Phase statistics of interferograms with applications ... - Semantic Scholar
Phase statistics of interferograms with applications to synthetic aperture radar Dieter Just and Richard Bamler
Interferometric methods are well established in optics and radio astronomy. In recent years, interferometric concepts have been applied successfully to synthetic aperture radar (SAR) and have opened up new
possibilities in the area of earth remote sensing. However interferometric SAR applications require thorough phase control through the imaging process. The phase accuracy of SAR images is affected by decorrelation effects between the individual surveys. We analyze quantitatively the influence of decorrelation on the phase statistics of SAR interferograms. In particular, phase aberrations as they occur in typical SAR processors are studied in detail. The dependence of the resulting phase bias and variance on processor parameters is presented in several diagrams. Key words: Synthetic aperture radar, interferometry, phase statistics, decorrelation, aberrations.
1.
Introduction
Interferometry has a long history in optics with applications ranging from nondestructive testing to stellar imaging. These techniques have been adopted in the microwave regime by radioastronomers. During recent years strong interest in advanced methods of earth remote sensing has stimulated various investigations on interferometric synthetic aperture radar (SAR).14 This development has been promoted by the availability of a wealth of high-quality SAR data acquired by modern spaceborne sensors. SAR is a two-step imaging process that requires coherent radar echo acquisition and coherent processing of raw data. The intended results are highresolution complex images that carry not only intensity (pictorial) information but also exhibit a phase structure that can be used for interferometry. Two or more complex images of the same ground area, acquired at different times and/or from different (nominally parallel) orbits, can be used for various purposes, e.g. interferograms, samples of the scene coherence function,5 and superresolved images.6 Interferograms are usually further processed to yield terrain height models of the imaged areas. For these applications of the SAR image is a delicate The authors are with the Remote Sensing Data Center, German Aerospace Research Establishment,
82230 Oberpfaffenhofen,
Ger-
many. Received 2 February November 1993.
1993; revised manuscript
0003-6935/94/204361-08$06.00/0. o 1994 Optical Society of America.
received 10
feature and should be thoroughly controlled through the imaging process. This is the background for recent discussions on phase-preserving SAR processing algorithms.7 The achievable phase accuracy of SAR images is affected by decorrelation caused by temporal scene decorrelation (e.g., changes of water surface or vegetation), radar receiver noise, phase aberrations introduced during data acquisition or processing, and spectral misalignment of transfer functions caused by different aspect angles. We investigate the phase statistics of interferograms suffering from the latter three effects. We restrict ourselves to Rayleigh scattering scenes of homogeneous (spatially constant) backscatter coefficients whose reflectivity can be modeled as a complex, circular, stationary Gaussian process. The organization of the paper is as follows: in Section 2 we describe data acquisition and processing by use of linear transfer functions that include possible aberration effects; thermal noise is added for each of the observations taken. The interferogram is defined as the complex conjugate product of two complex images. In Section 3 we discuss the probability density function of the interferometric phase for
arbitrary transfer functions. It turns out that the phase statistics are completely determined by the complex correlation coefficient y, which is a function of the transfer functions and signal-to-noise ratios 10 July 1994 / Vol. 33, No. 20 / APPLIED OPTICS
4361
(SNR's) of the interferometric channels. We discuss the effect of specific SAR transfer functions on the achievable phase estimate accuracy in Section 4. 2.
System Model
In this paper we adopt the common two-step linear filter model to describe the SAR imaging system [see Fig. 1(a)]: transfer function Hi, represents data acquisition, H12 models the processing system. The input signal process x represents the scene reflectivity. Thermal noise n is added at the raw data stage. Both the signal and the noise are modeled by complex, stationary, white, circular Gaussian processes with power spectral densities of 2o2 and 2U,2, respectively. The real and imaginary parts of x and n are denoted by xr, xi, n, and n. They are assumed to be statistically independent, zero-mean random processes. The autocorrelation functions of x and n are then given by R_,r) = 20r28(T)
(1)
Rnn(T) = 2o' 2 (Qr).
(2)
For simplicity we restrict ourselves to one-dimensional signals and transfer functions throughout Sections 2 and 3. Generalization to multidimensional signals is straightforward. Without loss of generality we can combine the two transfer functions into a single end-to-end filter function H1 and add thermal noise after signal x has passed the filter [see Fig. 1(b)]. In this case we must replace the white-noise process n by ni with Rnln,(T) = 2cr12
f
IH 2 (f)
2
exp(-2rjfT)df.
image
rawdata
scene reflectivity H11(f)
(3)
Now we consider an interferometric system with a general layout as shown in Fig. 1(c). The transfer functions H, and H2 that describe the two imaging processes are in general different because of the different orbits and of possible differential processor aberrations. The output signals y, and Y2 of the two filters form a complex stationary joint Gaussian process.
8
We also allow for different thermal noise processes n, and n2 in the two surveys with autocorrelation functions according to Eq. (3), where for n 2 the transfer function H1 2 is replaced by H2 2 and or1 by or2. Clearly, n and n 2 are statistically independent of each other and ofy, andy 2. The interferometric phase is composed of two parts: a geometric-induced phase that bears useful information and a disturbance (bias, variance). In this paper we are interested only in a quantitive estimate of the disturbance. Hence we assume that the geometric part of the interferometric phase has been removed by an appropriate factor. Therefore in the model of Fig. 1(c) the white input process x is the same for both channels. Note an interesting difference between SAR interferometry on extended scatterers in the Rayleigh regime as described in this paper and certain other applications in classical, optical interferometry or holography9 : in the first case, the signal itself exhibits speckle and is appropriately modeled by a random process. In addition, there is random thermal noise. The latter case deals with the sum of a constant, coherent background, which is regarded as the signal, and a speckle pattern.' 0 Both, cases have distinctly different phase statistics. For SAR the statistics of the latter case is applicable only for deterministic, pointlike scatterers.
