An empirical model of the Earth’s bow shock based on an artificial neural network

Abstract

All of the past empirical models of the Earth’s bow shock shape were obtained by best-fitting some given surfaces to large collections of observed crossings. However, the issue of bow shock modelling can be addressed by means of artificial neural networks (ANN) as well. The ANN approach is powerful and flexible since an ANN can capture the hidden relation between the bow shock position and a set of given inputs and forecast its response on the basis of the inputs only.In this paper we present a perceptron, a simple feedforward network, which computes the bow shock radial position, along a given direction, using as inputs: the two angular coordinates of that direction; the bow shock radial distance RF79 provided by Formisano’s model (F79) (Formisano, 1979) and the upstream Alfvénic Mach’s number Ma.The perceptron output can be regarded as a correction to the F79 representation of the bow shock shape. A statistical analysis, performed over a test data set of 944 bow shock crossings from several spacecraft, demonstrates that the ANN predictions are effectively more accurate than F79 ones. Indeed, the ANN mean value of the ratio between predicted and observed shock radial distance r̄ANN is generally closer to the expected value μr ​= ​1 than the corresponding r̄F79. Such improvement on F79 is partly due to the addition of Ma to the model inputs. However, the statistical error σrANN is practically the same as that from an identical network but with no Ma input line. In this regard, we discuss the possibility that an irreducible uncertainty in predictions originates from the bow shock motions related to the impacts of interplanetary discontinuities on the magnetosphere.