Thesis (Ph. D.)--Joint Program in Oceanography (Massachusetts Institute of Technology, Dept. of Earth, Atmospheric, and Planetary Sciences; and the Woods Hole Oceanographic Institution), June 1990."April 1990."At scale lengths less than 100 km or so, statistical descriptions of seafloor morphology can be usefully employed to characterize processes which form and reshape abyssal hills, including ridge crest volcanism, off-axis tectonics and volcanism, mass wasting, sedimentation, and post-depositional transport. The objectives of this thesis are threefold: (1) to identify stochastic parameterizations of small-scale topography that are geologically useful, (2) to implement procedures for estimating these parameters from multibeam and side-scan sonar surveys that take into account the finite precision, resolution, and sampling of real data sets, and (3) to apply these techniques to the study of marine geological problems. The seafloor is initially modeled as a stationary, zero-mean, Gaussian random field completely specified by its two-point covariance function. An anisotropic two-point covariance function is introduced that has five free parameters describing the amplitude, orientation, characteristic width and length, and Hausdorff (fractal) dimension of seafloor topography. The general forward problem is then formulated relating this model to the statistics of an ideal multibeam echo sounder, in particular the along-track auto-covariance functions of individual beams and the cross-covariance functions between beams of arbitrary separation. Using these second moments as data functionals, we then pose the inverse problem of estimating the seafloor parameters from realistic, noisy data sets with finite sampling and beamwidth, and we solve this inverse problem by an iterative, linearized, least squares method. Resolution of this algorithm is tested against ship variables such as length of data, the orientation of ship track with respect to topographic grain, and the beamwidth. This analysis is conducted by inverting sets of synthetic data with known statistics. The mean and standard deviation of the inverted parameters can be directly compared with the input parameters and the standard errors output from the inversion. The experiments conducted in this study show that the rms seafloor height can be estimated to within -15% and anisotropic orientation to within ~5* (for a strong lineation) using very short track lengths (down to 3 characteristic lengths, or -10 to 100 km), and characteristic lengths of seafloor topography can be estimated to within -25% using fairly short track lengths (down to 5 or 6 characteristic lengths, or 10's of km to -200 kin). The number of characteristic lengths sampled by a ship track, and hence the accuracy of the estimation, is maximized when the ship track runs perpendicular to abyssal hill lineation. Using the assumed beamwidth, the measured noise values, and the seafloor parameters recovered from the inversion, Sea Beam "synthetics" are generated whose statistical character can be directly compared with raw Sea Beam data. However, these comparisons are spatially limited in the athwart ship direction. A recent SeaMARC II survey along the flanks and crest of the East Pacific Rise between 130 and 15* N included sufficient off-axis topography to permit a comparison of a complete 2-D synthetic topographic field with a region of abyssal-hill terrain that has close to 100% data coverage. Synthetic data is compared to both Sea Beam swaths and SeaMARC II survey data. These comparisons generally indicate that we are successful in characterizing the second order properties of the seafloor. They also indicate the directions we will need to take to improve our modeling, including generalization of the second-order model and characterization of higher moments. The inversion procedure is applied to a data set of 64 near-ridge Sea Beam swaths to characterize near ridge abyssal hill morphology and its relationship to ridge properties. Much of the data (27 swaths) comes from cruises to the Pacific-Cocos spreading section of the East Pacific Rise between 9* and 15* N. These data provide very good abyssal hill coverage of this well-mapped and studied ridge section and form the basis of a regional analysis of the correlation between ridge morphology and stochastic abyssal hill parameters. This regional analysis suggests a strong relationship between magma supply and the character of abyssal hills. We also have data from near the Rivera (9) and Nazca (7) spreading sections of the East Pacific Rise, the Mid-Atlantic Ridge (18), and the Indian- African Ridge (3). Though spotty, this constitutes a good initial data set for the analysis of correlations among covariance parameters and between parameters and ridge characteristics, especially spreading rate. A working hypothesis is introduced to explain the observations within a geological framework. This hypothesis contends 1) that the maximum size of abyssal hills is related to the lithosphere's ability to elastically support the load, 2) that fissuring and horst and graben formation dominate abyssal hill formation at fast spreading ridges, and 3) that volcanic edifice formation, modified by faulting driven by lithospheric necking, dominates abyssal hill formation at slow spreading ridges. To quantify abyssal hill characteristics such as vertical and lateral asymmetry and "peakiness" we must appeal to higher statistical moments than order two. A mathematical framework is introduced for the study of higher moments of a topographic field. This framework is built upon the concept that lower-order moment provide the groundwork for studying the higher-order moments. A simple 1-D parameterized model is proposed for moments up to order 4. This model includes two parameters for the third moment, describing vertical and lateral asymmetries, and one for the fourth moment, which describes the peakiness of topography. Initial methods are developed for estimating these parameters from bathymetric profiles. Results from the near ridge data set are presented and interpreted with regard to abyssal hill forming processes.by John Anson Goff.Ph.D.