Earth, Moon and Planets

DOI: 10.1007/BF00055161



A mathematical theory and a corresponding algorithm have been developed to derive topographic maps from radar images as photometric arrays. Thus, as radargrammetry is to photogrammetry, so radarclinometry is to photoclinometry. Photoclinometry is endowed with a fundamental indeterminacy principle even for terrain homogeneous in normal albedo. This arises from the fact that the geometric locus of orientations of the local surface normal that is consistent with a given reflected specific-intensity of radiation is more complicated than a fixed line in space. For a radar image, the locus is a cone whose half-angle is the incidence angle and whose axis contains the radar. The indeterminacy is removed throughout a region if one possesses a control profile as a boundary-condition. In the absence of such ground-truth, a point-boundary-condition will suffice only in conjunction with a heuristic assumption, such as that the strike-line runs perpendicularly to the line-of-sight. In the present study I have implemented a more reasonable assumption which I call 'the hypothesis of local cylindricity'. Firstly, a general theory is derived, based solely on the implicit mathematical determinacy. This theory would be directly indicative of procedure if images were completely devoid of systematic error and noise. The theory produces topography by an area integration of radar brightness, starting from a control profile, without need of additional idealistic assumptions. But we have also theorized separately a method of forming this control profile, which method does require an additional assumption about the terrain. That assumption is that the curvature properties of the terrain are locally those of a cylinder of inferable orientation, within a second-order mathematical neighborhood of every point of the terrain. While local strike-and-dip completely determine the radar brightness itself, the terrain curvature determines the brightness-gradient in the radar image. Therefore, the control profile is formed as a line integration of brightness and its local gradient starting from a single point of the terrain where the local orientation of the strike-line is estimated by eye. Secondly, and independently, the calibration curve for pixel brightness versus incidence-angle is produced. I assume that an applicable curve can be found from the literature or elsewhere so that our problem is condensed to that of properly scaling the brightness-axis of the calibration curve. A first estimate is found by equating the average image brightness to the point on the brightness axis corresponding to the complement of the effective radar depression-angle, an angle assumed given. A statistical analysis is then used to correct, on the one hand, for the fact that the average brightness is not the brightness that corresponds to the average incidence angle, as a result of the non-linearity of the calibration curve; and on the other hand, we correct for the fact that the average incidence angle is not the same for a rough surface as it is for a flat surface (and therefore not the complement of the depression angle). Lastly, the practical modifications that were interactively evolved to produce an operational algorithm for treating real data are developed. They are by no means considered optimized at present. Such a possibility is thus far precluded by excessive computer-time. Most noteworthy in this respect is the abandonment of area integration away from a control profile. Instead, the topography is produced as a set of independent line integrations down each of the parallel range lines of the image, using the theory for control-profile formation. An adaptive technique, which now appears excessive, was also employed so that SEASAT images of sand dunes could be processed. In this, the radiometric calibration was iterated to force the endpoints of each profile to zero elevation. A secondary algorithm then employed line-averages of appropriate quantities to adjust the mean t

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Earth, Moon and Planets
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Kluwer Academic Publishers
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