Some methods for reducing propagation-induced phase errors in coherent imaging systems. I. Formalism

924 J. Opt. Soc. Am. A/Vol. 5, No. 6/June 1988

Some methods for reducing propagation-induced phaseerrors in coherent imaging systems. I. Formalism

Warren D. Brown and Dennis C. Ghiglia

Sandia National Laboratories, Albuquerque, New Mexico 87185

Received October 30, 1987; accepted February 23, 1988

The random phase errors induced in an electromagnetic wave propagating through turbulence in the troposphereand/or the ionosphere can degrade the performance of a coherent imaging system such as a synthetic-apertureradar. Several methods for removing or reducing these phase errors are described. Some of these methods, such ascubic-spline approximations with map drift and a deterministic technique based on a Legendre-series expansion ofthe phase error, are not new but are included for comparison. At least three techniques are (to our knowledge) new;a binary multiplexing scheme, a method using the Wigner-Ville distribution, and an optimal estimator based on thecovariance matrix of the Legendre-series expansion coefficients. The method based on the Wigner-Ville distribu-tion promises to be the most effective in reducing the deleterious effects of the phase errors on image quality.

1. INTRODUCTION

Phase errors produced in electromagnetic waves propagat-ing through turbulence in the troposphere and/or the iono-sphere can degrade the performance of a variety of imagingand remote-sensing systems, including, for example, syn-thetic-aperture radars (SAR's), coherent optical sensors, ra-dio telescopes, and various ground-based sensors for spacesurveillance and space-based sensors for ground surveil-lance. The turbulence can occur in the ambient tropo-sphere or ionosphere (e.g., turbulence in the lower atmo-sphere produced by wind shear or convective heating fromthe ground), or it can be generated by artificial disturbancessuch as low- or high-altitude nuclear explosions.1-4

There are a number of methods available for removing thelow-frequency components of the phase errors produced bypropagation through continuous random media.5' 6 Studieshave demonstrated, however, that these methods are ofteninadequate because the residual high-frequency compo-nents of the phase errors can still significantly degrade ordisrupt performance. 3' 4 For example, the performance of anominal satelliteborne SAR operating through either thehigh-latitude ionosphere or the ionosphere perturbed byhigh-altitude nuclear explosions would sometimes be severe-ly degraded because of phase errors generated by propaga-tion through turbulence.4

The purpose of this paper is to describe, analyze, andcompare several phase-error-mitigation techniques. Threeof these techniques are based on a Legendre-series expan-sion of the phase error, and one is particular to a power-lawphase-error spectrum. In Section 2, some of the character-istics of the phase errors generated in propagating throughcontinuous random media are described. An analytical ex-pression for the covariance matrix of the Legendre expan-sion coefficients is introduced. This expression is the basisfor a phase-error-correction scheme developed with mean-square-estimation theory.

The formalism describing the postprocessing phase-error-removal schemes applied to SAR imagery is presented in

Section 3. Cubic-spline approximations of the phase errorbased on the map drift and on the mean-frequency differ-ence are described first. Mean-square-estimation theory isthen used to determine the optimal estimate for the Legen-dre expansion coefficients, given a few measurements withthe map-drift method or the mean-frequency method andknowledge of the covariance matrix for the random phaseerror. The statistics of the map-drift-measurement errorand of other additive-noise sources are also taken into ac-count. The results of decomposing the ensemble-averagedimpulse response into contributions from the terms in theLegendre-series expansion are presented. A multiplexingscheme based on optimal binary maskings of the aperturedata is then formulated. Finally, the Wigner-Ville distribu-tion, which is a mixed representation of the aperture data inboth time and frequency, is briefly reviewed, and a methodbased on this mixed representation is introduced. Thismethod appears to be better than the other methods exam-ined in this paper in that maximal use of the image informa-tion is made in estimating the phase error. This behavior ismanifested in the numerical results presented in part II ofthis series. 7

Conclusions and recommendations are made in Section 5.Technical details are provided in Appendix A.

2. CHARACTERIZATION OF PHASE ERRORSPRODUCED BY PROPAGATION THROUGHTURBULENCE

Turbulence is usually conceptualized as a random process,primarily because specifying the detailed dynamics is cum-bersome and often unnecessary. * The index of refraction ofeither turbulent neutral air in the troposphere or turbulentplasma in the ionosphere is also a random process. Becausethe phase velocity of electromagnetic waves is inversely pro-portional to the index of refraction, propagation of micro-waves through turbulence produces a phase perturbationthat is also a random process. In that the phase error is arandom process, it must be described in terms of moments.

0740-3232/88/060924-18$02.00 © 1988 Optical Society of America

W. D. Brown and D. C. Ghiglia

Vol. 5, No. 6/June 1988/J. Opt. Soc. Am. A 925

Spectrum of Phase ErrorsConsider the phase error along some linear aperture or an-tenna. Let A be the length of the aperture, and let the originbe the center of the aperture (Fig. 1). Let 0(x) be therandom phase error at the point x along the aperture. Theautocorrelation function of the phase error is defined as

Ce(xx') = (0(x )0 ), (2.1)

where the brackets (.) denote an ensemble average. If it isassumed that the phase error is stationary in the wide sense,then the mean is a constant, and its autocorrelation dependsonly on (x - x') (Ref. 8):

( 0(x)0(x') ) = C(x - x'). (2.2)

This function of a single scalar variable can be one-dimen-sionally Fourier transformed:

c - x') = J dw cos[w(x - x')]G(w), (2.3)

where G(w) is the power spectrum of the phase errors andG(-ac) = GM.)

In some treatments of turbulence-produced phase errors(including the treatment in this paper), 0(x) is assumed to benonstationary but has stationary first increments. In thiscase the autocorrelation function does not exist, and thelower-order statistics are described by the structure func-tion, defined as

GW)Log-log plot

Co-

Troposphere p=8/3 (Kolmogorov)

Ionosphere 2< p< 3

Two Parameters Characterize Spectrum:

d = strengthp = spectral Index (-slope on log-log plot)

Do x)= dyJP 1 , Y = A

Fig. 2. The spectrum of propagation-induced phase errors has apower-law dependence on wave number.

if the integral in Eq. (2.3) diverges, the integral in Eq. (2.6)converges, provided that

lim z3G(w) = 0,co 0

(2.7)

D (x, x') = ([0(x) - O(x,)]2) = D,(x -x'), (2.4) Ilim wG(M) = 0.ages.

where the last equality is a result of the (wide-sense) station-arity of the first increments. Of course, the reason that thestructure function exists although the autocorrelation func-tion does not is that in the structure function the ensembleaverage is of the first difference of the phase error at twopoints. If the phase errors are stationary, then the autocor-relation and structure functions are related by

D(x -x') = 2(02) - 2C,(x - x'). (2.5)

The phase-structure function is related to the power spec-trum of the phase error by

D,,(x - x') = 2 1 dc1 - cos[co(x - x')]jG(w). (2.6)

Equation (2.6) is consistent with Eqs. (2.3) and (2.5). Even

-A/2 6

i I 0 W~~~~~~~~~~xx A/2

Structure Function

D (xx') _ x <[¢(x) -(x 2 >

D (x-x') = 2f dat[1-cos(cu(x-x'))]G(oA0

GW) = spectrum of Phase Errors

Fig. 1. Nonstationary phase error with a stationary first incre-ments is described by the structure function defined in Eq. (2.4).

[It is assumed the G(w) is well behaved such that the diver-gence of the integral in Eq. (2.6) would occur only at theendpoints (0, a).]

The spectrum of the phase errors produced by propaga-tion through turbulence is often represented by a power lawover the wave-number region of interest or by several contig-uous power-law regimes (see Fig. 2). All the mitigationtechniques considered in this paper are formally indepen-dent of the shape of the phase-error spectrum except for theoptimal estimator method, in which a power-law spectrum isassumed (more precisely, a closed-form analytical expres-sion for the covariance matrix is derived for a power-lawspectrum). For simplicity, the optimal-estimator method ispresented for a single-regime power-law spectrum, althoughit can be generalized readily'to a multiple-regime spectrum.Only two real parameters are necessary to characterize asingle-regime power-law spectrum:

d is the strength,

p is the spectral index (negative slope on log-log plot).

