Speckle Cross-Correlation in Multilook SAR Images of Dynamic Sea Surface Processed with Partially Overlapped Sub-Reference Signals Haipeng Wang and Kazuo Ouchi Department of Environmental Systems Engineering, Kochi University of Technology, Tosayamada-cho, Kochi 782-8502, Japan Email:
[email protected], and
[email protected] Abstract— A general expression is derived and discussed for the interlook cross-correlation function (CCF) of speckle patterns in synthetic aperture radar (SAR) multilook images of dynamic ocean surfaces processed with partially overlapped sub-reference signals at arbitrary Doppler center times. Provided that the backscattered field is statistically “white”, the CCF of the speckle intensity is given by the squared modulus of the autocorrelation function of the amplitude weighting functions of sub-reference signal where the time lag is the center time difference. For such Gaussian speckle patterns, the CCF of the interlook speckle patterns is independent of the surface coherence time of randomly moving dynamic sea surfaces. From the integral equation, an analytic expression is derived for a rectangular sub-reference weighting. Comparisons of the theoretical CCFs with those estimated from the RADARSAT-1 C-band and JERS-1 L-band images of the coastal waters of Japan show good agreement.
I. INTRODUCTION In the multilook processing of SAR data, the full synthetic aperture is divided into two or more subapertures which do not overlap. The intensity speckle patterns of the interlook images are, therefore, uncorrelated, and the addition of the images on an intensity basis reduces speckle noise (e.g. [1], [2], [3]). However, very little, to the authors’ knowledge, is reported on the theoretical and experimental studies of the interlook speckle correlation between the multilook images produced with partially overlapped sub-reference signals (or subapertures), except the theoretical study of Ouchi and Burge [4] on the CCF for Gaussian amplitude subaperture weightings. Their results also indicate that if the backscattered field is modeled as a Gaussian white noise, the CCF is independent of a scene coherence time of randomly fluctuating sea surfaces. The theory is supported by the recent paper using RADARSAT-1 data [5]. In this short paper, a general integral expression is derived and discussed for the intensity CCF of speckle patterns produced by subapertures of partially overlapped bandwidth at arbitrary center times. An analytic expression is then derived for a rectangular weighting function and compared with the RADARSAT-1 C-band and JERS-1 L-band data. For simplicity, consideration is given only in the azimuth direction where the multilook processing is applied and motion effects generally appear. 0-7803-9050-4/05/$20.00 ©2005 IEEE.
II. BASIC THEORY A. Siegert Relation When a statistically homogeneous random rough surface is imaged by a SAR, a classical “Gaussian” speckle pattern is formed, provided that more than about 5 scatterers exist within a resolution cell to satisfy the central limit theorem. The amplitude and intensity of such speckle patterns are described by the Rayleigh and negative exponential probability density functions respectively [1], [2]. It is well known [3] that the intensity CCF of the Gaussian speckle in multilook images can be described by the Siegart relation I(X; Tn ) I(X; Tn ) = I2 +|A(X; Tn )A∗ (X; Tn )|2 (1) where the angular brackets indicate taking an ensemble average, I(X; Tn ) is the intensity at an image position X produced by the subaperture of center time Tn , and A is the image complex amplitude. B. Multilook Image Complex Amplitude In order to evaluate the intensity CCF of (1), it is necessary first to derive an explicit expression for the image complex amplitude A(X; Tn ) as follows. A SAR antenna transmits a series of pulses and receives return signal as it propagates in the azimuth direction. A signal received at time t from a point scatterer at an azimuth surface position x is given by Es (x, t) = as (x)at (t|x) exp (−i2kr(x, t))
(2)
where k = 2π/λ is the wavenumber, λ is the wavelength, and r is the distance between the radar and the scatterer. The scattering amplitude is factorized to the spatial field as (x) and its temporal change at (t|x), where the temporal change is conditional upon the scatter being observed at x, and it is a function of time only [6]. The slant-range distance r can be approximated as follows. r(x, t) = R2 +(V t−x)2 (V t−x)2 (3) R+ 2R In the expression, R is the slant-range distance when the radar is directly above (abeam of) the scatterer, and V is the platform
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velocity. The return signal from an extended surface can then be given by the integral ∞ kV 2 2 (t−x/V ) dx (4) as (x)at (t|x) exp −i Es (t) = R −∞ The complex amplitude of the look n image can be produced by correlating the return signal of (4) with a sub-reference signal Er as ∞ A(X; Tn ) = Es (X/V +t)Er (t)dt (5) −∞
where the sub-reference signal is given by kV 2 2 t dt Er (t; Tn ) = W ((t−Tn )/T ) exp i R
−∞
−∞
(7)
where A0 is a normalizing constant.
