Next: LINEAR ESTIMATION OF MISSING Up: Nichols: Estimation of missing Previous: Nichols: Estimation of missing

# Introduction

We can estimate missing data from the surrounding data values if we make some assumptions about the characteristics of the interpolated output dataset. One commonly made assumption is that the data is locally smooth i.e. the output spectrum is low-pass. Other possible assumptions are: the data can be well approximated by a polynomial function, the data can be modeled as a small number of linear events, the data has a known output spectrum or the output spectrum can be estimated from the input spectrum.

In this paper I use our knowledge of the output data characteristics to design an error operator. The output of the operator is the residual part of the data that does not conform to our model. For example, if the output data is assumed to be smooth then a high pass operator should produce an output with little energy. Given such an operator I choose the interpolated data values to minimize the energy in the output. I cast this problem as a linear least squares problem that can be solved by standard numerical techniques.

If the parameters characterizing the output data are unknown there are two alternatives. The first is to estimate the parameters from the input data. This then reduces to the linear problem described above. However, there may be problems if the missing data points influence the estimation of the parameters or if the parameters are estimated far from the missing data points. The second alternative is to perform a joint estimation of the output data parameters and the missing data values. I do this by estimating the missing data values and the parameters describing the output data in the same least squares problem. This produces a nonlinear problem that is more expensive to solve but also more stable.

When non-linear least-squares problems are solved it is important to have reasonable initial values for the model parameters. I demonstrate a missing data estimation technique that can be used to interpolate aliased data if good initial estimates of the dips present in the data are available. I also show how these initial estimates can be obtained from a smoothed version of the data. The combination of these two methods can be used as an automatic interpolation technique for aliased data.

Next: LINEAR ESTIMATION OF MISSING Up: Nichols: Estimation of missing Previous: Nichols: Estimation of missing
Stanford Exploration Project
1/13/1998