The strength of the spectrum is defined such that it appearsin the structure function in a simple manner (for 1 < p < 3):

D,,(x) = dlyIP-', y = x/A,

r(p)cos P(3) dG~W = - - rA(' (d P

(2.9)

(2.10)

The restriction on p is needed to satisfy Eqs. (2.7) and (2.8),i.e., so that the integral in Eq. (2.6) converges and the struc-

(2.8)



ture function exists. Values of p greater than or equal to 3can be accommodated in the formalism by describing thestatistics of the phase error in terms of a function that is theensemble average of differences in the phase error higherthan the first difference.

For tropospheric turbulence, p is exactly 8/3. Kol-mogorov9 proved this in 1941, using dimensional analysis(principle of similitude). The proof is based on the assump-tion that turbulence in the troposphere is locally homoge-neous and locally isotropic. For turbulence in the iono-sphere, the isotropy is broken by the geomagnetic field be-cause the ionosphere is a plasma; i.e., the electron-densityirregularities in the ionosphere tend to align along the geo-magnetic field. The Kolmogorov proof is invalid for iono-spheric turbulence because of the anisotropy. There is am-ple empirical evidence that phase errors produced by propa-gation through turbulence in the ionosphere have a power-law spectrum, although the spectrum may consist of two ormore wave-number regimes with different spectral indices.10

A minor point concerning the phase-error spectrum is nowaddressed. The power-law phase-error spectrum must rolloff at sufficiently small and large wave numbers (Fig. 3).The roll-off at small wave numbers is a manifestation of thefact that there must be a largest size for the eddies in theturbulence. This outer-scale size is designated Lo. Thespectrum will roll off at wave numbers smaller than about27r/Lo. The portion of the phase-error spectrum that hasdeleterious effects on system performance is for wave num-bers larger than 27r/A, where A is the aperture length. Thiscan be understood heuristically by thinking of the turbulenteddies as lenses that focus or defocus a microwave beam. Ifthe lens is much larger than the aperture, then its maineffect is to produce a constant phase error across the aper-ture, which has no effect on the systems of interest here.For the systems of interest, the aperture length is muchsmaller than the outer-scale size (A << Lo), and therefore thelow-wave-number roll-off has no impact on their perfor-mance.

Consider the example of an orbiting SAR such as SEA-SAT.1' The relevant length is the projection of the outer-scale size onto the orbit from a point on the ground. Theouter-scale size of the ionospheric turbulence is at least 10km in the shortest dimension (ionospheric turbulence isnonisotropic).12 The altitude of SEASAT is about 800 km,

G(co)

2ir/L0 2n/A 2n/ho

Lo >> A>>> 10

I - outer scale size

Fig. 3. Low- and high-wave-number roll-offs of the phase-errorspectrum do not affect system performance. Lo >> A >> bo.

and the altitude of the ionospheric turbulence is about 350km. Therefore the outer-scale size projected onto the SEA-SAT orbit is at least 20 km, which is significantly larger thanthe synthetic-aperture length of 4 km. The outer-scale sizefor tropospheric turbulence is about 1 km, and it existsprimarily below an altitude of 2 km.'3 Therefore the projec-tion of the outer-scale size onto the orbit is at least 400 km,which is certainly much greater than the synthetic-aperturelength.

There is also a high-wave-number roll-off; e.g., for turbu-lence in the troposphere, large eddies cascade downwardinto smaller eddies until eventually viscosity becomes im-portant and the turbulent energy is dissipated as heat. Thevalue of the inner-scale size lo for the tropospheric turbu-lence ranges from about a millimeter to a few centimeters. 13

The inner-scale size for turbulence in the ionospheric plas-ma is of the order of a meter.'2 The high-wave-number roll-off for both the troposphere and the ionosphere is at such ahigh wave number, 27r/la, relative to 2r/A that it can beneglected. In other words, since lo << A, the differencebetween rolling off the spectrum at 2-7r/l and extrapolatingthe power-law behavior beyond 27r/lo is insignificant interms of effects on the system.

Even though the spectrum of the phase errors resultingfrom propagation through turbulence has low- and high-wave-number roll-offs, these can be neglected for the sys-tems of interest, and the spectrum can be assumed to bepower law for all wave numbers.

Legendre-Series Expansion of Phase ErrorsConsider a Legendre-series expansion of the phase erroralong an aperture of length A:

>(x) = E b> Yn An=0 W

--A X <A-,2 2

where

J1/2dYYn(Y)Ym(Y) = 6nm' y = x/A.

-1/2(2.12)

The Legendre polynomials, Yn(y), are a little different fromthe standard Legendre polynomials in that they are ortho-normal and are defined on the interval (-1/2 < y < 1/2)rather than on (-1 < y < 1). Of course, this difference isonly a matter of scaling. The Yn are described in more detailin Ref. 14.

The coefficients bn in Eq. (2.11) are random processes inthat the phase error 0(x) is a random process, but the Legen-dre polynomials are ordinary functions. Because the coeffi-cients are random processes, they must be characterized bytheir moments.

It is demonstrated in Ref. 14 that the weak stationarity ofthe first increments of ¢(x) implies that the mean of everycoefficient in the Legendre-series expansion vanishes exceptfor the zeroth-order coefficient; i.e.,

(bn)=0, n>1. (2.13)

The following expression for the covariance matrix of thecoefficients is derived in Appendix A for an arbitrary powerspectrum of the phase errors, G(w):



1

A

P=2.5

p =2.9

-C210 I l I

2 4 6 8 10 12 14 16Number of Terms in Series (n)

Fig. 4. Ratio of rms values of the nth coefficient to the first coeffi-cient as function of the number of terms in the Legendre series.

(bnbm) = (-1)1(_j)(n+m)/ 2 (2n + 1)1/2(2m + 1)1/2 f dwG(w)

X inA( 2 )0 (12)

=O0

for n + m even,

for n + m odd,

where

0^1 for n and m even11 for n and m odd

and jn is the spherical Bessel function.Substituting Eq. (2.10) into Eq. (2.14) for a power-law

phase-error spectrum and performing the integration overthe wave number (see Appendix A) results in the followingrecursion relation for the mean square of the Legendre-series expansion coefficients (the diagonal terms of the co-variance matrix):

(bn2) = 2n - 2n + 1 + P (bn-12),

(bl 2 ) = 3d . (2.15)(p + 3)(p + 1)

A derivation of a closed-form expression for the off-diagonalelements (bnbm) in terms of p, d, n, and m is given in Appen-dix A. The restriction 1 < p < 3 is explicitly manifest in Eqs.(2.15). Thus, if p = 1, (b12) = 0, and the recursion does notstart. If p < 1, then (b1

2) is negative, which is impossible.

If p = 3, then (b22) (and all higher-order terms) vanishes,

and if p > 3, then (b22) is negative, which again is impossible.

If the strength d and the spectral index p of a power-lawphase-error spectrum are known, then the covariance matrix(bnbm) for the Legendre expansion coefficients can be com-puted readily. Knowledge of the covariance matrix formsthe basis for one of the phase-error-removal schemes de-scribed in Section 3.

Figure 4 displays the ratio of the rms value of the nthcoefficient to the rms of the first coefficient as a function ofthe number of terms in the series for several values of thespectral index ranging from 1.1 to 2.9. The series convergesmore rapidly for larger values of the spectral index (steeperspectra).

3. PHASE-ERROR-MITIGATION METHODSAPPLIED TO SYNTHETIC-APERTURE RADARIMAGERY

The formalism describing several phase-error-mitigationmethods is presented in this section. Although the formal-ism could be generalized to include many coherent systems,it applied here specifically to the azimuthal dimension ofSAR imagery. Cubic-spline approximations of the phaseerror based on map drift and on the mean-frequency differ-ence are described first; then some methods based on theLegendre-series expansion of the phase error are presented,including deterministic methods and a method using mean-square-estimation theory. A technique is then describedthat is based on binary multiplexing. Finally, a methodbased on the Wigner-Ville mixed distribution is introduced.A brief description of map drift is now presented because itis important to some of the mitigation methods.

Map-Drift MethodThe map-drift method 5 is based on two properties of SARdata. The first is that SAR data in the azimuthal dimensionresemble a laser hologram in that if the synthetic aperture isdivided into a number of subapertures and an image is con-structed from each subaperture, these images will be of thesame terrain except that the resolution will be reduced. Thesecond property is that a linear phase error across the syn-thetic aperture shifts the entire scene in the azimuthal direc-tion because a linear phase error is equivalent to a constantfrequency shift (frequency is the derivative of phase). Be-cause of the Doppler processing, a constant frequency shiftsimply translates every pixel by a constant amount in theazimuthal direction.

