The remote reference as an instrumental variable in model space: time-series magnetotelluric inversion with noisy magnetic fields

Noise in the local magnetic field biases magnetotelluric (MT) responses, and a relevant, valid remote magnetic record removes the bias. For an earth model inverted from time series, remote referencing becomes estimating equations: the residual formed with the local field is orthogonal to the sensitivities driven by its projection on the remote field. Their solution, relevance and validity must be checked separately, and their scores over blocks of time supply the covariance of the root and a test of the remote site. Minimizing the misfit with the projected field as driver does not solve them. On observatory records rebuilt without noise it misses the model by up to 0.9 percent of the impedance, and the root, found by fixed point and Newton iterations, by 0.02 percent at most. The standard error shows neither failure of the instrument. A weak instrument draws the root toward the single-site response, as the share of the local field explained by the projection predicts. A remote site that the noise reaches moves the root toward the relation of the noise, and the difference of the roots with two remote sites, judged with the covariance of their common blocks, detects it. Near 25-kV railways, survey sites 1.5 to 12.5 km away fail this test in 49 of 61 testable pairs while trains run and in none of 63 at night, and a remote field 1090 km away passes it against four observatories in all 71 testable cases at the one percent level. With railway noise over whole days, the root and a standard robust code followed by a model recover the response on 61 and 58 of 80 records, so the method is not a better remote reference in itself. Its free-transfer-function form gives the full tensor and tipper of a railway record from service hours.