For substantially large area, the integration of (11) with respect to x leads to the Dirac delta function δ(t1 − t2 ). Then, integrating with respect to t1 and putting t = t2 , (11) can be expressed in the following simple integral. A(X; Tn )A∗ (X; Tn ) ∞ 2 = |A0 | σ W ((t−Tn )/T ) W ∗ ((t−Tn )/T ) dt (12) where all constants have been absorbed in |A0 |2 . Thus, the CCF of speckle complex amplitudes in multilook images centered at arbitrary center times is given by taking the autocorrelation of the amplitude weighting function of subreference signals. The CCF of the intensity speckle patterns of (1) is therefore proportional to the modulus square of (12). The expression of (12), which is the central result of this paper, indicates that the CCF of the interlook intensity speckle patterns is independent of the temporal random fluctuation of a scattering surface. For a rectangular weighting function W ((t−Tn )/T )) = rect ((t−Tn ))/T ) = 1 : −T /2 ≤ t−Tn ≤ T /2
C. Speckle Cross-Correlation Function The intensity CCF given by the Siegart relation of (1) requires the CCF of the complex amplitude. From (7), the integral form of the amplitude CCF is given by ∞ ∞ A(X; Tn )A∗ (X; Tn ) = |A0 |2 dx1 dx2 dt1 dt2 −∞
−∞
·as (x1 )a∗s (x2 )at (X/V + t1 )a∗t (X/V −t2 ) k (X −x1 )2 −(X −x2 )2 · exp −i R 2kV ((X −x1 )t1 −(X −x2 )t2 ) · exp −i R
(8)
The assumption of a white noise approximation for the backscattered field implies that the random phases are distributed uniformly over [0, 2π) and they are uncorrelated to each other and to the amplitudes. The ensemble average of the backscattered field can then be approximated by as (x1 )a∗s (x2 ) = σ δ(x1 −x2 )
(9)
where δ is the Dirac delta function and σ is the (mean) radar cross section assumed constant. The temporal correlation function can be defined as [6] at (t1 |x1 )a∗t (t2 |x2 ) = γ(t1 −t2 )|x1 =x2 0-7803-9050-4/05/$20.00 ©2005 IEEE.
(11)
−∞
−∞
·W ((t1 −Tn )/T ) W ∗ ((t2 −Tn )/T )
−∞
·W ((t1 −Tn )/T ) W ∗ ((t2 −Tn )/T ) 2kV (X −x)(t1 −t2 ) ·γ(t1 −t2 ) exp −i R
(6)
In this equation, W is the amplitude weighting function centered at the center time Tn , and T is the integration time to synthesize the subaperture. Substituting (4) and (6) into (5) yields the complex amplitude of the look n image ∞ ∞ A(X; Tn ) = A0 dx dt W ((t−Tn )/T ) ·as (x)at (X/V +t|x) 2kV k 2 (X −x)t · exp −i (X −x) exp −i R R
Using (9) the integral form of (8) can be simplified by integrating with respect to x1 and setting x2 = x as ∞ ∞ ∗ 2 A(X; Tn )A (X; Tn ) = |A0 | σ dx dt1 dt2
(10)
0 : otherwise
(13)
the intensity CCF is given by I(X; T1 )I(X; T2 )/I2 = 1+|1−∆T /T |2
(14)
where ∆T = |T1 − T2 | is the center time difference between the look 1 and look 2 subapertures. The two speckle patterns in the multilook images are identical when ∆T = 0, and they are, of course, perfectly correlated. They start to decorrelate as the center time difference increases and the overlapping area between the two subapertures decreases. The speckle patterns completely decorrelate when the center time difference equals or greater than the integration time (∆T = T ) because there is no overlapping area between the two subapertures. In spaceborne SARs such as RADARSAT-1 and JERS-1 SAR, the integration times of full single look are approximately 0.5-2 s so that the integration times of N -look range from 0.5/N to 2/N , while the coherence (decorrelation) time of sea surfaces is order of 0.1-0.05 s or less [7], [8]. An extra look could then be extracted if the multilook speckle patterns were to decorrelate according to the sea surface coherence time [9]. However, this cannot be done because, from the result of (12) and (14), the CCF of intensity speckle patterns is independent of the scene coherence time. The situation is different for deterministic targets and scatterers that give rise to non-Gaussian intensity fluctuations [3],
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Fig. 2. The probability density function (PDF) of speckle amplitude corresponding to the area ”A” in Fig.1. The vertical axis is the relative number of pixels and the horizontal axis is the amplitude in digital number. The measured amplitude PDF represented by the step line closely follows the Rayleigh distribution of the smooth curve.