Now consider a quadratic phase error across the syntheticaperture. A quadratic phase error corresponds to a linearfrequency shift. If the aperture is divided into two subaper-tures and an image is formed from each subaperture, the twoimages will be shifted in the azimuthal direction relative toeach other by an amount proportional to the quadratic phaseerror (the maps drift relative to each other). The quadraticphase error can be determined by cross correlating the twoimages in the azimuthal dimension and measuring the lag inthe correlation peak. Figure 5 illustrates the map-driftmethod for determining the quadratic phase error. A qua-dratic phase error across the full aperture is equivalent tolinear phase errors with opposite slopes across the two half-



where f is a specific realization of the random phase. Simi-larly, the mean frequency for the (i + 1)th subaperture is

oi+i = (3.2)(AyiJ+2 - (Ayill)yi+2 -Yi+1

- A/2 0 A/

Fig. 5.rection.

Map-drift autofocus method for quadratic phase-error cor-

Bounary 1 Boundary i Boundy m

t1Y I __

Y11 Y2 Y3 Y1 Y,+i Yi+2 Ym Y. I t Y-2

Subapenture 1 Subaperture i Subaperture (ml)

6i =(Ay)2 ["("' i,2) - 201(Ay if,)+ O(Ayi)I

Fig. 6. Map drift of m subaperture pairs for nth-order phase-errorcorrection. The mean-frequency difference is defined in Eq. (3.4).

apertures plus quadratic residuals across the two subaper-tures of one fourth the magnitude. The linear phases, whichare of opposite slopes, cause the offset between the twoimages. The subaperture maps drift one pixel for every 7rrad of quadratic phase error on the full aperture.

The map-drift method can be extended to measure high-er-order phase errors by dividing the aperture into a largernumber of subapertures. For example, to measure the cubicphase error, the aperture is divided into three subapertures,and all cross correlations are performed among the threepairs of images. In general, n subapertures are required tomeasure an nth-order polynomial (such as Yn).

Consider applying the map-drift method to m + 1 sub-apertures, i.e., to m adjacent subaperture pairs (see Fig. 6).The map-drift method measures the normalized differenceof the mean frequencies from one subaperture to the next(adjacent) subaperture. The mean frequency for the ithsubaperture is

- = 1 Q d0(Ay) - ¢'(Ayil) - n(Ayi)Wi = -Yi fy dy dy -yi (3.1)

The normalized difference in mean frequencies between the2 (i + 1)th and ith subapertures is

= - ~Ai= Wi+l Zi

1/2(yi+2 - Yd

For subapertures of equal sizes, Eq. (3.3) becomes

wi = 1 [X(AYi+2) - 20(Ayijl) + 0(Ayi)].(Ay) 2

(3.3)

(3.4)

The map-drift method measures Awi. Notice that in thelimit that the subaperture size approaches zero, Eq. (3.4) issimply the second derivative of the phase error; i.e.,

lim Acoi = d y |A~y-.O dy2

Y=Yi~1

The difference in the mean frequencies of two subaper-tures can be computed directly without performing the crosscorrelations in the map-drift method. The normalizedmean frequency for the ith subaperture can be written as

(We) = fJ dw {lFi(c)12/J dWlFi(w)I2)1 (3.5a)

where Fi(c) is the Fourier transform of the data over the ithsubaperture,

(3.5b)

and fi(x) is the data signal in the ith subaperture. Thus themean frequency for the ith subaperture can be obtained inthe Fourier-transform (image) domain by computing thefirst moment of the squared modulus of the image data.The mean-frequency difference is, in terms of Eq. (3.5a),

A = (o0i+1) - (Wj). (3.5c)

Equations (3.5a)-(3.5c) are formally equivalent to Eqs. (3.3)and (3.4). However, there are numerical advantages to com-puting the mean-frequency difference with Eqs. (3.5a)-(3.5c) in that cross correlation requires more computationalsteps (an N 2 operation, where n is the number of pixels) thandoes computing the mean frequency and subtracting (a lin-ear-in-N operation). To distinguish it from the map-driftmethod, using Eqs. (3.5a)-(3.5c) is called the mean-frequen-cy-difference method.

Cubic-Spline ApproximationsIn a conventional method of phase-error correction, the fullaperture is divided into a number of subapertures. Thecompressed data from adjacent subaperture pairs are crosscorrelated to determine a relative displacement (map drift).This displacement is interpreted as a sample of the secondderivative of the phase error at the boundary between theadjacent subaperture pair as shown in Eq. (3.4). Equiva-lently, the second derivative can be estimated by computingthe difference of the mean frequencies of the two subaper-


Fi(co) = 5'ViWl,


tures. These second-derivative samples at all the bound-aries between adjacent subaperture pairs are used to con-struct a piecewise cubic spline, kest, which approximates thetrue phase error. This phase-error estimate is removedfrom the data by complex multiplication with exp(-jest),and the entire process is repeated until convergence is at-tained. A number of enhancements of this basic scheme arepossible:

(a) The subaperture pairs can be overlapping, in whichcase the second derivative is usually assumed to be evaluatedat the midpoint of the overlap region.

(b) Subaperture pairs do not have to be adjacent, inwhich case the second derivative is assumed to be evaluatedat the midpoint between the two subapertures. The addi-tional improvement resulting from the inclusion of nonadja-cent pairs is usually not significant.

In part II of this series, a description of numerical results,7

it is demonstrated that reasonable estimates of the phaseerror and reconstructions of the unperturbed phase historiescan be accomplished if the number of subapertures is not toolarge and if the high-frequency phase errors are not largerelative to the low-frequency phase errors. This deteriora-tion of the method for large high-frequency phase errors andcorrespondingly large numbers of subapertures is a conse-quence of the reduction in the accuracy of cross correlation(or in the accuracy of the calculations of the mean frequen-cies) as the subaperture size decreases.

Two additional problems arise from using map drift (ormean-frequency difference) to estimate the derivatives ofthe phase error for the cubic-spline fit. The first is that thesecond-derivative samples are not available at the extremeendpoints of the full aperture, and therefore some extrapola-tion scheme is necessary, which usually leads to errors. Thesecond problem applies to overlapping (or to nonadjacent)pairs. The second-derivative samples are customarily as-signed a position corresponding to the midpoint of the over-lap region. This position is theoretically correct only forpure quadratic phase error and a constant-magnitude signalacross the subaperture. The midpoint position assignmentis incorrect for higher-order phase errors, which results in anincorrect spline-function estimate.

Legendre-Series Expansion: Determined SolutionThe two problems mentioned in the preceding paragraphcan be eliminated by combining the Legendre-series expan-sion of the phase error with the map-drift (or the mean-frequency-difference) method to obtain a determined solu-tion5 of the phase error, i.e., to find an estimate of the phaseerror in which the number of unknown coefficients in theLegendre-series expansion is equal to the number of map-drift measurements. The underdetermined solution, forwhich the number of unknown Legendre coefficients is larg-er than the number of map-drift measurements, is discussedin the next subsection.

Let (x) represent a specific realization of the randomphase error across the synthetic aperture; e.g., 0(x) could bethe phase error along the azimuthal dimension correspond-ing to an individual SAR image. This specific realizationcan be approximated by the finite Legendre series,

(3.6)

Notice that the Legendre expansion coefficients bn are notrandom processes because 0(x) is a specific realization of arandom process.

Substituting the Legendre-series expansion [relation(3.6)] into Eq. (3.4) results in

i= 2 bn[Yn(Yi+2) - 2Yn(yi+1 ) + Yn(yi)],

i1,2,...,m(m=l-1). (3.7)

The summation starts at n = 2 because the first (linear) termvanishes. The map-drift (or mean-frequency-difference)method cannot sense a tilt across the aperture. It is as-sumed in this subsection that the number of unknown Le-gendre coefficients is equal to the number of subapertureboundaries.

Equation (3.7) can be written as a matrix equation. Let

(A)i = (Y)2 [1(Ayi) - 24(Ayi+i) + $(Ayi)],

1(P)in- =(A [Y n(Yi+2) - 2Y0(yi+1 ) + Y,(yi)],

where

i = 1, 2, .. ., m (m = I -1),

n = 2,3,..., 1.