+
3.
n:= n + i
Interferometric
Phase Statistics
It is not difficult to show that the joint probability density function (pdf), pdf(zi, Z 2 ), of processes z, = y, + n1 andz 2 =Y2 + n 2 is again Gaussian," i.e.,
(a) Xi)
pdf(Z1, Z2 )
=
pdf(zl,,
Zli, Z2r, Z2 i)
(b)
(2 ,r)2
n,
W I 1, 2
_2
T
ll
where W is the covariance matrix of processes Zr,
Z1 KIZ2
4 Zli
and IWI, W-1are the determinant and the inverse of respectively. Vector u is defined as 2r, Z2i
',
[Zir] (c) Fig. 1.
I
n1
System models of (a) the two-step SAR imaging process, (b)
an equivalent description of the same process, (c)the formation of a SAR interferogram.
4362
APPLIED OPTICS / Vol. 33, No. 20 / 10 July 1994
(5) 1Z2r1 LZ2iJ
The elements of
-'
are given by
with y = p/q
=r2 (q - lp 12 )
(17)
-o = arg{y} = argtp}.
-Retp} ql
q2
-Re{pJ
-Im{pl
0
-ImIp}
Im~p}
0
It is easy to show that y is identical to the complex correlation coefficient of Z1, z2 :
-Re{p} '
Im{p}
q2
0
-Rejpj
E{zz 2 *1 = (Etl 1{2} E 1Z2 121)1/2'
(18)
(6)
and IWI is given by
(7)
12 )2
WI=ul(q -
Ip
Hj(f)H
(f)*df,
with p=
2
(8) (9)
q = qlq2,
ql = (1 + SNR,-')
1
IH,(f) 2df,
(10)
where E{. . denotes the expectation value. The pdf of Eq. (15) has been derived before in the more general context of second-order speckle statistics.13 Also, the statistics of the copolarized phase difference in polarimetric measurements 4 implies the same pdf as that presented here. and has to be Note that pdf(+) is periodic with 2wT referenced to a base interval of the phase. An elegant choice is the interval
(o -
r, Po + r) since
pdf(4) is symmetric around its maximum at +O. In this case the phase variance becomes independent of 4+O. This choice applies to the phase of a properly unwrapped interferogram. Then we obtain (assuming ergodicity) E{+} = h~o= argfy}
(19)
I+
71 2
1 HUM(
12df
SNR,-' =
(11) Uj2
0` = E((
IH,(f) 2 df
_
-
(4>)2} =
-
SNR 2 -1
SNR2 -') 1
1H2 (f)I 2 df,
(12)
(f) 12 df
22 f
IH
LT2
IH2 (f) 2 df
22
=
Now we consider the interferogram phase process (14)
Its probability density function pfd(+) is obtained from Eq. (4).12 The result is yI 2
1-
2w
pdf(4) =
1- l
1 1yl2 cos 2 ( -
o)
ii
X
+
2
ko)2 pdf(k)d-
+ (0 )d4.
(20)
1 1.
The pdf for phase + in the case of a constant signal plus noise mentioned at the end of Section 2 is markedly different' 0 from Eq. (15) and the resulting phase standard deviation is given by [(1 - y2)/(2y2)]1/2 for values of 1yI close to one.15 This formula leads to
much smaller values for the phase standard deviation as when determined by Eq. (20). For example, a correlation coefficient 1y I = 0.99 results in a phase
1yIcos(4 - 4O)arccos[- I y Icos( 2
-
It follows from Eq. (15) that pdf(4 + +O)is independent of +O. The latter integral has been numerically evaluated for the examples below. Figure 2 shows pdf(+) versus + for l0 = 0 and four values of | y1. The probability density function pdf(+) is uniform, pdf(+) = 1/2,r, for completely decorrelated signals z1 and Z2 ( I = 0), and approaches a delta function for complete correlation ( I = 1). Figure 3 shows the phase standard deviation ug, versus
( = arg{z1 2*'}
(k
no~~~~~-v
2 pdf(+
q2 = (1 +
f
1
[1 - I I cos (+ - ()o)] /2
-
+0)]
noise of ry,= 15.04° when we apply Eq. (20) but only 5.77° for the constant signal and noise case. For extended scatterers in the Rayleigh regime the phase noise should be determined according to Eq. (20),
10 July 1994 / Vol. 33, No. 20 / APPLIED OPTICS
4363
pdf(Q) 1.5
in turn can be expressed by functionals of the transfer functions H1 , H2 and does not depends on the SNR. The variance U+2 of + depends only on the magnitude of the complex correlation coefficient and can be determined for any given H1, H2 , and SNR.
lYI= 0.95
4.
In this section we evaluate quantitatively the effects of specifictransfer functions and thermal noise on the interferogram statistics. In particular we discuss the pure noise case, phase aberrations that are typical of SAR, and the effect of spectral misalignment on the transfer functions. As was shown in Section 3, it is sufficient to determine y in magnitude and phase. For illustration and ease of interpretation, plots are given for phase bias and standard deviation.
1.o
0.5
iy
Quantitative Examples
= 0.75
A. Thermal Noise
iY =0.5 I =0.0
For the case of identical transfer functions H, = H2 but finite SNR's we have
n
p =
Fig. 2. The pdf() of the interferometric phase for +o = 0 and different values of the magnitude of the correlation coefficient 1,y1. The pdf is constant for 1 I = 0 and converges toward a delta function as 1 approaches one.
whereas the high coherence approximation is adequate only for deterministic, pointlike scatterers. We summarize our results as follows: The mean value +o of the interferometric phase is equal to the phase of the complex correlation coefficient y, which
+ IH(f)
2df,
(21)
and, hence, 1
= [(1+ SNR 1-')(1 + 'Y= 1h'I
SNR 2
2 '1)]1/
(22)
As expected, thermal noise does not introduce a phase bias (y is real). If the SNR's of both surveys are identical, y reduces to 1
(23)
= 1 + SNR-1'
which is a well-known expression.'5 "16 Figure 4 shows the standard deviation of + as a function of SNR. B.
