The December 13, 1995, Eos “In Brief” column refers to the “new” establishment of knowledge of the geometrical dependence of the Moon's diffuse reflectivity for phase angles under 1.5°, normally inaccessible to Earth‐based observation because of lunar eclipse, as derived by Bonnie Buratti using Clementine data. This knowledge was first “new” 26 years ago when my colleagues and I derived it as members of the Apollo Orbital Photographic Reconnaissance Team [Pohn et al., 1969; Wildey and Pohn, 1969]. Although a good deal larger than Clementine and in a lower lunar orbit, the Apollo Command Module was also a negligible source of shadow as seen from the lunar surface against a 1/2° wide Sun and thus permitted the derivation without worry of eclipse by the 2° wide Earth.

Astronomers routinely violate the directive to sample surface brightness with at least twice the frequency of the highest spatial frequency of the Fourier transform of the continuous image, when doing direct CCD imaging. It is reasonably speculated that this practice is rationalized on the basis that the CCD does not actually sample the surface brightness at periodic intervals, but instead integrates the surface brightness over contiguous regions (the CCD pixels). It is herein derived that this mode of sampling changes the form of aliasing error, but the aliasing error is nevertheless present when undersampling occurs. The very nature of the error betrays the possibility of detecting its presence, a priori. It would be of value to develop an active optical apodizer to accommodate a given CCD, in terms of the Nyquist criterion, without the need to abandon either the full light-gathering power of the telescope or the plate scale at the chosen observing station.

The project to develop a line-integral approach to 2-dimensional radarclinometry and to bring it to the status of producing topographic maps from real radar images has been concluded. The final developments of the theory itself have involved a trial-and-error resolution of the curvature decision process at each integration step over range as follows: (1) Locally Indeterminate Azimuth-Azimuth Curvature is invoked if the range-directed path of integration is within 1 ‡ in angle of the tangent to a local characteristic curve of the partial differential equation of radarclinometry (equivalent to a lapse in the necessity for an auxiliary curvature assumption); (2) Local Cylindricity is invoked if the local image isophote has a radius-of-curvature greater than 50 pixels; (3) Least-Squared Local Sphericity is invoked if the characteristic curve trends at greater than 70 ‡ to the range direction (the auxiliary curvature assumption is becoming a sufficiently strong influence as to warrant the overconstraint), and (4) the default hypothesis, which is invoked most often, is the localization through the Euler/Lagrange equation from the calculus of variations of the global principle of minimization of the surface area of the terrain. The development of the set of line integrals into a 2-dimensional topographic surface is not practically achieved by branching the line integral at the range threshold, because the radarclinometry equations are too frequently coupled but weakly to the slope component in the direction of radar-azimuth, and under circumstances for which the powerfully influential auxiliary curvature assumption is too unrealistic. In other words, a line integration in radar-azimuth is far more frequently directed orthogonally to the local characteristic curve than is one carried out over range. Such orthogonality results in stepping the strike under the exclusive control of the curvature assumption. Instead, a quasi-surface-integration step is taken by modeling the dependence on initial strike of the gravitational potential energy of the vertical slab of terrain under the range-profile. The adopted starting strike for the range integral is the one which minimizes the gravitational potential energy. This radarclinometric method, in combination with my recently published method for determining an effective radar back-scattering function from one-dimensional slope statistics and image pixel-signal statistics, was applied to three images. First, to separate theoretical difficulties from experimental impediments, an artificial radar image was generated from a topographic map of the Lake Champlain West quadrangle in the Adirondack Mountains. Except for the regional trend in elevation, to which radarclinometry is insensitive by design, the agreement between the original and derived topography appears good. The morphologies agree and the range of relief is the same to within 4%. As an example of data of the highest quality available from space-borne radar at the present time, a SIR-B image of very rugged terrain in the coastal mountains of Oregon was similarly processed. The result, after filtering to redistribute photoclinometric errors about the two-dimensional spatial spectrum, agrees with ground truth almost as well. As an example of the worst possible data, in terms of signal-to-noise ratio and radar incidence angle (no detraction from the praise due the first high resolution space-borne radar-imaging of Venus intended), a Venera-15 image segment in Sedna Planitia just north-east of Sapho was processed, using Venera altimetry and Pioneer roughness data for slope statistics, in spite of the resolution mis-match. Considerably more trial-and-error filtering was required. The result appears plausible, but an error check is, of course, impossible.

A widely deployed experiment in freshman physics tests the hypothesis that the magnitude of the central force that accomplishes uniform circular motion is mv2/r (equivalently mrω2). This is accomplished by using a spring rotating about one end as the central force applied to a vertically hanging bob of known mass, tracking a circle of measured radius, at a speed deduced by counting revolutions over time. The spring force is independently measured by determining the gravitational force that produces the same degree of extension of the spring. Because the inertia of the spring is unimportant to the latter measurement, but does matter in the dynamical phase of the experiment, an error is introduced. Students have always been admonished to neglect the effects of the mass of the spring; however, typical parameters used in this experiment imply an error greater than 1%, which exceeds the probable error of which the experiment is capable.