Thus Eq. (3.7) becomes

Aw = Pb,

(3.8)

(3.9)

(3.10)

where Aco and b are column vectors of dimension (1 - 1) andP is a square matrix. The elements of Aw are determined bythe map-drift (or mean-frequency-difference) method, andthe elements of P can be obtained analytically with Eq. (3.9).

The formal solution of Eq. (3.10) is straightforward:

b = P-Aw (3.11)

which is, of course, the solution of m linear equations in munknowns. If nonadjacent pairs are also cross correlated,then for (m + 1) subapertures a total of m(m + 1)/2 crosscorrelations can be performed. In this case the solution isoverdetermined, and the least-squares solution is appropri-ate:

b = PAw. (3.12)

P is now an m(m + 1)/2 X m rectangular matrix, and P is thepseudoinverse of P. In the presence of the inevitable mea-surement error in cross correlation, it is sometimes advanta-geous to overdetermine the problem by cross correlating allsubaperture pairs.

Linear Mean-Square-Estimation TechniquesIn this subsection, the underdetermined problem in whichthe number of subaperture cross correlations is less than thenumber of unknown Legendre expansion coefficients is ad-dressed. Linear mean-square-estimation theory is used


I

� W --� �', b,, Yn (x 1A).n=1


along with knowledge of the covariance matrix to derive theoptimal estimate of the Legendre coefficients.

Equation (3.10) is generalized to include measurementnoise:

Ale = Pb + 71, (3.13)

where P is a rectangular matrix (underdetermined problem)and ij is a column vector that represents the numerical errorin cross correlating the subapertures. In applications to realSAR data, wq could also include other sources of additivenoise.

Correcting for the phase error means obtaining a bestestimate for the elements of the column vector b (the Legen-dre-series expansion coefficients) in Eq. (3.13). The infor-mation available consists of several map-drift measure-ments (AW), the statistical properties of the phase errorsinduced by turbulence (as expressed by the covariance ma-trix (bibj)), and the statistical properties of the measure-ment errors. According to linear least-squares-estimationtheory, a reasonable estimate of b, taking into account themeasurements Aw and the statistical properties of the turbu-lence and measurement errors, is the weighted least-squaresestimate 8, where'5

6 = (B-' + pTR-lp)-lpTR-l'Aw (3.14)

In the limit that the numerical error and the additive noisevanish, Eq. (3.14) reduces to Eq. (3.12).

The solution of most interest here is Eq. (3.14) when 1 >(m + 1), i.e., the underdetermined solution for which thenumber of map-drift measurements is less than the numberof terms in the Legendre-series expansion. Equation (3.14)provides a prescription for the best choice of Legendre-seriesexpansion coefficients in the underdetermined problem, giv-en knowledge of the covariance matrices for the randomphase error and the measurement noise.

Equation (3.14) simplifies the expression for the case inwhich the errors in the measurement of the differences in themean frequencies of adjacent subapertures are uncorrelatedand approximately the same at each subaperture boundary.The error matrix R is then diagonal and proportional to theidentity matrix:

ratio. The variance of the measurement errors can presum-ably be estimated from an analysis of the map-drift tech-nique. A value for d can be obtained from the map driftsused in computing Aw; i.e., the map drifts between the madjacent pairs of subapertures will yield m estimates of theturbulence strength, and the value of d to use in Eq. (3.18)can be computed by averaging these m estimates. It ispossible that some sort of fine tuning of (aN2/d) could beused to obtain the best compensated image.

In the more-general case in which the spectral index p isalso unknown, an estimate of p can also be obtained from themap-drift measurements if there are more than two sub-apertures (at least two map-drift measurements are neededto determine both p and d).

The covariance matrix for the error in the estimate of theLegendre coefficients is given by' 5

D = ((b - 8)(b - &)T) = IN 2(N2B-1 + pTp)-l (3.19)

or

(3.20)D-'=B-_ + 1 pTp

UN

If a normalized covariance matrix for the error, D, is definedby factoring out the strength of turbulence, d, then Eq.(3.20) can be written as

D-1 = A-1 + (d 2 ) pTp. (3.21)

Equation (3.21) suggests an iterative scheme for estimatingthe Legendre expansion coefficients. The first iterationprovides the estimate in Eq. (3.18):

6 [= TP+ (i2-) ]-1P, TAWU (3.22)

where the first-iteration quantities are labeled by the sub-script 1. The number of subaperture boundaries used in thefirst iteration is ml. The first-iteration estimate of thephase error is

R = IN2 I, (3.15)

where UN2 is the variance of the measurement errors at each

subaperture boundary. Substituting Eq. (3.15) into Eq.(3.14) results in

6 = (N 2 'B- + pTp)-lpTAWX (3.16)

In the initial numerical simulations, it is assumed that thespectral index p is known, as for tropospheric turbulence.The strength of the turbulence d is unknown but appearslinearly in the covariance matrix B; i.e.,

B = dP, (3.17)

where B is known. Substituting Eq. (3.17) into Eq. (3.16),we obtain

6 = [(O ) )-1 + PTP] PTAW (3.18)

Notice that the strength of the turbulence and the varianceof the measurement errors only appear in Eq. (3.18) as a

(3.23)f (X) = > 8 1jYj(x/A).j=2

This can be subtracted from the phase history to obtain aresidual phase error,

501W = ¢(x) - fl(x). (3.24)

The second iteration is performed with this residual phaseerror. Dividing the aperture into (M2 + 1) subapertures andusing the covariance matrix for the error in the first-itera-tion estimate results in the second-iteration estimate,

62 = [P2 TP2 + (Ud2 ) )_J P2TAd, (3.25)

where U22 is the variance of the measurement noise associat-

ed with the m2 map-drift measurements of the second itera-tion. The elements of the column vector AcW2 are the resultsof these second-iteration map-drift measurements, and P2 isthe M2 X (I - 1) rectangular matrix given by Eq. (3.9).Substituting Eq. (3.21) into Eq. (3.25) yields



2 = [P 2TP 2 + ( 2 ) plTpl+ ( d2) 3_1 p2TAw2

(3.26)

If the variance of the measurement noise is the same for thetwo iterations (U12

= 2 2= UN

2), Eq. (3.26) reduces to

82 = [P2TP2 + PlTPl + (ud ) n-j P2TAW2. (3.27)

Equation (3.27) can be generalized to n iterations:

n= PjT pJ + ( dN) f3] PnTAOn. (3.28)

The normalized mean square of the residual phase error canalso be generalized to n iterationsl 5 :

(n) = Tr{ dN E p.Tp. + (_) S-1] } (3.29)

Notice that since PjTPj is at least a positive semidefinitematrix, the mean-square residual phase error cannot in-crease in successive iterations; i.e., on average, the iterationprocedure will not decrease the accuracy of the estimate ofthe Legendre coefficients.

There are two advantages to performing two (or perhapsmore) iterations with ml and M2 subapertures in contrast toperforming a single iteration with ml or M2 subapertures:

(1) In part II of this series7 it is demonstrated that theoptimal estimate based on an even number of subapertures(m odd) provides a good estimate for the coefficients of even-parity Legendre polynomials but a poor estimate of some ofthe coefficients of odd-parity Legendre polynomials and viceversa for an odd number of subapertures. Recall that theparity of Legendre polynomials is

Yi(-Y) = (- 1)IYJ(y); (3.30)

i.e., the parity is even or odd according to whether the orderis even or odd. The numerical results of part II demonstratethat if an even (odd) number of subapertures is used in thefirst iteration and an odd (even) number of subapertures isused in the second iteration, then the estimate is dramatical-ly improved over the case in which ml or M2 subapertures areused in a single iteration or the case in which an even (or odd)number of subapertures is used in both iterations. Thus thefirst iteration estimates primarily the coefficients of theeven-parity (odd-parity) Legendre polynomials and the sec-ond iteration estimates primarily the coefficients of the odd-parity (even-parity) Legendre polynomials.

(2) The accuracy of the map-drift measurements de-creases in direct proportion to the decrease in subaperturesize. In other words, since the resolution of the map corre-sponding to a subaperture is inversely proportional to thelength of the subaperture, the accuracy of the cross correla-tion of two adjacent maps implicit in the map-drift measure-ment decreases with decreasing subaperture length. Sincemultiple iterations are based on larger subapertures than arethe corresponding single iterations (same total number ofsubapertures), the map-drift measurements correspondingto multiple iterations are intrinsically more accurate. This

is also true for the mean-frequency-difference measure-ments.

