomenon and aliasing effects and the term and q([,n)q'([,n) can also be written as where x13d /2 is the pixel area in the (u,v) plane, and d is the spacing between antennas, in wavelengths. Now amplitude and phase errors can be introduced by writing the appropriat expression of V r w in (A1). If the module of the visibility function has an error, it translates to the recovered temperature of the scene according to (A1): d a 2 1 (A2) Now replacing the value of the visibility function by its expression, (A2) can be rewritten as Its expected value will be (,n)q*(,n)> = I d 2 2 Then + d 2 q(, ) -- 1 (A3) (A7) where the expectation inside the summation is different from zero only for and s or m-m and s=-n, since e ors e assumed to be unco elated (in this theoretical expression) and the he ticity resoa has been used. Moreover, in order to simplify the equations, we can define f d T IFd,,01 (A4) Then (AS) and the error in the expected brightness temperature distribution is: A (l[,q) < (l[,q) -T(l,q)] (l[,q)-t(l[,q)]*> = <q(,n)q(,n)*>+t'r(,n)-<'i'(,n)>l -<q(,n)> (A5)

12 640 TORRES ET AL.: RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS The first double summation accounts for rn = r and Camps, A., J. Bari, I. Corbella, and F. Torres, Inversion n = s, and the second one accounts for rn = -r and n algorithms over hexagonal sampling grids in =-s (with rn and n different from zero). Equation (A9) interferometric aperture synthesis radiometers, paper can be rewritten as presented at the Progress in Electromagnetics Research Symposium, Univ. of Wash., Seattle, 1995b. European Space Agency, Conclusions and recommendations from the soil moisture and ocean salinity measurements and radiometer techniques consultative meeting, ESA WPP-87, ESTEC, Noordwijk, Netherlands, Hagen, J., and D. Farley, Digital correlation techniques in radio science, Radio Sci., 8, , Laursen, B., and N. Skou, A spaceborne synthetic aperture radiometer simulated by the TUD demonstration model, paper presented at the Int. Geosci. and Remote Sens. Inserting this expression into equation (A4), the Symp., Calif. Inst. of Technol., Pasadena, Aug. 8-12, radiometric sensitivity due to residual module errors is given by LeVine, D.M., The sensitivity of synthetic aperture radiometers for remote sensing from space, Radio Sci. 25, , Martin-Neira, M., Y. Menard, J.M. Goutoule, and U. Kraft, MIRAS, a two-dimensional aperture synthesis radiometer, paper presented at the Int. Geosci. and Remote Sens. Symp., Calif. Inst. of Technol., Pasadena, Aug. 8-12, Martin-Neira, M. et al., Integration of MIRAS breadboard and future activities, paper presented at the Int. Geosci. (All) and Remote Sens. Symp., Lincoln, Nebr., May Mersereau, R.M., The processing of hexagonally sampled signals, Proc. IEEE, 67 (6), , The last term in (All) accounts for Gibbs National Radio Astronomy Observatory, A collection of phenomenon and aliasing effects. Now a similar Lectures from the Third NRAO Synthesis Imaging expression can be developed for low in-phaserrors, Summer School, vol. 6, Astron. Soc. of the Pacific, San just by replacing (l+avmn) in (A2) by (l+ja0mn). Francisco, California, Ruf, C.S., Error analysis of image reconstruction by a Acknowledgments. This work has been supported by the shynthetic aperture interferometric radiometer, Radio European Space Agency within the MIRAS Rider 2 Sci., 26, , activities, Matra Marconi Space being the main contractor. Ruf, C.S., C.T. Swift, A.B. Tanner, and D.M. Le Vine, We would like to thank Mr. Martfn-Neira, of ESA, for his Interferometric synthesis aperture microwave radiometry helpful discussions throughout this work. for remote sensing of the Earth, IEEE Trans. Geosci. Remote Sens., 16(5), , References Scientific Components, Minicircuits RF/IF Designer Handbook, Minicircuits Div., Brooklyn, N.Y., Barj, J., A. Camps, I. Corbella, and F. Torres, Bi- Tanner, A.B., and C.T. Swift, Calibration of a synthetic dimensional discrete formulation aperture synthesis aperture radiometer, IEEE Trans. Geosci. Remote Sens., radiometers, 9777/92/NL/PB, final report, Eur. Space 31(1), , Agency, ESTEC, Noordwijk, Netherlands, Thompson, A.R., J. Moran, and G. Swenson, Camps, A., J. Bari, I. Corbella, and F. Torres, Visibility Interferometry and Synthesis in Radio Astronomy, John inversion algorithms over hexagonal samplingrids, in Wiley, New York, Proceedings of the Soil Moisture and Ocean Salinity Thompson, A.R., and L.R. D'Addario, Frequency response Measurements and Radiometer Techniques Consultative of a synthesis array: Performance limitations and design Meeting, pp , Eur. Space Agency, ESTEC, tolerances, Radio Sci., 17, , Noordwijk, Netherlands, 1995a. Torres, F., A. Camps, J. Bar, I. Corbella, and R. Ferrero,

13 TORRES ET AL.: RECEIVER ERRORS IN RADIOMETER INTERFEROMETERS 641 On-board phase and modulus calibration of large Universitat Polit cnica de Catalunya, Barcelona, aperture synthesis radiometers. Study applied to MIRAS, IEEE Trans. Geosci. Remote Sens., 34(4), , J. Bara, A. Camps, I. Corbella, and F. Torres, (Received June 14, 1996; revised September 19, 1996; Department of Signal Theory and Communications, accepted September 30, 1996.)