Phase Aberrations
So far we have not made any assumption about the transfer functions H1 , H2 . Now we focus on the effects of particular two-dimensional SAR processor aberrations. We assume that the spectral envelopes of H1 and H2 are identical; however, their phases are different. Without loss of generality we can model H1 as real valued and introduce the phase difference, which we call differential phase aberration, into H2 . For the examples given in this subsection H1 and H2 are two-dimensional low-pass filters:
H,(tt, v) = rect( )rectt v)I H 2 (p,, v) = H 1(ji, v)exp{j*(pi,
Fig. 3. Standard deviation Taof the interferometric phase versus magnitude of the correlation Zoefficient y 1. The standard deviation has a maximum value of approximately 104° for completely uncorrelated signals (I y I = 0). 4364
APPLIED OPTICS / Vol. 33, No. 20 / 10 July 1994
(24) v)},
(25)
where p. is the azimuth frequency, v is the range frequency, B. is the azimuth bandwidth, B, is the range bandwidth, and i(p., v) represents the differential phase aberration between the two imaging processes. 1 7 We assume that the aberrations are small
2.
qap or
WiHO, Relative Geometric Misregistration in
Azimuth or Range A possible error in coregistration of the two complex images can be modeled as a differential linear phase aberration. Without loss of generality, we consider only azimuth misregistration. If a denotes the relative shift between the two images in fractions of a resolution cell 1/B., i.e., 1o = ot/Bp, then we obtain
600
*(p, v) = 27+1op = 2,rrot R
(28)
p = BvB,, sinc(a).
(29)
400,
Hence k = 0,
0
s
10
15 SNR20dB
Fig. 4. Standard deviation qWof the interferometric phase versus SNR. Only the influence of the thermal receiver noise has been considered.
enough that we can expand qj(ji,v) into a Taylor series: *1(p.,v) = 2r{4o
+ 110 p +
+ '20p2 +
Therefore misregistration does not introduce a phase bias phase but does introduce phase variance. Figure 5 shows the phase standard deviation versus shift at. For example, a residual misregistration of 1/8 resolution cell leads to standard deviations of approximately 230 and 42° for SNR's of dB and 10 dB, respectively. 3. i20or q102,Defocusing in Azimuth and Range Again we consider only the azimuth case. Let IVbe the phase error at the edge of bandwidth p. = +BL/2, i.e., '1)20 = 2/(rrB 2)P. Then we have
olv + Aillav
2V2 + '121 p.2v + . . ..
41(p,, v) = 2r41 2 4L2 = 42/Bp2.
(31)
(26)
The coefficients represent the following processor aberrations:
aI
t 10ois the constant phase error, q1jois the geometric misregistration in azimuth, 410l is the geometric misregistration in range, '120 is the defocusing in azimuth (wrong FM rate), '102 is the defocusing in range (wrong chirp rate or secondary range compression coefficient), q1,,is the uncompensated linear range migration, and 4121 is the uncompensated quadratic range migration.
SNR = 10
In the following we evaluate Iy j, phase bias, and variance for each phase aberration individually. We assume that the signal energies are properly normalized (q, = q2 = 1). For the functional plots two different SNR's (10 dB, o dB) are considered.
(30)
AyI = sinc(ao).
SNR = -
A discus-
sion of SAR processor aberrations can be found in the literature.' 8
1. i/g,0 ConstantPhaseError
It is obvious that 40o simply describes a constant phase factor that adds to any interferometric phase but does not introduce a phase variance. Hence, To = 2wT00, and IyI = 1.
(27)
0
0.2
0.4
0.6
0.8
a
1
Fig. 5. Standard deviation a(1, of the interferometric phase versus misregistration acfor two different SNR's. The case of a = 1 is tantamount to a misregistration of one resolution cell and results in complete decorrelation. 10 July 1994 / Vol. 33, No. 20 / APPLIED OPTICS
4365
at
From Eq. (31) we obtain
p
BV7J
exp j
2
d[L
SNR = 10
1P/ [W( I) +jY(221/)]
(32)
integrals. theFresnel where 9(.. .)and (.. .)denote The phase bias is then
SNR
40 = arctan
=
(33)
9'U2P/ r)
200.
and the magnitude of the correlation coefficient is
I=
1
1( [
2 (2p/)
+
y2(j2P/r)]1/
2
(34)
Unlike the other aberrations, defocusing is rE,presented by an even phase aberration function and introduces a phase bias. Figures 6 and 7 sho F go and the phase standard deviation as a functio n of
phase error T at the edgeof the bandwidth. 4.
il, UncompensatedLinear RangeMigratior
0
0.ln
0.2c
v
(35)
I 0.4x
Y/
0.5n
Fig. 7. Standard deviation oa,of the interferometric phase versus defocusingerror. The standaid deviation is shown for two different SNR's.
We consider the residual range walk' 9 in fractions IP of a range resolution cell 1/Bv, i.e, 11 = 1/(BIpBv). Hence we obtain +(1x,v) = 2'rr1,jjtv = 2p.
0.3n
rp Si( /2),
p =
(36)
where Si(... .) is the integral sine. We conclude that 2
2 o
=
P/
sin(p)
J
1z' = A
p.
Si(iTr,/2) =
dp
(37)
Figure 8 shows the phase standard deviation for uncorrected linear range migration P between 0 and 1 resolution cells.
5. qi21,UncompensatedQuadraticRangeMigration The residual range curvature' 9 is expessed in fractions E of a range resolution element 1/By, i.e., 121 =
4e/B 2 B,. Hence
*(p, v) = 2*
p
2 21 p v
=
4
1
P2v B 2B
V;.1 sinc([L2 )dl..
=
(38)
(39)
Therefore we have no phase bias and I yIis given by
isc 1
0.1x
0.2x
0.3n
0.4xs
V
.Sn
Fig. 6. Phase bias +o of the interferometric phase versus defocusing error. The defocusing error is expressed as the maximum phase error P at the edge of the bandwidth. 4odoes not depend on the SNR. 4366
APPLIED OPTICS / Vol. 33, No. 20 / 10 July 1994
~~~~~1 = ;472
)d
sine( p2 )dji.
(40)
Figure 9 shows the phase standard deviation for uncompensated quadratic range migration between 0 and 0.5 resolution cells.
