High-precision geomagnetic measurement depends on eliminating variations of ionspheric and magnatospheric origin. The commonly used technique of simply taking differences between total field magnetometers is only partially sucessful. Our method involves finding the multichannel Wiener filters which predict the field variation at a given total field magnetometer of an array from the fields of the remaining magnetometers and a three-component magnetometer. The difference is then taken between the total field and the predicted field, leaving a cleaned total field. Filter lengths and number and choice of input channels are determined using methods of statistical parametric model fitting. The resulting filters which are defined on short noise free record sections are found to be effective over the remainder of the record. Detailed analysis of data from the University of California, Los Angeles, array in southern California, for which we have the best vector data, shows that for optimal predictive cleaning it is essential to use vector components and that only the two components orthogonal to the total field direction are necessary. This means that the field-aligned component may be omitted from a vector station, which simplifies the instrumentation considerably. Analysis of the standard deviation of the residuals after cleaning (0.1 nT for hourly averages) shows that it is 8 times that expected from digitization noise alone. The most probable explanation for this is that measurement noise is about 3 times the digitization interval (0.25 nT) and that higher-precision measurements are required for further improvement. This conclusion has been confirmed by tests on the instruments at close spacing. Application of the method to data taken in central California and Hawaii reveals tectonomagnetic effects which are otherwise hidden in noise. Wiener filters are especially suitable for real-time analysis, which is an important factor in earthquake prediction.