Multiple realizations and data variance: Successes and failures

Robert G. Clapp

bob@sep.stanford.edu

ABSTRACT

Geophysical inversion usually produces a single solution to a given problem. Often it is desirable to have some error bounds on our estimate. We can produce a range of models by first realizing that the single solution approach produces the minimum energy, minimum variance solution. By adding appropriately scaled random noise to our residual vector we change the minimum energy solution. Multiple random vectors produce multiple new estimates for our model. These various solutions can be used to assess error in our model parameters. This methodology strongly relies on having a decorrelated residual vector and, previously, was used primarily on the model styling portion of our inversion problem because it came closer to honoring the decorrelation requirement. With an appropriate description of the noise covariance, multiple realizations can be estimated. Examples of perturbing the data fitting portion of the standard inversion are shown on a 2-D deconvolution and 1-D velocity estimation problems. Results indicate that methodology has potential but is not well enough understood to be generally applied.

Risk assessment is a key component to any business decision. Geostatistics has recognized this need and has introduced methods such as simulation to attempt to assess uncertainty in their estimates of earth properties Isaaks and Srivastava (1989). The problem is that the geostatistical methods are generally concerned with local, rather than global, solutions to problems and therefore can not be easily applied to global inversions problems that are common in geophysics.

In previous works Clapp (2000, 2001a,b), I showed how we can modify standard geophysical inverse techniques by adding random noise into the model styling goal to obtain multiple realizations. In Clapp (2002) and Chen and Clapp (2002), these multiple realizations were used to produce a series of equiprobable velocity models. The velocity models were used in a series of migrations, and the effect on Amplitude vs. Angle (AVA) was analyzed. In previous papers on the subject, the concentration was on modifying the model styling goal and only briefly mentioned that it should be feasible to apply the same methodology to the data fitting goal.

In this paper, I show the data fitting goal can also be changed to produce equiprobable results. I show that it is a much more difficult problem than modifying our model styling goal because of the difficulties in building a realistic noise covariance operator. I begin by reviewing the methodology of multiple realizations with operators. I introduce a simple inverse filtering example to illustrate that for simple cases we can create pseudo-datasets with realistic noise distribution. These datasets can be used as input to an inversion process to get some preliminary boundaries on model errors. I conclude by showing an example of creating multiple reasonable interval velocity models from a single velocity scan.