A method is derived for determining the dependence of radar backscatter on incidence angle that is applicable to the region corresponding to a particular radar image. The method is based on enforcing mathematical consistency between the frequency distribution of the image's pixel signals (histogram of DN values with suitable normalizations) and a one-dimensional frequency distribution of slope component, as might be obtained from a radar or laser altimetry profile in or near the area imaged. In order to achieve a unique solution, the auxiliary assumption is made that the two-dimensional frequency distribution of slope is isotropic. The backscatter is not derived in absolute units. The method is developed in such a way as to separate the reflectance function from the pixel-signal transfer characteristic. However, these two sources of variation are distinguishable only on the basis of a weak dependence on the azimuthal component of slope; therefore such an approach can be expected to be ill-conditioned unless the revision of the transfer characteristic is limited to the determination of an additive instrumental background level. The altimetry profile does not have to be registered in the image, and the statistical nature of the approach minimizes pixel noise effects and the effects of a disparity between the resolutions of the image and the altimetry profile, except in the wings of the distribution where low-number statistics preclude accuracy anyway. The problem of dealing with unknown slope components perpendicular to the profiling traverse, which besets the one-to-one comparison between individual slope components and pixel-signal values, disappears in the present approach.

Because radarclinometry is fundamentally describable in terms of a nonlinear, first-order, partial differential equation, one expects that it can, in principle, be carried out by direct deterministic integration beginning at a given threshold profile along the azimuthal coordinate. Such a boundary condition could be provided by the altimetry profile obtained on a preceding or succeeding orbital revolution of the radar-bearing spacecraft. Notwithstanding the mismatched resolutions of the radar altimeter and the radar imaging system as planned for the Megallan mission to Venus, there are fundamental considerations, not involving system noise, that influence the possibility of success of this approach. From the topographic map of the Lake Champlain West quadrangle in the Adirondack Mountains of the U.S., a radar image is synthesized. Radarclinometry, in surface integral form, recaptures the topographic map when the applicable radar reflectance function is weakly variable over the range of application, but it diverges beyond a certain point for nominally variable reflectance functions. The effect can be understood by using results from the “shape-from-shading” literature. (This literature is produced by a group within the artificial intelligence community who have been independently attacking, for all practical purposes, photoclinometry, except that they have not given primacy to images of terrain.) The ubiquity of the instability suggests that the value of the surface integral approach is much in doubt.

Radarclinometry, the invention of which has been previously reported, is a technique for deriving a topographic map from a single radar image by using the dependence upon terrain-surface orientation of the integrated signal of an individual image pixel. The radiometric calibration required for precise operation and testing does not yet exist, but the imminence of important applications justifies parallel, rather than serial, development of radarclinometry and radiometrically calibrated radar. The present investigation reports three developmental advances: (1) The solid angle of integration of back-scattered specific intensity constituting a pixel signal is more accurately accounted for in its dependence on surface orientation than in previous work. (2) The local curvature hypothesis, which removes the requirement of a ground-truth profile as a boundary condition and enables the formulation of the theory in terms of a line integral, has been expanded to include the three possibilities of Local Cylindricity, Local Biaxial Ellipsoidal Hyperbolicity, and Least-Squares Local Sphericity. (3) The theory is integrated in the cross-ground-range direction, which is ill-conditioned compared to the ground-range direction, whereas the original formulation was based on enforced isotropy in the two-dimensional power spectrum of the topography. It was found necessary to prohibit the hypothesis of Local Biaxial Ellipsoidal Hyperbolicity in the cross-range stepping, for reasons not completely clear. Variation in the proportioning between curvature assumptions had produced topographic maps that are in good mutual agreement but not realistic in appearance. They are severely banded parallel to the ground-range direction, most especially at small radar zenith angles. Numerical experimentation with the falsification of topography through incorrect decalibration as performed on a Gaussian hill suggests that the banding and its exaggeration at high radar incidence angles could easily be due to our lack of radiometric calibration.

The question of adapting to radar images the existing hardware that form topographic maps through stereo-photogrammetric models, is examined in principle. Such hardware utilizes a human/computer hybrid. Although the problem of brightness differentials between corresponding landmarks can be dealt with pseudo-photoclinometrically, the main problem is whether the perspective in a radar image can be conceived to mimic that of a photographic image obtained by a suitably positioned camera. This conception is found to be possible, providing the characteristic relief subtends a very small angle at the radar and at the fictitious camera. The photogrammetric model parameters must be determined a priori.