RADARSAT-1 single-look intensity image of the Kumano Sea. The Size is 24 km and 55 km in azimuth and range respectively. The white boxed areas marked “A”, “B”, “C” and “D” of size 100 × 100 pixels are used for the analysis.
Fig. 1.
[4], [10], [11]. It can easily be shown that the images of corner reflectors, for example, have high degrees of interlook correlation even if the center time difference is greater than the integration time of a subaperture. The suppression of the background Gaussian speckle while keeping the intercorrelated targets is indeed the basis of the image enhancement by multilook processing. If the received signal from an active reflector has temporal fluctuations, the multilook images of the target decorrelate with increasing center time difference. If, therefore, the assumption is made that the backscattered field is statistically “non-white” arising from a few such randomly fluctuating scatterers, then the scene coherence time should enter the noise statistics. This latter case can be found in the sea surface under moderate to strong wind conditions, where the randomly varying images show the non-Gaussian behavior. III. EXPERIMENTAL RESULTS AND DISCUSSION
B. Statistical Analysis
A. RADARSAT-1 and Meteorological Data Fig.1 shows the RADARSAT-1 single-look intensity image (Standard 1 mode) over the coast of the Yoshino-Kumano National Park and the Kumano Sea (scene center: 33◦ 34’ N, 136◦ 00’ acquired at 21:05 GMT on the 28th of October 1998. The size is approximately 24 km in azimuth and 60 km in range. The white boxed areas marked as “A”, “B”, “C” and “D” of each size 100 × 100 pixels are used for analysis. The meteorological data from the Japan Weather Association at the time three hours prior to the RADARSAT-1 pass indicate the 0-7803-9050-4/05/$20.00 ©2005 IEEE.
wind speed of 3 m/s from the northwest and 0.9 m significant waveheight with period of 8 s, propagating from the southeast. The wave data correspond to the deep sea approximately 60 km in the southeast away from the center of Fig.1 [5], and no clear wave images are seen in the coastal sea shown in Fig.1, except some wind and frontal features and some ships with their wake lines. In the present analysis, 11 sets of images of the same area shown in Fig.1 are processed from the raw data using the subreference signal having the bandwidth of 500 Hz and the center frequencies ranging from -250 Hz to 250 Hz with the center frequency interval of 50 Hz. The radar wavelength is 5.6 cm (5.3 GHz at C-band HH polarization), the relative platform velocity is 7.0 km/s and the slant-range distance is 983.5 km with the incidence angle of 27◦ . Hence, from the relation t = λR fD /(2V 2 ) between the Doppler frequency fD and azimuth time t, the integration time of each look and interlook time difference can be calculated as 0.28 s and 0.028 s respectively. The corresponding azimuth resolution cell is 14 m with range resolution cell of 25 m. The intensity images which appear to be statistically homogeneous, i.e., “A”, “B”, “C” and “D” in Fig.1, are extracted and cross-correlated.
Before taking interlook cross-correlation, the probability density function (PDF) is calculated to ensure that the amplitude fluctuations constitute the Gaussian speckle pattern. As can be seen in Fig.2, the estimated PDF of the area “A” is very close to the Rayleigh distribution. Similar good agreement is obtained for the other areas (not shown in this paper). Fig.3 shows the cross-correlation function of intensity speckle patterns as a function of center time difference. The solid line is the theoretical curve of (14). It can be seen that the degree of cross-correlation decreases with increasing
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Fig. 3. The interlook cross-correlation function of the area ”A”, ”B” , ”C” and ”D” in Fig.1 as a function of the center time difference. The solid line is the theoretical curve based on (14).