For the rectangular transfer function of Eq. (24) we have SNR = 10
-
17I =
1POI BL
1 -
1VOI , B,,
to= 0.
(42)
IThe physical reason for spectral misalignment is the
different aspect angles of a scene element in the two surveys. Different squint angles result in different Doppler centroid (DC) frequencies, fDc, and fDC,2, of the two data sets. If p.represents Doppler (azimuth) frequencies in hertz, then
SNR = o
0
0.8
0.6
0.4
0.2
#
I
Fig. 8. Standard deviation q 4 of the interferometric phase versus uncompensated linear range migration for two SNR's. The linear range migration is expressed in fractions p of a range resolution cell.
C.
For cross-track interferometry the DC difference po can be made to approach zero by controlling the antenna squint to be the same for both surveys. Spectral misalignment in the range frequency dimension, however, is an inherent feature of crosstrack interferometry. 6 Let us assume that an area of interest on the ground is seen at incident angles of 0, and 02 in the two surveys. Then a certain harmonic ground structure transforms into different echo frequency components, depending on the incident angles. This effect can be interpreted conversely as a relative shift of H, I and H2 1in the v direction by the amount of
Spectral Envelope Misalignment
Even in noise-free and aberration-free cases, signal decorrelation will occur if the envelopes of H, and H2 are shifted relative to each other, i.e., IH2 (p., v) I
=
IHi(p
(41)
p.o,v - vo)I*
-
(43)
fDC,l-
[LO= fDC,2-
1
11
sin 0,](
V0 rac ars n O sin 02
(44) 4
where 0 = (0 + 02)/2 and fradar is the carrier frequency of the radar. For horizontal baselines of length A that are much smaller than range R, relation (44) assumes the following form: cos2 v0 -fracar sin
0A
R
(45)
In fact, v0 is the local frequency of the interferometric fringe pattern. Decorrelation caused by spectral misalignment can be avoided if the two data sets are properly bandpass filtered in order to retain only those spectral compo-
nents that pass through both H, and H2 . 5. Conclusions
0.1
0.2
0.3
0.5
0.4
E
Fig. 9. Standard deviation or of the interferometric phase versus uncompensated quadratic range migration for two SNR's. The quadratic range migration is expressed in fractions e of a range resolution cell.
We have developed a system theoretical approach to determine the phase statistics of interferograms of distributed scatterers. Our approach is quite general and can be applied to all systems that are characterized by linear transfer functions. We have applied this technique to interferometric SAR. In particular, we were interested in the effects of SAR processor aberrations on phase biases and phase variations. We evaluated the first two phase moments for typical processing aberrations, e.g., geometric misregistration, defocusing, and uncompensated range migration under the influence of thermal noise. Only one of these aberrations, defocusing, leads to a phase bias, but all of them introduce phase variance, 10 July 1994 / Vol. 33, No. 20 / APPLIED OPTICS
4367
as has been shown in several figures. More generally, only even phase aberration functions introduce phase biases. The phase variance limits the phase accuracy that can be obtained, so that an evaluation of processor designs for interferometric SAR applications is an implicit result of our study. References 1. L. C. Graham,
"Synthetic
interferometer
radar
for topo-
graphic mapping," Proc. IEEE 62, 763-768 (1974). 2. H. A. Zebker and R. M. Goldstein, "Topographic mapping from interferometric synthetic aperture radar observations," J. Geophys. Res. 91, 4993-4999 (1986). 3. C. Prati, F. Rocca, A. M. Guarnieri, and E. Damonti, "Seismic
migration for SAR focusing: interferometrical applications," IEEE Trans. Geosci.Remote Sensing 28, 627-639 (1990). 4. F. K. Li and R. M. Goldstein,
"Studies
of multibaseline
spaceborne interferometric synthetic aperture radars," IEEE Trans. Geosci. Remote Sensing 28, 88-96 (1990). 5. C. Prati and F. Rocca, "Limits to the resolution of elevation maps from stereo SAR images," Int. J. Remote Sensing 11, 2215-2235
(1990).
6. C. Prati and F. Rocca, "Range resolution enhancement with multiple SAR surveys combination," presented at the International Geoscience and Remote Sensing Symposium, Houston, Tex., 1992).
7. K.Raney and P. Vachon, "A phase preserving SAR processor," Proc. Inst. Electr. Eng. Part F 139, 147 (1992). 8. A. Papoulis, Probability, Random Variables, and Stochastic
4368
APPLIED OPTICS / Vol. 33, No. 20 / 10 July 1994
Processes, 1st ed. (McGraw-Hill, New York, 1965), Chap. 14, p. 475.
9. J. W. Goodman, "Film-grain noise in wavefront-reconstruction imaging," J. Opt. Soc.Am. 57, 493-502 (1967). 10. J. W. Goodman, "Statistical properties of laser speckle patterns," inLaserSpeckle andRelated Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 29-35. 11. Ref. 8, Chap. 7, p. 222. 12. W. B. Davenport, Jr., and W. L. Root, An Introduction
to the
Theory of Random Signals and Noise, 1st ed. (Institute of Electrical and Electronics Engineers, New York, 1987), Chap. 8, pp. 158-165. 13. Ref. 10, pp. 42-46.
14. K Saribandi, "Derivaton of phase statistics from the Mueller matrix," Radio Sci.27, 553-560 (1992). 15. E. Rodriguez and J. M. Martin, "Theory and design of interferometric synthetic aperture radars," Proc. Inst. Electr. Eng. Part F 139, 147-159 (1992).
16. H. A.Zebker and J. Villasenor, "Decorrelation in interferometric radar echoes," IEEE Trans. Geosci. Remote Sensing 30, 950-959(1992). 17. M. Born and E. Wolf,Principles of Optics, 6th ed. (Pergamon, London, 1980), Chap. 5, pp. 203-207.
18. R. Bamler, "A comparison of range-Doppler and wavenumber domain SAR focusing algorithms," IEEE Trans. Geosci. Remote Sensing 30, 706-713 (1992). 19. C. Wu, K. Y. Liu and M. Jin, "Modeling and correlation
algorithm for spaceborne SAR signals," IEEE Trans. Aerosp. Electron. Syst. AES-18, 563-574 (1982).
