MULTI‐OFFSET ACOUSTIC INVERSION OF A LATERALLY INVARIANT MEDIUM: APPLICATION TO REAL DATA1
Authors: HELGESEN JAN , KOLB PIERRE 
Published: April 2006

Abstract

The aim of seismic inversion methods is to obtain quantitative information on the subsurface properties from seismic measurements. However, the potential accuracy of such methods depends strongly on the physical correctness of the mathematical equations used to model the propagation of the seismic waves. In general, the most accurate models involve the full non‐linear acoustic or elastic wave equations. Inversion algorithms based on these equations are very CPU intensive. The application of such an algorithm on a real marine CMP gather is demonstrated. The earth model is assumed to be laterally invariant and only acoustic wave phenomena are modelled. A complete acoustic earth model (P‐wave velocity and reflectivity as functions of vertical traveltime) is estimated. The inversion algorithm assumes that the seismic waves propagate in 2D. Therefore, an exact method for transforming the real data from 3D to 2D is derived and applied to the data. The time function of the source is estimated from a vertical far‐field signature and its applicability is demonstrated by comparing synthetic and real water‐bottom reflections. The source scaling factor is chosen such that the false reflection coefficient due to the first water‐bottom multiple disappears from the inversion result. In order to speed up the convergence of the algorithm, the following inversion strategy is adopted: an initial smooth velocity model (macromodel) is obtained by applying Dix's equation to the result of a classical velocity analysis, followed by a smoothing operation. The initial reflectivity model is then computed using Gardner's empirical relationship between densities and velocities. In a first inversion step, reflectivity is estimated from small‐offset data, keeping the velocity model fixed. In a second step, the initial smooth velocity model, and possibly the reflectivity model, is refined by using larger‐offset data. This strategy is very efficient. In the first step, only ten iterations with a quasi‐Newton algorithm are necessary in order to obtain an excellent convergence. The data window was 0–2.8 s, the maximum offset was 250 m, and the residual energy after the first inversion step was only 5% of the energy of the observed data. When the earth model estimated in the first inversion step is used to model data at moderate offsets (900 m, time window 0.0–1.1 s), the data fit is very good. In the second step, only a small improvement in the data fit could be obtained, and the convergence was slow. This is probably due to the strong non‐linearity of the inversion problem with respect to the velocity model. Nevertheless, the final residual energy for the moderate offsets was only 11%.

The estimated model was compared to sonic and density logs obtained from a nearby well. The comparison indicated that the present algorithm can be used to estimate normal incidence reflectivity from real data with good accuracy, provided that absorption phenomena play a minor role in the depth interval considered. If details in the velocity model are required, large offsets and an elastic inversion algorithm should be used.