Applications of the generalized norm solver

Introduction

Currently, many geophysics problems are solved with least-squares (L2) model fitting because of its fast convergence, simple parametrization, and easy to understand numerical analysis. However, L2 minimization places disproportionate emphasis on large residual values. Therefore, an L1-type norm inversion technique is more appropiate for solving geophysical problems that have a ``blocky" model space. One example is to do adaptive subtraction of multiples Guitton (2005) using IRLS (iterative re-weighted least squares). Other possible applications are tomography (Bube and Langan (1997)) and deconvolution of noisy data (Chapman and Barrodale 1983).

In geophysics, a popular way to run L1-type inversions is with IRLS (Gersztenkorn et al. (1986)) . Running with IRLS often improves the results. A drawback is that its computational time is considerably higher because of repeated application of the costly forward and adjoint data-fitting operator within two iterative loops. To overcome this computational deficiency while retaining the benefit of L1-type inversion, we came up with another solver that does a better job than IRLS. Claerbout (2009) developed an algorithm of that we call the generalized norm solver, which steps with conjugate-direction and using the Taylor series expansion. Such a solver allows us to perform inversion using the L2 or the mixed L1/L2 norm Maysami and Moussa (2009). For convenience, we will use the term 'norm' to refers to all kind of measures and norms. It is understood that a norm has a strict definition in mathematics. For the theoretical description of the solver, please refer to the report by Claerbout (2009).

Maysami and Moussa (2009) have implemented such a solver, which allows us to test its robustness in this paper. Three norms will be used in our study: the least-squares (L2), Hybrid, and Huber norms. It is worth mentioning that the theory for using the least-squares norm with our solver is exactly the same as the theory for solving the least-squares problem with conventional conjugate-direction algorithm. For the rest of this paper, we will refer the generalized norm solver with the Hybrid norm as the hybrid solver and the generalized norm solver with the Huber norm as the Huber solver.

We have applied the generalized norm solver to two test cases. The first test case recovers the equation of a straight line given data that are corrupted with Gaussian noise and spikes. We find that the generalized norm solver recovers the equation of a straight line when using either the hybrid solver or the Huber solver. The second test case is called the 1-D Galilee problem. This problem is a simplified version of a real depth sounding experiment of the Sea of Galilee and a standard test problem in SEP textbooks, (Claerbout (2008); Claerbout and Fomel (2008)).The synthetic data for this case are measurements between the water surface and the bottom of the lake. The first part of the test simply aims to recover the true depth of a 1-D lake given that the data contain occasional large spikes. The second part of the test has data corresponding to a water level that change in a step-like manner. We will refer to these kind of jumps as ``drift." We aim to recover the true depth and sudden drift in data. We find that the hybrid solver always gives the best result as compared to the Huber solver and the least-squares solver.

Applications of the generalized norm solver