Interferometric methods are well established in optics and radio astronomy. In recent years, interferometric concepts have been applied successfully to synthetic aperture radar (SAR) and have opened up new
possibilities in the area of earth remote sensing. However interferometric SAR applications require thorough phase control through the imaging process. The phase accuracy of SAR images is affected by decorrelation effects between the individual surveys. We analyze quantitatively the influence of decorrelation on the phase statistics of SAR interferograms. In particular, phase aberrations as they occur in typical SAR processors are studied in detail. The dependence of the resulting phase bias and variance on processor parameters is presented in several diagrams. Key words: Synthetic aperture radar, interferometry, phase statistics, decorrelation, aberrations.
1.
Introduction
Interferometry has a long history in optics with applications ranging from nondestructive testing to stellar imaging. These techniques have been adopted in the microwave regime by radioastronomers. During recent years strong interest in advanced methods of earth remote sensing has stimulated various investigations on interferometric synthetic aperture radar (SAR).14 This development has been promoted by the availability of a wealth of high-quality SAR data acquired by modern spaceborne sensors. SAR is a two-step imaging process that requires coherent radar echo acquisition and coherent processing of raw data. The intended results are highresolution complex images that carry not only intensity (pictorial) information but also exhibit a phase structure that can be used for interferometry. Two or more complex images of the same ground area, acquired at different times and/or from different (nominally parallel) orbits, can be used for various purposes, e.g. interferograms, samples of the scene coherence function,5 and superresolved images.6 Interferograms are usually further processed to yield terrain height models of the imaged areas. For these applications of the SAR image is a delicate The authors are with the Remote Sensing Data Center, German Aerospace Research Establishment,
82230 Oberpfaffenhofen,
Ger-
many. Received 2 February November 1993.
1993; revised manuscript
0003-6935/94/204361-08$06.00/0. o 1994 Optical Society of America.
received 10
feature and should be thoroughly controlled through the imaging process. This is the background for recent discussions on phase-preserving SAR processing algorithms.7 The achievable phase accuracy of SAR images is affected by decorrelation caused by temporal scene decorrelation (e.g., changes of water surface or vegetation), radar receiver noise, phase aberrations introduced during data acquisition or processing, and spectral misalignment of transfer functions caused by different aspect angles. We investigate the phase statistics of interferograms suffering from the latter three effects. We restrict ourselves to Rayleigh scattering scenes of homogeneous (spatially constant) backscatter coefficients whose reflectivity can be modeled as a complex, circular, stationary Gaussian process. The organization of the paper is as follows: in Section 2 we describe data acquisition and processing by use of linear transfer functions that include possible aberration effects; thermal noise is added for each of the observations taken. The interferogram is defined as the complex conjugate product of two complex images. In Section 3 we discuss the probability density function of the interferometric phase for
arbitrary transfer functions. It turns out that the phase statistics are completely determined by the complex correlation coefficient y, which is a function of the transfer functions and signal-to-noise ratios 10 July 1994 / Vol. 33, No. 20 / APPLIED OPTICS
4361
(SNR's) of the interferometric channels. We discuss the effect of specific SAR transfer functions on the achievable phase estimate accuracy in Section 4. 2.
System Model
In this paper we adopt the common two-step linear filter model to describe the SAR imaging system [see Fig. 1(a)]: transfer function Hi, represents data acquisition, H12 models the processing system. The input signal process x represents the scene reflectivity. Thermal noise n is added at the raw data stage. Both the signal and the noise are modeled by complex, stationary, white, circular Gaussian processes with power spectral densities of 2o2 and 2U,2, respectively. The real and imaginary parts of x and n are denoted by xr, xi, n, and n. They are assumed to be statistically independent, zero-mean random processes. The autocorrelation functions of x and n are then given by R_,r) = 20r28(T)
(1)
Rnn(T) = 2o' 2 (Qr).
(2)
For simplicity we restrict ourselves to one-dimensional signals and transfer functions throughout Sections 2 and 3. Generalization to multidimensional signals is straightforward. Without loss of generality we can combine the two transfer functions into a single end-to-end filter function H1 and add thermal noise after signal x has passed the filter [see Fig. 1(b)]. In this case we must replace the white-noise process n by ni with Rnln,(T) = 2cr12
f
IH 2 (f)
2
exp(-2rjfT)df.
image
rawdata
scene reflectivity H11(f)
(3)
Now we consider an interferometric system with a general layout as shown in Fig. 1(c). The transfer functions H, and H2 that describe the two imaging processes are in general different because of the different orbits and of possible differential processor aberrations. The output signals y, and Y2 of the two filters form a complex stationary joint Gaussian process.
8
We also allow for different thermal noise processes n, and n2 in the two surveys with autocorrelation functions according to Eq. (3), where for n 2 the transfer function H1 2 is replaced by H2 2 and or1 by or2. Clearly, n and n 2 are statistically independent of each other and ofy, andy 2. The interferometric phase is composed of two parts: a geometric-induced phase that bears useful information and a disturbance (bias, variance). In this paper we are interested only in a quantitive estimate of the disturbance. Hence we assume that the geometric part of the interferometric phase has been removed by an appropriate factor. Therefore in the model of Fig. 1(c) the white input process x is the same for both channels. Note an interesting difference between SAR interferometry on extended scatterers in the Rayleigh regime as described in this paper and certain other applications in classical, optical interferometry or holography9 : in the first case, the signal itself exhibits speckle and is appropriately modeled by a random process. In addition, there is random thermal noise. The latter case deals with the sum of a constant, coherent background, which is regarded as the signal, and a speckle pattern.' 0 Both, cases have distinctly different phase statistics. For SAR the statistics of the latter case is applicable only for deterministic, pointlike scatterers.
