Statistical biases and errors inherent in photoclinometric surface slope estimation with natural light

Photoclinometry is the most common method used to obtain high-resolution topographic maps of planetary terrain. We derive the likelihood function of photoclinometric surface slope from (1) the probability distribution of the measured photon count of natural sunlight through a Charge-Coupled Device (CCD) including uncertainty due to camera shot noise, camera read noise, small-scale albedo fluctuation and atmospheric haze, and (2) a photometric model relating photocount to surface orientation. We then use classical estimation theory to determine the theoretically exact biases and errors inherent in photoclinometric surface slope and show when they may be approximated by asymptotic expressions for sufficiently high sample size. We show how small-scale albedo variability often dominates biases and errors, which may become an order of magnitude larger than surface slopes when surface reflectance has a weak dependence on surface tilt. We provide bounds on the minimum possible error of any unbiased photoclinometric surface slope estimate, and compute the sample sizes necessary to constrain errors within desired design thresholds

Higher order asymptotic inference in remote sensing of oceanic and planetary environments

Thesis (Ph. D. in Ocean Engineering)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 223-230).An inference method based on higher order asymptotic expansions of the bias and covariance of the Maximum Likelihood Estimate (MLE) is used to investigate the accuracy of parameter estimates obtained from remote sensing measurements in oceanic and planetary environments. We consider the problems of (1) planetary terrain surface slope estimation, (2) Lambertian surface orientation and albedo resolution and (3) passive source localization in a fluctuating waveguide containing random internal waves. In these and other applications, measurements are typically corrupted by signal-independent ambient noise, as well as signal-dependent noise arising from fluctuations in the propagation medium, relative motion between source and receiver, scattering from rough surfaces, propagation through random inhomogeneities, and source incoherence. We provide a methodology for incorporating such uncertainties, quantifying their effects and ensuring that statistical biases and errors meet desired thresholds. The method employed here was developed by Naftali and Makris[84] to determine necessary conditions on sample size or Signal to Noise Ratio (SNR) to obtain estimates that attain minimum variance, the Cramer-Rao Lower Bound (CRLB), as well as practical design thresholds. These conditions are derived by first expanding the bias and covariance of the MLE in inverse orders of sample size or SNR, where the firstorder covariance term is the CRLB. The necessary sample sizes and SNRs are then computed by requiring that (i) the first-order bias and second-order covariance terms are much smaller than the true parameter value and the CRLB, respectively, and (ii) the CRLB falls within desired error thresholds. Analytical expressions have been derived for the asymptotic orders of the bias and covariance of the MLE obtained from general complex Gaussian vectors,[68, 109] which can then be used in many practical problems since (i) data distributions can often be assumed to be Gaussian by virtue of the central limit theorem, and (ii) they allow for both the mean and variance of the measurement to be functions of the estimation parameters, as is the case in the presence of signal-dependent noise. In the first part of this thesis, we investigate the problem of planetary terrain surface slope estimation from satellite images. For this case, we consider the probability distribution of the measured photo count of natural sunlight through a Charge- Coupled Device (CCD) and also include small-scale albedo fluctuation and atmospheric haze, besides signal-dependent (or camera shot) noise and signal-independent (or camera read) noise. We determine the theoretically exact biases and errors inherent in photoclinometric surface slope and show when they may be approximated by asymptotic expressions for sufficiently high sample size. We then determine the sample sizes necessary to yield surface slope estimates that have tolerable errors. We show how small-scale albedo variability often dominates biases and errors, which may become an order of magnitude larger than surface slopes when surface reflectance has a weak dependence on surface tilt. The method described above is also used to determine the errors of Lambertian surface orientation and albedo estimates obtained from remote multi-static acoustic, optical, radar or laser measurements of fluctuating radiance. Such measurements are typically corrupted by signal-dependent noise, known as speckle, which arises from complex Gaussian field fluctuations. We find that single-sample orientation estimates have biases and errors that vary dramatically depending on illumination direction measurement diversity due to the signal-dependent nature of speckle noise and the nonlinear relationship between surface orientation, illumination direction and fluctuating radiance. We also provide the sample sizes necessary to obtain surface orientation and albedo estimates that attain desired error thresholds. Next, we consider the problem of source localization in a fluctuating ocean waveguide containing random internal waves. Propagation through such a fluctuating environment leads to both the mean and covariance of the received acoustic field being parameter-dependent, which is typically the case in practice. We again make use of the new expression for the second-order covariance of the multivariate Gaussian MLE,[68 which allows us to take advantage of the parameter dependence in both the mean and the variance to obtain more accurate estimates. The degradation in localization accuracy due to scattering by internal waves is quantified by computing the asymptotic biases and variances of source localization estimates. We show that the sample sizes and SNRs necessary to attain practical localization thresholds can become prohibitively large compared to a static waveguide. The results presented here can be used to quantify the effects of environmental uncertainties on passive source localization techniques, such as matched-field processing (MFP) and focalization. Finally, a method is developed for simultaneously estimating the instantaneous mean velocity and position of a group of randomly moving targets as well as the respective standard deviations across the group by Doppler analysis of acoustic remote sensing measurements in free space and in a stratified ocean waveguide. It is shown that the variance of the field scattered from the swarm typically dominates the rangevelocity ambiguity function, but cross-spectral coherence remains and enables high resolution Doppler velocity and position estimation. It is shown that if pseudo-random signals are used, the mean and variance of the swarms' velocity and position can be expressed in terms of the first two moments of the measured range-velocity ambiguity function. This is shown analytically for free space and with Monte-Carlo simulations for an ocean waveguide. It is shown that these expressions can be used to obtain accurate, with less than 10% error, of a large swarm's instantaneous velocity and position means and standard deviations for long-range remote sensing applications.by loannis Bertsatos.Ph.D.in Ocean Engineerin