The raw data (Figure ), irregularly distributed across the surface, is 132,044 triples, (xi,yi,zi), where xi ranges over about 12 km and where yi ranges over about 20 km. We want to interpolate the data to a regular grid using inversion. The pertinence of this dataset to to our daily geophysical problems is three fold. First, we often have to do interpolation of seismic maps Britze (1998), potential field data Guspi and Introcaso (2000) or other measurements to compensate for the sparseness of acquisition geometries. Second, as seen in the raw data in Figure , some noise bursts need to be accounted for in the inversion scheme. This problem is common, for example, in tomography Bube and Langan (1997), deconvolution of noisy data Chapman and Barrodale (1983) and velocity analysis Guitton and Symes (1999) where outliers can degrade the final model if we assume a Gaussian distribution of the noise. And third, the final image of the Sea of Galilee will display the vessel tracks because the measurements on the lake were made on different days, with different weather and human conditions. We can directly link this problem to the goal of removing the acquisition footprint with 3-D seismic data Chemingui and Biondi (2002); Duijndam et al. (2000); Schuster and Liu (2001). Therefore, the interpolation of the data from the Sea of Galilee becomes a spiky noise and a coherent noise attenuation problem. We solve both problems with a finely tuned inversion scheme that should be usable for other geophysical applications.
There is a long list of students at the Stanford Exploration Project who attempted to produce a satisfying map of the sea bottom. Fomel and Claerbout (1995) introduced the norm via Iteratively Reweighted Least Squares (IRLS) to get rid of the noise bursts present in the data. Recently, Brown (2001) attempted to remove acquisition tracks by estimating the systematic error between tracks at crossing points. Karpushin and Brown (2001) used a bank of prediction-error filters (PEFs) to whiten the residual along tracks. However, in most of these results, there is a loss of resolution that hampers our goal of seeing small features in the final image. In this paper, we borrow ideas from these authors with three new approaches. The first important twist is preconditioning Fomel (2001), the second one is modeling of the ship's track instead of filtering it from the residual and the third is estimating the model covariance with a PEF. We show that preconditioning with an IRLS method removes the glitches and noise bursts very well. In addition, the modeling of the ship tracks within the inversion removes almost entirely the acquisition footprint. The PEF unravels small details in the middle of the Sea of Galilee.
We first examine the preconditioning trick along with the IRLS method. Next we introduce a new fitting equation that takes into account the inconsistency between different tracks inside the inversion. Finally we demonstrate that the model covariance can be estimated with a PEF to better preserve small features at the bottom of the sea.