+
3.
n:= n + i
Interferometric
Phase Statistics
It is not difficult to show that the joint probability density function (pdf), pdf(zi, Z 2 ), of processes z, = y, + n1 andz 2 =Y2 + n 2 is again Gaussian," i.e.,
(a) Xi)
pdf(Z1, Z2 )
=
pdf(zl,,
Zli, Z2r, Z2 i)
(b)
(2 ,r)2
n,
W I 1, 2
_2
T
ll
where W is the covariance matrix of processes Zr,
Z1 KIZ2
4 Zli
and IWI, W-1are the determinant and the inverse of respectively. Vector u is defined as 2r, Z2i
',
[Zir] (c) Fig. 1.
I
n1
System models of (a) the two-step SAR imaging process, (b)
an equivalent description of the same process, (c)the formation of a SAR interferogram.
4362
APPLIED OPTICS / Vol. 33, No. 20 / 10 July 1994
(5) 1Z2r1 LZ2iJ
The elements of
-'
are given by
with y = p/q
=r2 (q - lp 12 )
(17)
-o = arg{y} = argtp}.
-Retp} ql
q2
-Re{pJ
-Im{pl
0
-ImIp}
Im~p}
0
It is easy to show that y is identical to the complex correlation coefficient of Z1, z2 :
-Re{p} '
Im{p}
q2
0
-Rejpj
E{zz 2 *1 = (Etl 1{2} E 1Z2 121)1/2'
(18)
(6)
and IWI is given by
(7)
12 )2
WI=ul(q -
Ip
Hj(f)H
(f)*df,
with p=
2
(8) (9)
q = qlq2,
ql = (1 + SNR,-')
1
IH,(f) 2df,
(10)
where E{. . denotes the expectation value. The pdf of Eq. (15) has been derived before in the more general context of second-order speckle statistics.13 Also, the statistics of the copolarized phase difference in polarimetric measurements 4 implies the same pdf as that presented here. and has to be Note that pdf(+) is periodic with 2wT referenced to a base interval of the phase. An elegant choice is the interval
(o -
r, Po + r) since
pdf(4) is symmetric around its maximum at +O. In this case the phase variance becomes independent of 4+O. This choice applies to the phase of a properly unwrapped interferogram. Then we obtain (assuming ergodicity) E{+} = h~o= argfy}
(19)
I+
71 2
1 HUM(
12df
SNR,-' =
(11) Uj2
0` = E((
IH,(f) 2 df
_
-
(4>)2} =
-
SNR 2 -1
SNR2 -') 1
1H2 (f)I 2 df,
(12)
(f) 12 df
22 f
IH
LT2
IH2 (f) 2 df
22
=
Now we consider the interferogram phase process (14)
Its probability density function pfd(+) is obtained from Eq. (4).12 The result is yI 2
1-
2w
pdf(4) =
1- l
1 1yl2 cos 2 ( -
o)
ii
X
+
2
ko)2 pdf(k)d-
+ (0 )d4.
(20)
1 1.
The pdf for phase + in the case of a constant signal plus noise mentioned at the end of Section 2 is markedly different' 0 from Eq. (15) and the resulting phase standard deviation is given by [(1 - y2)/(2y2)]1/2 for values of 1yI close to one.15 This formula leads to
much smaller values for the phase standard deviation as when determined by Eq. (20). For example, a correlation coefficient 1y I = 0.99 results in a phase
1yIcos(4 - 4O)arccos[- I y Icos( 2
-
It follows from Eq. (15) that pdf(4 + +O)is independent of +O. The latter integral has been numerically evaluated for the examples below. Figure 2 shows pdf(+) versus + for l0 = 0 and four values of | y1. The probability density function pdf(+) is uniform, pdf(+) = 1/2,r, for completely decorrelated signals z1 and Z2 ( I = 0), and approaches a delta function for complete correlation ( I = 1). Figure 3 shows the phase standard deviation ug, versus
( = arg{z1 2*'}
(k
no~~~~~-v
2 pdf(+
q2 = (1 +
f
1
[1 - I I cos (+ - ()o)] /2
-
+0)]
noise of ry,= 15.04° when we apply Eq. (20) but only 5.77° for the constant signal and noise case. For extended scatterers in the Rayleigh regime the phase noise should be determined according to Eq. (20),
10 July 1994 / Vol. 33, No. 20 / APPLIED OPTICS
4363
pdf(Q) 1.5
in turn can be expressed by functionals of the transfer functions H1 , H2 and does not depends on the SNR. The variance U+2 of + depends only on the magnitude of the complex correlation coefficient and can be determined for any given H1, H2 , and SNR.
lYI= 0.95
4.
In this section we evaluate quantitatively the effects of specifictransfer functions and thermal noise on the interferogram statistics. In particular we discuss the pure noise case, phase aberrations that are typical of SAR, and the effect of spectral misalignment on the transfer functions. As was shown in Section 3, it is sufficient to determine y in magnitude and phase. For illustration and ease of interpretation, plots are given for phase bias and standard deviation.
1.o
0.5
iy
Quantitative Examples
= 0.75
A. Thermal Noise
iY =0.5 I =0.0
For the case of identical transfer functions H, = H2 but finite SNR's we have
n
p =
Fig. 2. The pdf() of the interferometric phase for +o = 0 and different values of the magnitude of the correlation coefficient 1,y1. The pdf is constant for 1 I = 0 and converges toward a delta function as 1 approaches one.
whereas the high coherence approximation is adequate only for deterministic, pointlike scatterers. We summarize our results as follows: The mean value +o of the interferometric phase is equal to the phase of the complex correlation coefficient y, which
+ IH(f)
2df,
(21)
and, hence, 1
= [(1+ SNR 1-')(1 + 'Y= 1h'I
SNR 2
2 '1)]1/
(22)
As expected, thermal noise does not introduce a phase bias (y is real). If the SNR's of both surveys are identical, y reduces to 1
(23)
= 1 + SNR-1'
which is a well-known expression.'5 "16 Figure 4 shows the standard deviation of + as a function of SNR. B.
