The objective is to develop and test a robust theory to invert seismic data. The theory can cope with arbitrarily dipping, locally plane, seismic 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.

The theory of seismic inversion that was developed in this project is data-driven, rather than model driven. It makes the same assumption about the data from migration and imaging as every theory of seismic migration; that is, the data consist of primary reflection only and all internal multiples and free-surface effects are ignored. In the processing scheme these effects, and the extraction of primary reflection coefficients are handled after imaging. The new theory developed here maps all the seismic data into the double Radon transform (po,ps) domain, in which the transforms are taken with respect to offset and source coordinates. The dip information is discretised in this domain and all horizontal information is mapped into the ps=O plane, in which it is very easy to determine a robust r.m.s. velocity model for the whole data set. Imaging and migration is done by selecting data on an "imaging surface" in this domain, defined partly by the velocity model, and by transforming these selected data back to space-time, by performing an inverse Radon transform over the ps coordinate. With some corrections, the resulting data are correctly time migrated and have the correct amplitudes.

To extract the reflection coefficients, the source signature must be deconvolved, the data must be calibrated to obtain the correct units, and the free surface effects and internal multiples must be removed, using a theory that was developed in a previous EC project. For the dynamite data we obtained in this project, the source signature was obtained using two different size charges at each shot point and relating the source signatures by a scaling law. The method was tested using a third shot at each shot point to make a seismogram that should be predictable from the other two. The method was totally successful. It is the only method we know of to determine the dynamite signature that makes no assumpions about the geology and relies on a theory that has survived a rigorous test.

The line was shot so as to intersect a logged well. To avoid cross-line interference effects, the line was shot perpendicular to the geological strike. The well was deviated and mostly in a single plane at an angle to the plane of the seismic section. The position of the line was chosen so as to intersect the well at the target. The velocity log was complete from the surface down to the target, below 2000 m. The densities had been logged only below 1600 m, and therefore the reflection coefficients estimated from these logs could be trusted only where the density log was good. Above this level densities were estimated from the velocities using an assumed conventional power law relation. The match of the inverted reflection coefficients to those obtained from the well was very good.

We also processed the data using a conventional processing scheme, using common mid-point stacking and finite difference migration after stack.

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 must be known. The inversion yields reflection coefficients at the interfaces, interval velocities, and densities.

Fokkema and Van den Berg 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, 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 Tubbergen in the Netherlands, using dynamite as the source. The line was chosen to intersect a logged well, the logs of which were provided by the Netherlands Aardolie Maatschappij (NAM). The source signature was obtained using two shots at each shot point and the source scaling law.

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 takes time to be understood. Provided the data have been adequately sampled in the field, this is the ideal domain in which to perform certain data processing tasks such as the imaging of the reflecting interfaces. 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 in the 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, 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 done as the application of a filter on each trace in the double Radon transform domain, using the source signature and the known scaling factor.

The layer-stripping inversion in the double Radon transform domain works from the top down on each trace and 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 are also removed.