center time difference. However, the measured values tend to over-estimate the theoretical curve, mainly because of spatially correlated features. A close inspection of interlook CCF using a small moving window indeed confirms this [12]. There are no simultaneous sea truth data on the scene coherence time, but in general the coherence time is approximately 0.1 s or less [7], [8]. Therefore, if the scene coherence time were to enter the interlook cross-correlation, then the measured CCF in Fig.3 should have been lower than the theoretical curve. The result shown in Fig.3 is an evidence that this is not the case, and that the interlook CCF of speckle patterns is independent of the random temporal fluctuation of the sea surface under the condition that the backscattered spatial field is statistically white. Fig.4 shows the interlook CCF values of the four test sites selected from the JERS-1 L-band SAR data over the Tsugaru Straight in Japan as a function of the center time difference [12]. It can be seen that the CCF values computed from the JERS-1 L-band data show excellent agreement with the theory, except one set of data due again to the correlated features in the area. IV. C ONCLUSIONS In the present paper, we have derived a general integral expression for the intensity CCF of interlook speckle patterns in the multilook subimages processed with partially overlapped subapertures, and a simple analytical expression is derived for a rectangular subaperture weighting. The result shows that, for statistically white backscattered fields which gives rise to the classical Gaussian speckle, the CCF depends only on the amplitude weighting and center time difference of subapertures; and it is not dependent on the temporal fade process. This theoretical conclusion is supported by the Radarsat-1 Cband and JERS-1 L-band SAR images over the coastal waters of Japan. 0-7803-9050-4/05/$20.00 ©2005 IEEE.
The inter-look cross-correlation function of selected areas in the JERS-1 L-band data over the Tsugaru Straight in Japan as a function of the center time difference [12]. The solid line is the theoretical curve based on (14).
Fig. 4.
ACKNOWLEDGMENTS The Canadian Space Agency/ Agence spatiale canadienne are the owners of the RADARSAT-1 data used in this paper. The RADARSAT-1 raw data were purchased by the Mitsubishi Heavy Industries Ltd., and processed at Kochi University of Technology. R EFERENCES [1] J.C.Dainty (ed): Laser Spackle and Related Phenomena, New York: Springer, 1975. [2] J.S.Lee and I.Jurkevich, “Speckle filtering of synthetic aperture radar imagery: a review,” Remote Sens. Reviews, vol8, pp.313-340, 1994. [3] C.Oliver and S.Quegan, Understanding Synthetic Aperture Radar Images, Artech House: London, 1998. [4] K.Ouchi and R.E.Burge, “Speckle cross-correlation function in multilook SAR images of moving discrete scatterers,” Int. J. Remote Sens., vol.12, pp.1933-1946, 1991. [5] K.Ouchi, S.Tamaki, H.Yaguchi, and M.Iehara “Ship detection based on coherence image derived from cross-correlation of multilook SAR images” IEEE Trans. Geosci. Remote Sens. Lett., vol.42, pp.184-187, 2004. [6] R.K.Raney, “SAR response to partially coherent phenomena,” IEEE Trans. Antennas Propagat., vol.28, pp.777-787, 1980. [7] R.E.Crande, “Estimating ocean coherence time using dual-baseline interferometric synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens., vol.32, pp.846-854, 1994. [8] S.J.Frasier and A.J.Camps, “Dual-beam interferometry for ocean surface current vector mapping,” IEEE Trans. Geosci. Remote Sens., vol.39, pp.401-414, 2001. [9] R.K.Raney and P.W.Vachon, “Synthetic aperture radar imaging of ocean waves from an airborne platform: focus and tracking issues,” J. Geophy. Res., vol.93, C10, pp.12,475-12,486, 1988. [10] K.Ouchi: Principle of Synthetic Aperture Radar for Remote Sensing (in Japanese) Tokyo Denki University Press: Tokyo, 2004. [11] K.Ouchi and R.A.Cordey, “Statistical analysis of azimuth streaks observed in digitally processed CASSIE imagery of sea surface,” IEEE Trans. Geosci. Remote Sens., vol.29, pp.727-735, 1991. [12] K.Ouchi and H.Wang, “Interlook cross-correlation function of speckle in SAR images of sea surface processed with partially overlapped subapertures,” IEEE Trans. Geosci. Remote Sens., vol.43, pp.695-701, 2005.
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