Phase Aberrations
So far we have not made any assumption about the transfer functions H1 , H2 . Now we focus on the effects of particular two-dimensional SAR processor aberrations. We assume that the spectral envelopes of H1 and H2 are identical; however, their phases are different. Without loss of generality we can model H1 as real valued and introduce the phase difference, which we call differential phase aberration, into H2 . For the examples given in this subsection H1 and H2 are two-dimensional low-pass filters:
H,(tt, v) = rect( )rectt v)I H 2 (p,, v) = H 1(ji, v)exp{j*(pi,
Fig. 3. Standard deviation Taof the interferometric phase versus magnitude of the correlation Zoefficient y 1. The standard deviation has a maximum value of approximately 104° for completely uncorrelated signals (I y I = 0). 4364
APPLIED OPTICS / Vol. 33, No. 20 / 10 July 1994
(24) v)},
(25)
where p. is the azimuth frequency, v is the range frequency, B. is the azimuth bandwidth, B, is the range bandwidth, and i(p., v) represents the differential phase aberration between the two imaging processes. 1 7 We assume that the aberrations are small
2.
qap or
WiHO, Relative Geometric Misregistration in
Azimuth or Range A possible error in coregistration of the two complex images can be modeled as a differential linear phase aberration. Without loss of generality, we consider only azimuth misregistration. If a denotes the relative shift between the two images in fractions of a resolution cell 1/B., i.e., 1o = ot/Bp, then we obtain
600
*(p, v) = 27+1op = 2,rrot R
(28)
p = BvB,, sinc(a).
(29)
400,
Hence k = 0,
0
s
10
15 SNR20dB
Fig. 4. Standard deviation qWof the interferometric phase versus SNR. Only the influence of the thermal receiver noise has been considered.
enough that we can expand qj(ji,v) into a Taylor series: *1(p.,v) = 2r{4o
+ 110 p +
+ '20p2 +
Therefore misregistration does not introduce a phase bias phase but does introduce phase variance. Figure 5 shows the phase standard deviation versus shift at. For example, a residual misregistration of 1/8 resolution cell leads to standard deviations of approximately 230 and 42° for SNR's of dB and 10 dB, respectively. 3. i20or q102,Defocusing in Azimuth and Range Again we consider only the azimuth case. Let IVbe the phase error at the edge of bandwidth p. = +BL/2, i.e., '1)20 = 2/(rrB 2)P. Then we have
olv + Aillav
2V2 + '121 p.2v + . . ..
41(p,, v) = 2r41 2 4L2 = 42/Bp2.
(31)
(26)
The coefficients represent the following processor aberrations:
aI
t 10ois the constant phase error, q1jois the geometric misregistration in azimuth, 410l is the geometric misregistration in range, '120 is the defocusing in azimuth (wrong FM rate), '102 is the defocusing in range (wrong chirp rate or secondary range compression coefficient), q1,,is the uncompensated linear range migration, and 4121 is the uncompensated quadratic range migration.
SNR = 10
In the following we evaluate Iy j, phase bias, and variance for each phase aberration individually. We assume that the signal energies are properly normalized (q, = q2 = 1). For the functional plots two different SNR's (10 dB, o dB) are considered.
(30)
AyI = sinc(ao).
SNR = -
A discus-
sion of SAR processor aberrations can be found in the literature.' 8
1. i/g,0 ConstantPhaseError
It is obvious that 40o simply describes a constant phase factor that adds to any interferometric phase but does not introduce a phase variance. Hence, To = 2wT00, and IyI = 1.
(27)
0
0.2
0.4
0.6
0.8
a
1
Fig. 5. Standard deviation a(1, of the interferometric phase versus misregistration acfor two different SNR's. The case of a = 1 is tantamount to a misregistration of one resolution cell and results in complete decorrelation. 10 July 1994 / Vol. 33, No. 20 / APPLIED OPTICS
4365
at
From Eq. (31) we obtain
p
BV7J
exp j
2
d[L
SNR = 10
1P/ [W( I) +jY(221/)]
(32)
integrals. theFresnel where 9(.. .)and (.. .)denote The phase bias is then
SNR
40 = arctan
=
(33)
9'U2P/ r)
200.
and the magnitude of the correlation coefficient is
I=
1
1( [
2 (2p/)
+
y2(j2P/r)]1/
2
(34)
Unlike the other aberrations, defocusing is rE,presented by an even phase aberration function and introduces a phase bias. Figures 6 and 7 sho F go and the phase standard deviation as a functio n of
phase error T at the edgeof the bandwidth. 4.
il, UncompensatedLinear RangeMigratior
0
0.ln
0.2c
v
(35)
I 0.4x
Y/
0.5n
Fig. 7. Standard deviation oa,of the interferometric phase versus defocusingerror. The standaid deviation is shown for two different SNR's.
We consider the residual range walk' 9 in fractions IP of a range resolution cell 1/Bv, i.e, 11 = 1/(BIpBv). Hence we obtain +(1x,v) = 2'rr1,jjtv = 2p.
0.3n
rp Si( /2),
p =
(36)
where Si(... .) is the integral sine. We conclude that 2
2 o
=
P/
sin(p)
J
1z' = A
p.
Si(iTr,/2) =
dp
(37)
Figure 8 shows the phase standard deviation for uncorrected linear range migration P between 0 and 1 resolution cells.
5. qi21,UncompensatedQuadraticRangeMigration The residual range curvature' 9 is expessed in fractions E of a range resolution element 1/By, i.e., 121 =
4e/B 2 B,. Hence
*(p, v) = 2*
p
2 21 p v
=
4
1
P2v B 2B
V;.1 sinc([L2 )dl..
=
(38)
(39)
Therefore we have no phase bias and I yIis given by
isc 1
0.1x
0.2x
0.3n
0.4xs
V
.Sn
Fig. 6. Phase bias +o of the interferometric phase versus defocusing error. The defocusing error is expressed as the maximum phase error P at the edge of the bandwidth. 4odoes not depend on the SNR. 4366
APPLIED OPTICS / Vol. 33, No. 20 / 10 July 1994
~~~~~1 = ;472
)d
sine( p2 )dji.
(40)
Figure 9 shows the phase standard deviation for uncompensated quadratic range migration between 0 and 0.5 resolution cells.
For the rectangular transfer function of Eq. (24) we have SNR = 10
-
17I =
1POI BL
1 -
1VOI , B,,
to= 0.