The map-drift measurements can be applied to pairs ofadjacent subapertures [see Eq. (3.4)]. There is no restric-tion that the map drift be applied to adjacent subaperturesonly; any pair will do. For n subapertures, there are (n - 1)pairs of adjacent subapertures but n(n - 1)/2 total subaper-ture pairs; the total number of subaperture pairs is, muchlarger than the number of adjacent pairs for a large numberof subapertures. For example, for 10 subapertures, thereare 9 pairs of adjacent subapertures but a total of 45 pairs.Some improvement in the estimation of the Legendre ex-pansion coefficients is obtained by using all subaperturepairs in the map-drift measurements; the largest improve-ment is obtained for the larger values of the measurementnoise.

The generalization of the map-drift measurements to allsubaperture pairs is straightforward. The map-drift meth-od measures the normalized difference of the mean frequen-cies between any two subapertures. Consider any two sub-apertures labeled by i andj. The mean frequency for the ithsubaperture is

_ O(Ayi+l) - O(Ayi)Wi = Yi+l -

The normalized difference in mean frequencies between thejth and ith subapertures is

(11)_Y~l- yj + 2 +1 -2

(Y1+1 -YJ + Yi+l -1

X [(Ayj+,) - q(Ayj) _(Ay+i) - (Ayi)1L Yj+l - Yj Yi+l - Yi

For subapertures of equal sizes this reduces to

Acji (A y)2 [0(Ayj+l) -¢(Ayj) - O(Ayi+) + b(Ayi)].

(3.31)

Substituting the Legendre-series expansion of the phase er-ror into Eq. (3.31) results in

1 1'i[T.t) Y(Jj (Ay)2 E bkLYk(Y - - Yk(Yi+l) + Yk(Yi)]-

(3.32)

Let the subscript q denote the [n(n - 1)/2] pairs of i and j inan orderly way, i.e., with J > i, taking adjacent pairs first,then pairs formed from every other subaperture, then pairsfrom every third subaperture, etc., always working from leftto right. Thus

for n = 2 (2, 1),

for n = 3 (2, 1)(3, 2)(3, 1),

for n = 4 (2, 1)(3, 2)(4, 3)(3, 1)(4, 2)(4, 1).

Equation (3.32) can therefore be written as



AWq = E Pqkbk, q = 1, 2, ... , n(n - 1)/2, (3.33)k=2

or, in matrix notation,

Aw = Pb, (3.34)

where P is the rectangular matrix,

Pqk = [Yk(Yj+l) - Yk(Yj) - Yk(yi+l) + YkCVd)],

k = 2,.. .,1, q=1,2,...,n(n-1)/2. (3.35)

The derivation of the expression for the optimal estimate ofthe Legendre expansion coefficients is exactly as describedearlier in this section. The improvement in the estimates ofthe Legendre expansion coefficients resulting from using allsubaperture pairs instead of only the adjacent pairs is dis-played in part II.7

Impulse ResponsesEquations are derived in Refs. 14, 16, and 17 for the ensem-ble-averaged far-field antenna pattern (impulse response) ofa SAR when components of the phase error to any order inthe Legendre-series expansion are removed in the least-squares sense. In this subsection, the results of evaluatingthese equations with a VAX 11/780 computer are displayedand discussed as a function of the order of the approxima-tion for a number of combinations of values for the spectralstrength d and the spectral index p.

The voltage aperture-weighting function for all the im-pulse responses is a Taylor weighting resulting in five equi-amplitude sidelobes at -35 dB (relative to the main-lobepeak in the absence of phase errors) adjacent to the mainlobe on each side. Figure 7 displays the impulse responsecorresponding to the absence of phase errors.

It is mentioned in Section 2 that the Legendre-series ap-proximation to the phase error converges more rapidly forlarger values of the spectral index (steeper spectra), as indi-cated in Fig. 4. This is dramatically illustrated in Figs. 8and 9, which correspond to p = 2.9 and d = 3 and to p = 1.5and d = 1, respectively. Both cases produce approximatelythe same value for the integrated sidelobe ratio (ISLR) (-5.7dB) before any components of the phase error are removed.(The ISLR is defined in App. E of Ref. 14). For the steep

Cn

wCL

-J

0~

0

-10

-20

- 30

-40

-50

-600 1 2 3 4 5 6

NORMALIZED ANGLE FROM BORE SIGHT

Fig. 7. Unperturbed (no phase error) impulse response for Taylorweighting (-35 dB).

spectrum, p = 2.9, the impulse response is essentially re-stored to its unperturbed condition by the sixth order,whereas for the shallow spectrum, p = 1.5, the restorationdoes not occur until about order 16. As shown in Fig. 10,even when the perturbation is much stronger for the steepspectrum (d = 40), the impulse response is still restored toits essentially unperturbed condition at a lower order (N -14) than for the shallow spectrum. Notice the small separa-tion between the first few terms in the Legendre-series cor-rection for a shallow spectrum (p = 1.5) and the large separa-tion for the steep spectrum (p = 2.9).

The most widely used and accepted morphological modelof ionospheric turbulence was constructed and is currentlybeing upgraded by Fremouw and Lansinger.18 Although thespectral index for ionospheric turbulence appears to varybetween about 2 and 3 according to the strength of theturbulence, the geomagnetic location, the time, and perhapsother factors, the knowledge of this variation is incomplete.Northwest Research Associates has therefore fixed the spec-tral index at a value of 2.5 in the morphological model.Figures 11 and 12 display impulse responses for p = 2.5,representing the effects of nominal ionospheric turbulence.

Figure 11 represents the effects of moderately intenseturbulence in the ionosphere. The ISLR for no phase cor-rection is 0.6 dB, and in the quadratic approximation theISLR is -11.6 dB. The impulse response is almost restoredto the unperturbed condition by order 15. Figure 12 corre-sponds to severe turbulence in the ionosphere. The ISLRbefore any phase-error removal is 6.6 dB, whereas in thequadratic approximation it is equal to -5.6 dB. The corre-sponding image would be greatly distorted. Nevertheless,the impulse response is essentially restored to the unper-turbed state by order 16.

The lower-order terms affect the impulse response nearthe main lobe, and as the order increases the effect movesaway from the main lobe to the more distant sidelobes.Thus, for example, in Fig. 12, through order L = 7 theprimary effect is to reconstruct the first minimum in theimpulse response. The first sidelobe appears in order L = 8and is completely reconstructed by order L = 11. The sec-ond sidelobe appears in order 11 and is reconstructed byorder 12. This behavior is not surprising in that the nth-order Legendre polynomial has n zeros along the syntheticaperture. Higher-order Legendre polynomials correspondto higher-frequency phase errors across the synthetic aper-ture, which affect the more-distant sidelobes of the antennapattern. In fact, the nth-order Legendre polynomial corre-sponds to approximately p periods across the synthetic aper-ture, where

p = (n - 1)2.

It is helpful to relate the impulse responses displayed inFigs. 8-12 to ionospheric parameters for an orbiting SAR.The spectral strength parameter d can be written as'9

d _ 2 F(p/2)(Xre) 2 D(C L), (3.36)

where re is the classical electron radius, X is the radar wave-length, and r(x) is the gamma function.


W. D. Brown and D. C. Ghiglia Vol. 5, No. 6/June 1988/J. Opt. Soc. Am. A 933

LEGENDRE SERIES CORRECTION OF PHASE ERRORSDISTURBANCE STRENGTH 3.00, SPECTRAL INDEX 2.9

POINTING ERROR 0.28, OUBORATIC ERROR 0.25EFFECT THROUGH ORDER 16

1 1.5 2 2.5 3

U

I I I3.5 41 4.5

I - -- ---5 .5 6

Fig. 8. Impulse responses (IPR) for a spectral strength of 3 and a spectral index of 2.9, displayed with unperturbed impulse response.


