Transformation from total-field magnetic anomaly to the projection of the anomalous vector onto the normal geomagnetic field based on an optimization method

Abstract

The total-field magnetic anomaly [Formula: see text] is approximated as the component [Formula: see text] of the anomalous vector [Formula: see text] along the direction of the normal geomagnetic field. It is widely used in magnetic data processing and interpretation practices as a routine if [Formula: see text] is relatively small. But in highly magnetic environments, the distinction between [Formula: see text] and [Formula: see text] is often too large to be ignored. We carefully investigate the difference between [Formula: see text] and [Formula: see text] and find that it will increase rapidly in the trend of the quadratic function as [Formula: see text] strengthens. We also test the effects of approximation on the component transformation and reduction to the pole on a synthetic single-sphere model. As expected, the error caused by inaccurate information will propagate into subsequent data processing procedures and adversely affect the results. Therefore, we have developed an optimization strategy based on the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm to transform the total-field anomaly into [Formula: see text]. First, we have constructed an objective function after transforming [Formula: see text] into [Formula: see text] through the component transformation in the frequency domain. Then, using [Formula: see text] as the initial value of [Formula: see text], [Formula: see text] is calculated iteratively by the L-BFGS algorithm. To test the validity of the optimization algorithm, [Formula: see text] is transformed for noise-free and noise-corrupted models and models with a background field. The synthetics indicate that the transformed [Formula: see text] is almost the same as the model [Formula: see text], whose maximum error is approximately one-hundredth (30 nT) of the difference (8000 nT) between the modeled [Formula: see text] and [Formula: see text]. The synthetics and field data example from the Yangshan Iron Mine, Fujian Province, southern China, also indicate that the data transformation and forward-modeling results can benefit from the direct use of transformed [Formula: see text] instead of [Formula: see text].