(42)
IThe physical reason for spectral misalignment is the
different aspect angles of a scene element in the two surveys. Different squint angles result in different Doppler centroid (DC) frequencies, fDc, and fDC,2, of the two data sets. If p.represents Doppler (azimuth) frequencies in hertz, then
SNR = o
0
0.8
0.6
0.4
0.2
#
I
Fig. 8. Standard deviation q 4 of the interferometric phase versus uncompensated linear range migration for two SNR's. The linear range migration is expressed in fractions p of a range resolution cell.
C.
For cross-track interferometry the DC difference po can be made to approach zero by controlling the antenna squint to be the same for both surveys. Spectral misalignment in the range frequency dimension, however, is an inherent feature of crosstrack interferometry. 6 Let us assume that an area of interest on the ground is seen at incident angles of 0, and 02 in the two surveys. Then a certain harmonic ground structure transforms into different echo frequency components, depending on the incident angles. This effect can be interpreted conversely as a relative shift of H, I and H2 1in the v direction by the amount of
Spectral Envelope Misalignment
Even in noise-free and aberration-free cases, signal decorrelation will occur if the envelopes of H, and H2 are shifted relative to each other, i.e., IH2 (p., v) I
=
IHi(p
(41)
p.o,v - vo)I*
-
(43)
fDC,l-
[LO= fDC,2-
1
11
sin 0,](
V0 rac ars n O sin 02
(44) 4
where 0 = (0 + 02)/2 and fradar is the carrier frequency of the radar. For horizontal baselines of length A that are much smaller than range R, relation (44) assumes the following form: cos2 v0 -fracar sin
0A
R
(45)
In fact, v0 is the local frequency of the interferometric fringe pattern. Decorrelation caused by spectral misalignment can be avoided if the two data sets are properly bandpass filtered in order to retain only those spectral compo-
nents that pass through both H, and H2 . 5. Conclusions
0.1
0.2
0.3
0.5
0.4
E
Fig. 9. Standard deviation or of the interferometric phase versus uncompensated quadratic range migration for two SNR's. The quadratic range migration is expressed in fractions e of a range resolution cell.
We have developed a system theoretical approach to determine the phase statistics of interferograms of distributed scatterers. Our approach is quite general and can be applied to all systems that are characterized by linear transfer functions. We have applied this technique to interferometric SAR. In particular, we were interested in the effects of SAR processor aberrations on phase biases and phase variations. We evaluated the first two phase moments for typical processing aberrations, e.g., geometric misregistration, defocusing, and uncompensated range migration under the influence of thermal noise. Only one of these aberrations, defocusing, leads to a phase bias, but all of them introduce phase variance, 10 July 1994 / Vol. 33, No. 20 / APPLIED OPTICS
4367
as has been shown in several figures. More generally, only even phase aberration functions introduce phase biases. The phase variance limits the phase accuracy that can be obtained, so that an evaluation of processor designs for interferometric SAR applications is an implicit result of our study. References 1. L. C. Graham,
"Synthetic
interferometer
radar
for topo-
graphic mapping," Proc. IEEE 62, 763-768 (1974). 2. H. A. Zebker and R. M. Goldstein, "Topographic mapping from interferometric synthetic aperture radar observations," J. Geophys. Res. 91, 4993-4999 (1986). 3. C. Prati, F. Rocca, A. M. Guarnieri, and E. Damonti, "Seismic
migration for SAR focusing: interferometrical applications," IEEE Trans. Geosci.Remote Sensing 28, 627-639 (1990). 4. F. K. Li and R. M. Goldstein,
"Studies
of multibaseline
spaceborne interferometric synthetic aperture radars," IEEE Trans. Geosci. Remote Sensing 28, 88-96 (1990). 5. C. Prati and F. Rocca, "Limits to the resolution of elevation maps from stereo SAR images," Int. J. Remote Sensing 11, 2215-2235
(1990).
6. C. Prati and F. Rocca, "Range resolution enhancement with multiple SAR surveys combination," presented at the International Geoscience and Remote Sensing Symposium, Houston, Tex., 1992).
7. K.Raney and P. Vachon, "A phase preserving SAR processor," Proc. Inst. Electr. Eng. Part F 139, 147 (1992). 8. A. Papoulis, Probability, Random Variables, and Stochastic
4368
APPLIED OPTICS / Vol. 33, No. 20 / 10 July 1994
Processes, 1st ed. (McGraw-Hill, New York, 1965), Chap. 14, p. 475.
9. J. W. Goodman, "Film-grain noise in wavefront-reconstruction imaging," J. Opt. Soc.Am. 57, 493-502 (1967). 10. J. W. Goodman, "Statistical properties of laser speckle patterns," inLaserSpeckle andRelated Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 29-35. 11. Ref. 8, Chap. 7, p. 222. 12. W. B. Davenport, Jr., and W. L. Root, An Introduction
to the
Theory of Random Signals and Noise, 1st ed. (Institute of Electrical and Electronics Engineers, New York, 1987), Chap. 8, pp. 158-165. 13. Ref. 10, pp. 42-46.
14. K Saribandi, "Derivaton of phase statistics from the Mueller matrix," Radio Sci.27, 553-560 (1992). 15. E. Rodriguez and J. M. Martin, "Theory and design of interferometric synthetic aperture radars," Proc. Inst. Electr. Eng. Part F 139, 147-159 (1992).
16. H. A.Zebker and J. Villasenor, "Decorrelation in interferometric radar echoes," IEEE Trans. Geosci. Remote Sensing 30, 950-959(1992). 17. M. Born and E. Wolf,Principles of Optics, 6th ed. (Pergamon, London, 1980), Chap. 5, pp. 203-207.
18. R. Bamler, "A comparison of range-Doppler and wavenumber domain SAR focusing algorithms," IEEE Trans. Geosci. Remote Sensing 30, 706-713 (1992). 19. C. Wu, K. Y. Liu and M. Jin, "Modeling and correlation
algorithm for spaceborne SAR signals," IEEE Trans. Aerosp. Electron. Syst. AES-18, 563-574 (1982).