POINTING ERROR 0.16, OURDRATIC ERROR 0.62EFFECT THROUGH ORDER 16

I I 7 I 0.5 1 1.5 2 2.5 3

U

3.5 4 4.5 5 5.5 6


_ I

Go '

CL ,

C, II-I O 5

0-I

_ -

rn-I

C)3

0

IE-I

I

1--)



POINTING ERROR 1.02, 01IRDRATIC ERROR 0.90EFFECT THROUGH ORDER 16

0 0.5 1 1.5 2 2.5| TI -__- 1- F -I

3 3.5 4 4.5 5 5.5 6

U


The geometrical parameter D is given by

D = 9(VeffAIv)P-1, (3.37)

where A is the synthetic-aperture length, v is the magnitudeof the relative velocity of a radar beam with respect to elec-tron-density irregularities, & is a geometrical factor account-ing for nonisotropy of electron-density irregularities, andVeff is the effective scan velocity of the radar beam acrosssurfaces of constant correlation.

Expressions for 9 and Veff are given in Ref. 20. CL is theheight-integrated strength of ionospheric turbulence, and Lis the vertical thickness of the layer containing the iono-spheric irregularities (turbulence). C, is the turbulencestructure constant for the ionosphere:

C, = 82r3/2 2 1) ((6n 2X (3.38)

rp2- 1) ro p 2

where (GWn)2) is the mean square of the fluctuating compo-nent of the electron density and ro is the outer-scale size ofthe electron-density irregularities.

Typical values for a disturbed ionosphere are

C 8L - 2 X 1026 (mks),

C, _ 1021 (mks),

assuming that L = 200 km.Table 1 displays the values of CL and C, corresponding to

Figs. 8-12. The values for p = 2.5 and d = 40 represent aseverely disturbed high-latitude ionosphere, whereas p = 2.5and d = 10 correspond to a more-moderately disturbed iono-

sphere. Notice that SEASAT is more sensitive to iono-spheric disturbances with a steep spectrum (p = 2.9) thanwith a shallow spectrum (p = 1.5). This is a result of thelarger amount of low-wave-number components in the phaseerror for a steep spectrum. The Legendre-series expansionof the phase error converges more slowly for a shallow spec-trum, but the SAR is less sensitive to a given level of iono-spheric turbulence. For a steep spectrum, the Legendreseries converges more rapidly, but the SAR is more sensitiveto the ionospheric turbulence.

Binary MultiplexingA binary-mask multiplexing method for recovering phaseinformation is developed for optical systems in Ref. 21. Inthis subsection, the method is applied to the problem ofestimating the coefficients in the Legendre-series expansionof the phase error. This method essentially consists of ap-plying optimally chosen binary masks to the aperture data toobtain measurements of the phase error. There are severaladvantages to the multiplexing method over methods forestimating the phase error that are based on map drift or onthe difference in mean frequencies:

(a) Binary multiplexing provides independent combina-tions of data measurements over the full aperture.

(b) There are no subaperture resolution limitation prob-lems.

(c) A high signal-to-noise ratio is maintained becauseonly 50% of the full aperture data are masked at any onetime.

(d) Space-domain and transform-domain data are usedsimultaneously.

(e) Mean frequencies of transformed masked apertures

01

S!

C,

Ecm

0~'


Mo

n s

C,1P -

C,10 -I

C)


can be computed quite accurately (as opposed to mapdrifts).

(f) The basic multiplexing-phase-derivative equation isexact (no approximations are necessary).

(g) High-order estimates or multiplexing provides more-accurate results and is easily incorporated (contrast with themap-drift method on small subapertures).

(h) The solution structure is identical with the optimal-estimator formalism derived in Section 3. The matrix P inEq. (3.18) must be replaced by another matrix, Q, which is

the product of two easily generated matrices, and the mea-surement vector is computed differently.

The following equation is derived in Ref. 21 as the basis ofthe multiplexing method:

J dtSn(t)a2 (t)0(t) = 2 | dJ (coG(,

n = 0, 1, 2,..., N-1, (3.39)

-u

0::

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

UFig. 11. Impulse responses (IPR) for a spectral strength of 10 and a spectral index of 2.5, displayed with unperturbed impulse response, apointing error of 0.53, and a quadratic error of 1.08.

-u

2 I

0

te:-

T; C

Fig. 12. Impulse responses (IPR) for a spectral strength of 40 and a spectral index of 2.5, displayed with unperturbed impulse response, apointing error of 1.07, and a quadratic error of 2.16.

I 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6U



Table 1. Strength of Ionospheric Turbulencea

P d C5 L (mks) C, (mks)b

1.5 1.0 2.6 X 1028 1.3 X 10232.0 10.0 7.9 X 1027 4.0 X 1022

2.5 10.0 1.4 X 1026 6.8 X 1020

2.5 40.0 5.4 X 1026 2.7 X 1021

2.9 3.0 4.9 X 1023 2.5 X 1020

2.9 40.0 6.5 X 1024 3.3 X 1019

2.9 80.0 1.3 X 1025 6.5 X 1019

a For a typical disturbed ionosphere, C, 1021, and C8L - 2 X 1026.b Assuming that L = 200 km.

Equation (3.42) can be cast into the form of Eq. (3.13) byincluding the additive-measurement-noise vector,

(3.47)Z= Qb+ q,

where Q is the N X L matrix,

[Q]n = E [A]nm[!']mj.m

Equation (3.14) implies that the optimal-estimate solutionof Eq. (3.47) is

6 = (B-1 + QTR-lQ)-lQTR-lZ, (3.49)

(3.48)

where S0 (t) is the nth binary mask function, f(t) = a(t)eiO(t)represents the complex aperture data, a(t) = lf(t)l is magni-tude of the aperture data, and ¢(t) = db(t)/dt is the deriva-tive of the phase.

Gn(W) is the Fourier transform of the nth masked aper-ture; i.e.,

Gn(W) = '[Sn 0(t)f(t)]

The uniformly sampled form of Eq. (3.39) is appropriate tothe SAR problem:

M-1

E Sn(m)a2 (in~(i) = 2 >3kIGn(k)I2 * (3.40)m=_ k

m refers to the sample number across the data aperture, andN is the total number of optimal measurement masks; N <M. Substituting the Legendre-series expansion of a specificrealization of the phase error,

L+1

¢(y) = >3 bnYn(Y), (3.41)1=2

into Eq. (3.40) results in

Si(m)a2(M) b)(j m --I) =- L3 klGn((1k)I2,m=0 1=2 k

(3.42)

where

dYn(y)Yn(y) = dy

Equation (3.42) can be written as the matrix equation,

AYb = Z, (3.43)

where A is the N X M matrix,

where, as before, B = (bbT) is the covariance matrix for theexpansion coefficients and R = (nn7qT) is the covariance ma-trix for the additive noise. The previous special-case solu-tions of Eq. (3.14) are still valid; e.g., for the diagonal noisecovariance matrix,

R = UN2I,

Eq. (3.49) reduces to

6 = [(d)N2)1 + QTQ] QTZ

where B = dA. For the determined problem, N = L,

6= Q-1Z

6= QZ

(3.50)

(3.51)

(3.52)

where Q is the pseudoinverse of Q.The primary advantage of the binary multiplexing meth-

od is that half of the synthetic-aperture data are retained foreach measurement in determining the expansion coeffi-cients. This is analogous to using fixed-length, but notnecessarily contiguous, subapertures to obtain additionalmeasurements in support of higher-order coefficients. Themain disadvantage of the binary multiplexing method is thatthe intervals at which the multiplexing vectors are nonzeroare independent of the underlying data being multiplexed.In some circumstances, especially for the higher-order mul-tiplexing vectors, the magnitude of the synthetic-aperturedata may be quite small in regions of the aperture where themultiplexing vector is nonzero. There may be poor couplingof the multiplexed data into the matrix equations for theexpansion coefficients, and therefore the system matrix mayapproach a rank-deficient condition. These effects general-ly require maintaining double-precision arithmetic so as notto lose the small multiplexed contributions to the linearalgebraic equations.

[A]nm = S(n(m)a2(M), (3.44)

and Y~ is the M X L matrix,

ll = [Y1(y)]y =(M 1-[m =dy [Y()Y (M-1 2)

n = 0,1,2, . .. , M-1. (3.45)

The measurement vector Z is the N X 1 column matrix,

Z(n) = 2 7 wkIGn(Wh)12 * (3.46)k

Wigner-Ville Distribution MethodThe (auto) Wigner-Ville distribution is defined as22-24

Wf(t, W) = J drej-rf(t + Tr2)f*(t - r12),

where

f(t) = a(t)ej(t)

(3.53)

are the complex aperture data. Notice that the Wigner-Ville distribution is related to the ambiguity function inradar theory by a two-dimensional Fourier transform.