The understanding of quantum mechanical phenomena has come to rely heavily on theory framed in terms of operators and their eigenvalue equations. This paper investigates the utility of that technique as related to the reciprocity principle in diffuse reflection. The reciprocity operator is shown to be unitary and Hermitian; hence, its eigenvectors form a complete orthonormal basis. The relevant eigenvalue is found to be infinitely degenerate. A superposition of the eigenfunctions found from solution by separation of variables is inadequate to form a general solution that can be fitted to a one-dimensional boundary condition, because the difficulty of resolving the reciprocity operator into a superposition of independent one-dimensional operators has yet to be overcome. A particular lunar application in the form of a failed prediction of limb-darkening of the full Moon from brightness versus phase illustrates this problem. A general solution is derived which fully exploits the determinative powers of the reciprocity operator as an unresolved two-dimensional operator. However, a solution based on a sum of one-dimensional operators, if possible, would be much more powerful. A close association is found between the reciprocity operator and the particle-exchange operator of quantum mechanics, which may indicate the direction for further successful exploitation of the approach based on the operational calculus.

view Abstract Citations (2) References (5) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The determination of physical and dynamical parameters of Pluto/Charon and binary asteroids by least-square formation of a matched filter for a time series of images. I - The operational theory Wildey, R. L. Abstract A theory is derived for the determination of the masses, radii, and orbital elements of the Pluto/Charon, or similar, system based on the prediction of an image distribution over space and time and its comparison with observation. The comparison may be ultimately through the theory of least squares or the application of a matched filter to the observations as a three-dimensional signal stream at an initial or intermediate state. The theory is an approximation correct to fifth order in the diameters of celestial bodies. The theory of astronomical seeing that is used is based on Kolmogorov turbulence in the long-exposure limit. The images must be photometric. Linear tracking errors that can be removed are preferable to either automatic or manual guiding, in the collection of candidate observations. Publication: The Astronomical Journal Pub Date: September 1985 DOI: 10.1086/113892 Bibcode: 1985AJ.....90.1883W Keywords: Asteroids; Charon; Orbital Elements; Planetary Mass; Pluto (Planet); Differential Equations; Least Squares Method; Matched Filters; Planetology; Radii; Time Series Analysis; PLUTO; SATELLITES; CHARON; PLUTO-CHARON SYSTEM; PARAMETERS; PHYSICAL PROPERTIES; DYNAMICS; BINARY ASTEROIDS; ASTEROIDS; PROCEDURE; TECHNIQUES; THEORETICAL STUDIES; MASS; RADIUS; ORBITS; OBSERVATIONS; COMPARISONS; TURBULENCE; PHOTOMETRIC METHODS; IMAGE PROCESSING; IMAGERY; REMOTE SENSING; EARTH-BASED OBSERVATIONS; Astronomy; Satellites of Pluto full text sources ADS |

A mathematical theory and a corresponding numerical procedure have been developed to produce digital topography from radar images as digital photometric arrays. Thus, as radargrammetry is to photogrammetry, so radarclinometry is to photoclinometry. Photoclinometry encompasses a fundamental indeterminacy principle even for terrain that is homogeneous in normal albedo, because the surface normal consistent with a given reflected specific intensity is not unique. A geometric locus of such normal directions is implied, which generates a surface. For microwave backscatter, in specific application to radarclinometry, this surface is a cone whose half-angle is the incidence angle, whose axis contains the radar, and whose apex coincides with the terrain point. Although the indeterminacy can be removed if a properly directed profile of ground truth is available as a constraint, such is seldom the case. In its absence, an auxiliary assumption, such as that the strike line runs perpendicular to the illumination line, is needed. If metric integrity is a goal, then this is an absurd assumption. Herein, "the hypothesis of local cylindricity" has been assumed, a premise regarding the nature of topographic curvature that seems more realistic and that makes possible the production of topography as a set of parallel line integrals.

A digital file of the normal albedo of the Moon has been produced at a resolution of about 1/550 of a lunar diameter (about 6.3 km). The file was produced from five photographs taken with the 61-cm reflector of the Northern Arizona University Astrophysical Observatory. No mosaicking was necessary. Spatial control is selenodetic rather than landmark-morphologic. Photometric control is provided through a combination of electrography and regular photoelectric photometry. Pixel photometric function corrections are employed. The file was provided as data base for the Lunar Consortium. Brief discussion of the scientific implications of the frequency histogram is offered, and the negligibility of lunar limb darkening belowɛ = 77° is affirmed. It is specifically desired not to withhold these data from publication while more significant and detailed scientific interpretation is carried on.

A numerical analysis scheme is presented for a previously proposed theory of the temperature structure versus time of a diurnally insolated atmosphere of constant density wherein heat transfer occurs by both radiation and conduction. The approach is one of linearizing the difference equations corresponding to the analytical forms (rather than vice versa). Limitations of available computers and time there-on, coupled with an ill-conditioning characteristic of the one physical (lunar) situation attacked, limit the conclusions of the present paper to an affirmation of the method and an indication that a more approximate approach to the insolation of the lunar soil would be adequate. Explicit inclusion of the transfer equation, as done here, would be desirable for more rapidly rotating bodies such as asteroids, Galilean satellites and Saturn's rings.