Characterizing Data Measurement Errors with the EM Algorithm

The characterization of errors in measured data is important if one<br>wishes to condition reservoir models to diverse data sets because measurement/processing errors determine the proper relative weights of the data. In the literature, the measurement error for each data type is often estimated by some smoothing technique in the whole data domain, which often over-smoothes the data particularly around<br>points where the underlying true data change sharply and results in<br>over estimation of the measurement error. Here, we apply a modified<br>EM (Expectation-Maximization) algorithm to separate measured data<br>into groups based on the value of the measurement and the spatial<br>location. By applying a moving polynomial fit within each group, we generate estimates of the mean and covariance of measurement errors. The algorithm is applied to both synthetic and field time lapse seismic data as well as production data sets. For synthetic cases, the covariance functions estimated with the EM algorithm are superior to those obtained using smoothing with a moving window. The EM based procedure also appears to give more reasonable results for the field data.