大内和夫（高知工科大学）
Kazuo Ouchi(Kochi University of Technology)
E-mail: ouchi@env.kochi-tech.ac.jp

Abstract : Interferometric synthetic aperture radars (InSARs) can be operated in the cross-track and along-track modes. The former is used for topographic mapping and measurements of crust movements, and also investigating the spatial and temporal properties of scattering media. The latter can be applied to the velocity measurement of ocean currents and coherence (life) times of small-scale ocean waves. The analyses common in both InSAR data are based on the geometrical approach, in which the interferometric phase is calculated from the path difference between the first antenna from a scattering surface and the second antenna from the same surface. In practice, however, the interferogram is produced from the processed SAR images, but not from the received signals at the antennas. Accordingly, the geometrical approach, in strict sense, is not rigorous particularly for the analysis of coherence images, although it is a good approximation for interpreting the phase images. The present article emphasizes this point and presents a summary of the rigorous theory of understanding the interferogram by taking into account of the SAR image forming process.

干渉合成開口レーダ(InSAR)にはクロス・トラックとアロング・トラックの２種類があり、前者は地形図の作成や地殻変動等の計測と散乱体の時空間的変化の探査に利用され、後者は海流の流速と小さなスケールの海洋波（さざ波）のコヒーレンス（形状維持）時間の測定に使われる。これらのInSARデータの解析に共通な点は、２台のアンテナと散乱面の距離の違いから位相差を算出するといった幾何学的なアプローチを基礎としていることである。しかし、実際のインタフェログラムは、アンテナでの受信信号の位相差ではなく、生データから処理されたSAR複素画像を使って生成される。従って、幾何学的アプローチは位相画像の解析には良い近似ではあるが、コヒーレンス画像の解析には厳密な意味では適していないと言える。本プレゼンテーションでは、この点を強調し、インタフェログラムを定量的に理解するのに必要なSAR画像生成過程を考慮に入れた厳密理論の要約を提示する。

1. Cross-Track InSAR

The principles and applications of the cross-track InSAR are well known [e.g. 1] and well established for the practical use of routine operations. There is a space of further research and improvements to be made on the phase unwrapping techniques and the quantitative description of the relation between the interferometric coherence and the scattering media. The problem on the phase unwrapping is that there are a large number of proposed techniques, each of which is claimed as being superior over the others. At present, the path-following method appears to be popular, although other methods can be advantageous depending on the quality of interferogram [2]. The quantitative analysis of interferometric coherence in terms of the scattering media involves the EM scattering by rough surface and volumatic material. The coherence image does contain useful potential information on the spatial and temporal properties of scattering media, but the complex nature of the scattering theories tends to prevent the detailed understanding of the process.

  The general geometrical approach to relating the interferometric phase and surface topography has a limitation that it cannot provide the detailed analytic description for the coherence image, although it is a good approximation for the relation between the phase image and the spatial and temporal changes of surface height. In order to make further progress, it is necessary to understand the basics of conventional SAR image formation, and consequent processes of producing interfrograms. According to the convolution model [3], the SAR complex image is given by the convolution of the impulse response or point spread function (PSF) with the backscattered complex field. Once the SAR complex image is given, then the integral expression for the complex interfrogram can be derived based on several assumptions including the white noise approximation for the backscattered random fields. For surfaces of complicated undulations, analytic solutions are difficult to derive and a simulation can be useful for such cases. Analytic solutions for both the phase and coherence images are possible for simple surface undulations such as a linearly tilted surface. For such case, the following results for the interferometric phase can be shown.
 

1. The expression for the phase derived from the geometric approach is close to that of the solution derived from the convolution model.

2. The difference between the two approaches becomes large for the surface tilted towards
the radar at angles larger than the incidence angle.
 

1. The signal to noise ratio associated with the additive system noise: this term defines the lower limit of the achievable coherence.

2. The baseline or antenna separation (baseline decorrelation): this term is crucial for the design of SAR system.

3. The spatial random height changes in surface scattering (the coherence decreases with increasing surface roughness, but this term can be ignored if the roughness is small compared with the ground-range resolution scale).

4. The coherence decreases with increasing change in the random component of the backscattered fields between passes (temporal decorrelation): this can be due to the physical change of the surface roughness structure or the roughness structure inside the scattering media, and to the difference of the random components caused by the difference in incidence angles (although the viewing angle difference is very small, it could introduce substantial variations in the backscattered random fields in volume/multiple scattering). The loss of coherence becomes critical if the roughness change becomes comparable with the radar wavelength.

5. The loss of coherence increases as the deterministic change in surface height becomes comparable with the range resolution cell.

6. For range-tilted surfaces, the coherence increases as the tilt angle tends to increase away from the radar; and it decreases as the surface tends to tilt towards the radar. The total loss of coherence occurs when the tilt angle equals the incidence angle.

7. For azimuth-tilted surfaces, the coherence decreases with increasing tilt angles, but the loss of coherence is not significant in comparison with the range-tilted surfaces.

2. Along-Track InSAR

The along-track InSAR utilizes the change in the Doppler shift between the signals received by the front and back antennas placed in the along-track (azimuth) direction on an aircraft.. Its main application is to the measurements of ocean currents, life-times of ripple waves and the dynamics of ocean waves. The geometrical approach is again used to interpret the phase image, though several papers [e.g. 4] take account the image forming process. From the rigorous theory, the following results for the linearly varying ocean currents can be obtained.

 
1. When the slant-range current changes linearly in azimuth (current shear zone), the velocity gradient derived from the InSAR phase is different from the true gradient. This is caused by the spatially varying azimuth image shift.

2. The degree of coherence is governed by the bandwidth of the azimuth reference signal and the mean life-time of ripples.
 

References
 [1] R. Gens, and J.L. Van Genderen, "SAR interferometry-issues, techniques, applications,"  Int. J. Remote Sens., vol.17, pp.1803-1835, 1996.
 [2] D.C. Ghiglia, and M.D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithm, and Software, (New York: John Wiley and Sons), 1998.
 [3] J.C. Curlander, and R.N. McDonough, Synthetic Aperture Radar: Systems and  Signal Processing, (New York: Wiley), 1991.
 [4] L. Shemer, M. Marom, and D. Markman, "Estimates of currents in the nearshore ocean region using interferometric synthetic aperture radar,"  J. Geophy. Res., vol.98, pp.7001-7010, 1993.
 [5] K. Ouchi, "A theory on the distribution function of backscatter radar cross section from ocean waves of individual wavelength," IEEE Trans. Geosci. Remote Sens., (in press).