However, it is significant that the Wigner-Ville distributionis purely real (and hence measurable) for arbitrary complexfunctions, f(t), whereas the ambiguity function, even for areal-valued signal, is in general complex. If the support forthe aperture data is assumed to be the real interval (-T/2 •t < +T/2), then the support of the integrand of the Wigner-Ville distribution is the diamond-shaped region shown inFig. 13. The maximum overlap of the signal and its complexconjugate is for t = 0 (the center of the aperture), whereasthe overlap near the ends of the aperture (t - +T/2) is small.In a sense, the time-reversed complex conjugate of the signal(aperture data) acts as the mask (multiplexing vector) of thesignal in a multiplexing scheme. In this sense, there are asmany distinct masks as there are data points. In addition,using the data signal itself as a multiplexing vector essential-ly diagonalizes the system matrix for solving for the expan-sion coefficients. In other words, each expansion coefficientcan be determined by a mathematical operation on the Wig-ner-Ville distribution, and no matrix equations need besolved.

T-r+(T-21t1l)

W(to 4 )=f f(to+T/2)f *(toT/2)&j i1TdT

T--(T-21tol)

-T/2 to +T/2 t

(T/2)

-T +T

f(to+T/2)

-T-2t, +T-2to

f(to-T/2)

-T+2to +T+2to

t

II ~~~T1 2

T

T

T

to---I-+T/2

Region of Overlap(Support at t-to)

Fig. 13. Region of support for the integrand of the Wigner-Villedistribution.

Integrating the Wigner-Ville distribution over all fre-quencies yields the instantaneous power of the signal:

p(t)= = f dcWfQ, c)= If(t)12. (3.54)

The first moment of the Wigner-Ville distribution in fre-quency is proportional to the time derivative of the phase:

Qt = 2 2f(t) J dw[cWf(t, w)] = 0() (3.55)

The derivative of the phase can be expanded in a Legendre-polynomial series,

5 (t) = >3 CnYn(tIT), (3.56)

where1 1/2- 2 dtYn(t/T)Ym(t/) = 6'nm.T -1/2

(3.57)

Equation (3.56) can be inverted with Eq. (3.57), so that, byusing Eq. (3.55), the following expression can be written forthe expansion coefficients:

I 1 1'l2

Yn(tIT)

2 Ir 1 p2 (t y-.(3.58)

This expression for the Legendre coefficients in the expan-sion of the derivative of the phase in terms of a weighteddouble integral of the Wigner-Ville distribution implies thefollowing algorithm for phase-error correction:

(1) Given fi(t), generate Wfi(t, w) with Eq. (3.53).(2) Compute the coefficients of the Legendre-series ex-

pansion of ¢(t) with Eq. (3.58).(3) Calculate the estimated phase,

¢est(t) >3 fTI 2 dt'Yn(t'IT).n

(4) Correct the data (signal):

f1 +i(t) = fj(t)exp[-jkes 5 (t)](.

(5) Go to (1).

These steps are repeated until convergence is attained.

4. DISCUSSION AND CONCLUSIONS

A number of methods of varying complexity and elegance forreducing propagation-induced phase errors have been de-scribed. The most straightforward is the cubic-spline ap-proximation based on either map drift or the mean-frequen-cy difference to obtain estimates of the second derivative ofthe phase at the knots of the spline. The advantage of thepiecewise cubic spline is that measurement errors at oneknot do not propagate to the other knots, in contrast to someof the methods that are based on the Legendre polynomialexpansion, in which unstable behavior can sometimes beexhibited over the full aperture. The main disadvantage ofthe cubic-spline approximation is probably the loss of accu-racy in the estimates of the second derivative of the phase as


I -----------

II


the number of knots increases (the subaperture size de-creases), a problem common to all methods using map driftor the mean-frequency difference.

The remaining methods make use of the Legendre-seriesexpansion of the phase or of its derivative. In propagation-induced phase errors, most of the phase-error energy is con-tained in the low-frequency components, and therefore ex-pansion in an orthogonal basis set such as the Legendrepolynomials, in which lower-frequency components corre-spond to lower-order polynomials, is reasonable. In prac-tice, phase errors corresponding to each successive term inthe Legendre-series expansion would be estimated and re-moved sequentially, beginning with the lowest order, in thatthe lower orders contain the largest contributions. Thereason for choosing the Legendre series over-similar orthogo-nal expansions is that the first few terms of the Legendreseries have a close correspondence with the traditional (inSAR theory) linear, quadratic, etc. components of the phaseerror based on a Taylor-series expansion. Of course, theLegendre series is preferable to the Taylor series in that thelatter is not an orthogonal series.

The Legendre-series expansion of the phase error can beused, in conjunction with the map-drift method or the mean-frequency-difference method of estimating the second de-rivative of the phase at the subaperture boundaries, to re-duce the phase error. In the determined or overdeterminedcase in which the number of subaperture boundaries (mea-surements) is equal to or greater than the number of Legen-dre expansion coefficients, estimating the phase error con-sists of solving (least-squares solution in the overdeterminedcase) coupled linear algebraic equations for the expansioncoefficients. The primary advantage of this method is thatthe Legendre-series expansion permits a systematic ap-proach to progressing from low- to higher-frequency phaseerrors in the correction process. The main disadvantagesare the decrease in the accuracy of the estimate of the secondderivative of the phase as the order of the expansion in-creases (subaperture size decreases) and some possible nu-merical instabilities unless the problem is overdetermined(see part II of this series7 ).

Knowledge of the full covariance matrix of the Legendreexpansion coefficients for a power-law phase-error spectrumis useful for computing ensemble-averaged properties of co-herent systems such as the ensemble-averaged impulse-re-sponse function for a SAR. The decomposition of the im-pulse-response function into contributions from the individ-ual terms in the Legendre-series expansion of the phaseerror clearly demonstrates the relationship between thesidelobe structure of the impulse response and the order ofthe Legendre-series contribution. Low-order Legendre-se-ries terms, which consist primarily of low-frequency compo-nents, contribute to the main lobe and close-in sidelobestructure, and, as theorder of the Legendre-series termsincreases, the contribution moves to more-distant sidelobes.For a steep phase-error spectrum (large spectral index p),most of the energy is contained in the low-order Legendreterms and is manifested in the perturbed impulse-responsefunction as having most of the distortion in the main lobeand close-in sidelobes. In contrast, for a shallow spectrum(small spectral index), much of the phase-error energy re-sides in the higher-order Legendre terms and therefore inthe more-distant sidelobe structure. The implication for

phase-error-removal schemes based on the Legendre-seriesexpansion (or any other expansion whose higher-order termscorrelate with higher-frequency components of the phasesuch as the Taylor series) is that convergence of the series ismore rapid for a steep spectrum.

Linear mean-square-estimation theory is used to derivean optimal estimate of the Legendre expansion coefficients,given a few measurements with the map-drift method or themean-frequency method and knowing the covariance matrixfor the random phase error. The statistics of the measure-ment error and other additive-noise sources are also takeninto account. The estimate is optimal in that it is uncorre-lated with the error in the estimate, and, if the estimate andthe error have Gaussian probability distributions, they arealso independent.' 5 An iterative scheme based on the co-variance matrix for the error in the estimate of the Legendrecoefficients is presented. The optimal-estimator methodmay have applications in high-noise environments where theaccuracy of the measurements is severely limited or in esti-mating certain averaged properties of the phase error.

The primary advantage of the binary multiplexing meth-od appears to be that half of the synthetic-aperture data areretained for each measurement in determining the expan-sion coefficients. The main disadvantage is that the inter-vals at which the multiplexing vectors are nonzero are inde-pendent of the underlying data being multiplexed. Theremay be poor coupling of the multiplexed data into the ma-trix equation for the expansion coefficients, and thereforethe system matrix may approach a rank-deficient condition.

The Wigner-Ville distribution is a formally elegant math-ematical function that characterizes a signal in time andfrequency and has the advantage of being real even for com-plex signals. The Wigner-Ville method provides an esti-mate of the true phase derivative, which contains not onlythe assumed low- and medium-frequency phase errors butalso the high-frequency target structure. Some form of op-timal filtering or more-limited support of the integrandprobably needs to be performed within the Wigner-Villeformalism2 5 in order to limit the information content to thatconsistent with the order of the phase error to be estimated;otherwise, the high-frequency target-structure phase couldcause numerical problems in an actual implementation.

