The objective is to develop and test a robust theory to invert seismic data. The theory can cope with arbitrarily dipping, locally plane, sesimic reflectors. The test consists of (a) acquiring seismic reflection data along a line over a logged well, (b) inverting the data to obtain densities and interval velocities, (c) putting the theory at risk by comparing the inverted results with the data from the well.

The Earth parameters are determined from the top down in a recursive layer stripping scheme. This scheme is applied to the data which have been decomposed into their plane wave components using a double Radon transform with respect to offset and source coordinates. Both the source signature and the absolute calibration of the recording system are determined by measurements in order to perform the inversion. The reflecting interfaces are imaged using an RMS velocity function determined from the data and by transforming back to space coordinates with an inverse Radon transform.

When we began the project we thought we could extend our existing, proven, layer-stripping scheme for horizontal layers to seismic data from arbitrarily dipping layers. The theory we planned to use was based on ray theory. After working with it for more than a year we found wewere not able to get a proper grip on the amplitude information in the data, and realized that we needed to develop a wave theoretical approach to plane wave decomposition. This has now been provided by Fokkema and Van den Berg (1991). There ae tweo advantages to the project with this new theory. First, it accurately defines the plane wave decomposition of the data using the double Radon transform; we need this in order to perform the layer-stripping inversion. Second, it is primarily a theory of reflector imaging, or migration, which is needed to position the reflectors correctly in space.

Implementation and testing of the double Radon transform has now been completed (Vissinga, 1992) and two different versions of the imaging principle have been developed, implemented as computer programs, and tested on synthetic data. We are now applying the Tatalovic approach to the data acquired in this project and expect to obtain an improvement over conventional migration after stack. preliminary versions of both a conventional migration after tsack and the Tatlovic image of the field data now exist. The imaging problems, as far as the intial conception of the project was concerned, was almost a digression. However, we found that it was impossible to separate the imaging/migration problem from the inversion problems and we were unable to consider the application of the layer-stripping concept until the imaging problem had been solved. This is the reason for the delay in the completion of the project.

The seismic reflection data for the test were obtained along one line. At each shot point two shots of different sizes., 125 g and 500 g, were detonated to provide two shots records from which to calculate the source signature using the scaling law. A third shot of 250 g was used to test the scaling law theory : the seismic record from the third shot should be predictable from the other two. The errors in the prediction are small (< 10%),indicating that the source signatur can be obtained accurately in this way (Ziolkowski and Bokhorst, 1992). There is no other method for obtaining the source signature of a dynamite source which has been put at risk and tested as this one has.

We are now testing our layer-stripping inversion scheme in the double Radon transform domain on synthetic data calculated with a finite-difference program we have developed. Final adjustments to the frequency-dependent amplkitude factors caused by the Radon transform can then be checked and corrected, if necessary. The scheme will then be applied to the real data, followed by imaging of the rsults, an comparison with the well log data. The final report of this project will be submitted at the end of March 1992.

We have developed a theory for the inversion of seismograms obtained over a horizontally stratified acoustic earth model. The inversion is performed on the plane wave components of the data using a recursive layer-stripping approach from the top down. The source signature most be known. The inversion yields reflection coefficients at the interfaces, interval velocities, and densities.

Fokkema and Van den Berg (1991) have developed a theory to describe how seismic reflection data obtained over complicated structures may be decomposed into its plane wave components using a double Radon transform. The implementation of the transform is completed (Vissinga, 1992), and the layer-stripping inversion is then carried out in the same way. The double Radon transform and its implementation are new.

The seismic reflection data were obtained in March/April 1990 in Tubbegen in the Netherlands, using dynamic as the source. The source signature was obtained using two shots at each shop point andthe source scaling law (Ziolkowski and Bokhorst, 1992).

Determination of the source signature in this way is totally new.

Working with seismic data in the double Radon transformed domain is new. The domain has theoretical advantages which take time to be undertsood. Provide the data have been adequately sampled in the field (with respect to time, offset, and shot position), this is the ideal domain in which to perform certain data processing tasks such as the imaging of the reflecting interfaces (Fokkema and Van den Berg, 1992). we are now able to perform imaging, or correct positioning of the reflecting interfaces, in this domain with less effort and more accuracy than conventional migration after stack.

We consider it of fundamental importance that each interface be imaged before the wavefield is extrapolated downwards. Tatalovic et al. (1991) demonstrated the success of this approach on the famous synthetic Marmousi data set.

In order to perform the layer-stripping inversion inthe double Radon transform domain, both the source signature and the system calibration factor need to be known. The system calibration factor can be obtained from the range of plane wave components in which the deepest reflections are postcritically reflected, provided the source signature is known (Ziolkowski and Koster, 1991), and the data can be scaled to true amplitude in the correct units of particle velocity. The effects of the free surface must be removed before applying the layer-stripping inversion and can be down as the application of a filter on each trace in the double Radon transform domain, using the srource signature and the known scaling factor.

The layer-stripping inversion in the double Radon transform doamin works from the top down on each trace removes the effect of each layer. That is, the layer is removed by time-shifting the data downwards, and the internal multiples which are generated in the layer and which appear at later times in the remaining data arealso removed.