Simple tracked datasets, consisting of single-channel measurements (z) acquired by a moving instrument at surface points (x,y), have traditionally proven valuable test-beds for least squares estimation techniques at SEP, due to their small size and conceptual simplicity. Ben-Avraham et al.'s 1990 bathymetric survey of the Sea of Galilee (Lake Kinneret) has drawn a prolonged interest Claerbout (1999); Fomel and Claerbout (1995); Fomel (2001), primarily because errors in the data seriously inhibit the task of translating the data into a gridded map.
In practice, each point measurement contains random errors, due both to instrumental inaccuracy and to physical phenomena with time and spatial scales smaller than two neighboring measurements. Unfortunately, these ``random'' errors are often non-gaussian, violating a central assumption of estimation theory that data contain gaussian-distributed error. In the context of Galilee, the ship's pre-GPS radio location system often mis-positioned depth soundings on the earth's surface. The erroneous measurements are easily identified as spikes in locations where the true sea floor is nearly flat, but not where the sea floor dips steeply. Even more crucially, however, these data also contain systematic error, which I define as error which varies slowly-in time and space-over a single track of measurements. Causes may include tidal shifts, instrumental drift, and wind-induced bulging of the lake's surface.
Claerbout (1999) cast the translation of irregular point data into a regular gridded map as an ``inverse interpolation'' problem. The simplest 2-D formulation of this problem is sensitive both to non-gaussian and systematic errors as noted by Fomel and Claerbout (1995). To overcome both difficulties, they included a composite residual weighting operator consisting of: 1) diagonal weight estimated via Iteratively Reweighted Least Squares (IRLS) Guitton (2000); Nichols (1994) to handle non-gaussian noise, and 2) a finite-difference first derivative operator to suppress correlated components of the residual. A similar technique was applied successfully to process a Geosat dataset with similar errors by Ecker and Berlioux (1995).
While Fomel and Claerbout's 1995 approach suppresses both acquisition footprint and non-gaussian noise, the authors note a loss of resolution in the final map of Galilee, relative to the simple inverse interpolation result. As noted by Claerbout (1999), when the systematic error varies from one acquisition track to another, a bank of prediction error filters, one for each acquisition track, makes a far better residual decorrelator. Karpushin and Brown (2001) implement this approach and report good suppression of acquisition footprint, with preservation of underlying bathymetric features.
In this paper, I take a somewhat different tack to the problem. I measure the difference in measured sea floor depth at ``crossing points'', or points in space where two acquisition tracks nearly cross. I then solve a least squares missing data problem to estimate the systematic error at all points between the crossing points, subtract the estimated systematic error from the original data, and use the simple inverse interpolation methodology of Claerbout (1999), supplemented with IRLS diagonal residual weights to suppress spikes, in order to make a final map. My approach generates maps of Galilee's seafloor which are generally free of acquisition footprint, and which also exhibit excellent preservation of subsea geologic features. Furthermore, my new approach visibly unbiases the residual, proving that the result is optimal from the standpoint of estimation theory.