The elegance and the power of the Wigner-Ville distribu-tion are impressive in that all the information about thesignal structure is contained in the distribution. This com-plete information may be too much for determining thelower-order terms in the Legendre-series expansion; i.e., toreduce the computational burden and to improve conver-gence for the lower-order terms, some sort of low-pass filter-ing or smoothing, or, equivalently, a smaller region of sup-port for the integrand of the Wigner-Ville distribution,.maybe propitious. As higher-order terms are calculated, theregion of support could be appropriately enlarged.

The auto-Wigner-Ville distribution (both functions in theintegrand correspond to the same signal) is used in thispaper. A generalized cross-Wigner-Ville distribution per-mits two different functions in the integrand. Formally, theless-elegant correction methods (binary multiplexing, mean-frequency difference, etc.) should be derivable from thecross-Wigner-Ville distribution if the appropriate masking-type function is used along with the corrupted signal in theintegrand.



Conceptually, all the methods described in this papercould be applied to incoherent systems (where explicit phaseinformation is not available) when coupled to practicalphase-retrieval algorithms. 2 6' 27

APPENDIX A: DERIVATION OF EXPRESSIONFOR COVARIANCE MATRIX

Consider the random phase error 0(x) along an aperture oflength A,

¢(x) for -A/2 < x < A/2.

For now, it is assumed that the phase error is weakly station-ary and that it is defined with respect to its mean; i.e.,

(f(x)) = 0. (Al)

By the definition of weak stationarity, the autocorrelationfunction depends only on the difference (x - x'):

(0(x))(x')) = C/(x - x').

Expand the phase error in a Legendre series:

k(x) = > bnYn(y), y = x/A,n=O

(A2)

because Yn(-y) = (-1)nYn(y). Therefore

(bnbm) = 0 forn + m odd. (A10)

The autocorrelation function can be expressed in terms ofthe power spectrum of the phase errors by using the Fourier-transform relation,

CQY1 - Y2) = J dw cos[A(y, - y2 )]G(w). (All)

Substituting Eq. (All) into Eq. (A7) yields

I1/2 1/2(bnbm) = f dwG() | dylYn(y,) | dY2 Ym(y 2 )

Jo J -1/2 J -1/2

X cos[Aw(y, - Y2)]- (A12)

Expanding the cosine leads to

cos [Aw(yl - Y2)] = cos Awy, cos AwY2 + sin Aowy1 sin Acoy2.

(A13)

Substituting Eq. (A13)the form

A3) j2 dyYn(y)cos Acoy

-1/2

into Eq. (A12) leads to integrals of

where1/2

f-/ 2 dyn(Y)Y(Y) = n-

+1)/ jn(2 )(A4)

Equation (A3) can be inverted for the expansion coefficientsby using Eq. (A4):

1/2bn = | dyYn(y)q5(Ay).

J -1/2

Substituting Eq. (A3) into Eq. (Al) yields

(O(x)) = 0 = _ (bn)Yn(y).

(A5)

for n evenr n

for n odd(A14)

J1/2dyYn(y)sin Acwy

-1/2

+ 1) /i( 2 )

for n even. (A15)

for n odd

This can be true for all y if and only if

(bn) = 0 for all n. (A6)

From Eq. (A5), the second moments of the coefficients are1/2 1/2

(bnbm) 2 dYlYn(Y) j dY2 Ym(Y2 )C(Yl - Y2), (A7)

where

C(y1 - Y2) - C(x - x 2) = (W(AyO)O(Ay2)). (A8)

Weak stationarity implies that the parity of the autocorrela-tion function is even:

C(y 1 - Y2 ) = C(Y2 - Y1). (A9)

From Eq. (A.9) it follows that (bnbm) vanishes if n + m isodd. Thus, by letting y" - -Yi and Y2 - -Y2 in Eq. (A7), weobtain

1/2 (1/2

(bnbm) = dyYn(-Yl) dY2Ym(-y 2 )C(Y 2 - Y1 )J-1/2 J 1/2

r1/2 ,l/2= (-l)n+m I dy1Yn(yl) | dy 2Yn(y 2)C(yl - Y2)

J=-1/2 Jb-1/2

= (-l)n+m(bnbm)

Therefore, for n + m even, Eq. (A12) can be written as

(bnbm) = (_l)6(_l)(n+m)/2 (2n + 1)1/2 (2m + 1)1/2

(A16)

where

3={ 1 for n and m evenfor n and m odd

Equation (A16) is the same as Eq. (2.14) in Section 2. For m= n, Eq. (A16) reduces to

(bn2 ) = (2n + 1) J dWG(C)in (2 (c) (A17)

Equations (A16) and (A17) are now specialized to the caseof a power-law spectrum:

r(p)cos(r2p) d

GMc = - 7rAP- P (A18)

An objection could be raised at this point, in that Eq. (A16)is derived by assuming the existence of an autocorrelationfunction, whereas an autocorrelation function correspond-


X dwG(W)jn Aw - Awfo, ( 2 H 2 ),


ing to the spectrum, Eq. (A18), does not exist. The spec-trum in Eq. (A18) corresponds to a nonstationary randomphase error whose first increments are weakly stationary.Therefore the Fourier transform of the spectrum must beexpressed in terms of the structure function,

(bn2 ) =- 2P dr(p)cos 2P Ia Jn+1/22(0^ (A25)

where f = Aw/2.The integral in Eq. (A25) is taken from p. 692 of Ref. 28:

D "(X- x') = ([0(x) - (X)]2). (A19)

This objection can be handled by using a common artifice.A small-wave-number cutoff (an outer-scale size, ro) is intro-duced into the power-law spectrum in the derivation of Eq.(A16); i.e.,

G(ro; )= C(r0)d (A20)

where C(ro) is a normalization constant. This ensures weakstationarity and hence the existence of an autocorrelationfunction. The limit ro - is taken after the derivation ofEq. (A16).

Substituting Eq. (A18) into Eq. (A16) yields

r(p)cos 2p(bnbm) =--(Pcos(-)P d(2n + 1)1/2

X (2m + 1)1/2 J p+ Jn+l/ 2 ( 2 )Jm+l/2 ( 2)

(A21)

where use is made of the following relationship betweenspherical and cylindrical Bessel functions:

in( 2 -)~ (Ao) Jn+l/2( 2 ) (A22)

The integral in Eq. (A21) is found on p. 692 of Ref. 28 [Eq.(6.574.2)]:

J dwo (A\ (Acxw AP

o P+i Jn+l/ 2 2 )Jm+l/2 2 -2P+

X

F(p+ l)F(n+m+l-p)lm-n+p+2 \(n-m+p+2 /r(n+m+p+3 3

2 )2 2 )k2

(A23)

with the restriction that m + n > p - 1, or, since 1 < p < 3,that m + n 2 2. Substituting Eq. (A23) into Eq. (A21) andusing the relationships between gamma functions and theexpansions of gamma functions described, for example, inChap. 6 of Ref. 29 and in Sec. 8.3 of Ref. 28, results in thefollowing expression for the off-diagonal terms:

o /P+l Jn+1/22 (o =

r(p + l)r(2n + 1 - p)

2P+1P2(P + 2)r(2n + p + 3)'(A26)

with the restriction that n > 1. Substituting Eq. (A26) intoEq. (A25) and expanding the gamma functions results in thefollowing expression for the mean square of the Legendrecoefficients:

On2) = (2n + l)d

(2n-1-p)(2n-3-p) .*. (3-p)(p-1)(2n + p + 1)(2n + p -1)(2n + p -3) ... (p + l)p

(A27)

Equation (A27) can be written as a recursion relation:

(b2)= 2 1 2n + l +P(b 2),

(b, 2) = 3d p - I 1(p + 3)(p + l)p (A28)

Equations (A28) are identical to Eqs. (2.15) in Section 2.Equation (A24) can also be derived readily in coordinate

space.14

ACKNOWLEDGMENTS

The authors wish to express their sincere thanks to thereferees for their useful and constructive suggestions. Thisresearch was performed at Sandia National Laboratoriesand was supported by the U.S. Department of Energy undercontract no. DE-AC04-76 DP00789, by the Defense NuclearAgency under contract no. 87-827, and by the U.S. Air ForceSpace Technology Center under contract no. 87-32.

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Some methods for reducing propagation-induced phase errors in coherent imaging systems. I. Formalism

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