oa Base level (drift) estimation for frequency domain hem data using causality, an update on dispersion relations and the hilbert transform for em system response

As a check on frequency domain systems for calibration and drift, we aim to predict in-phase from quadrature data and vice versa. It is known that, for causal systems, the real and imaginary components of the response of the system under excitation are related through dispersion relations. The formal relationships between the components of any complex response function are the Rramers-Kronig relations, which are satisfied by any complex function which is analytic in the upper half plane, representing causality. Geophysical signals are inherently causal and as such it is in principle always possible to relate in-phase to quadrature measurements. In the frequency domain these dispersion relations take the form of the Hilbert transformation relationship between scaled in-phase and quadrature measurements which for specific ideal Earth structures are able to be solved analytically. However real Earth responses are rarely captured by these idealised models and numerical approaches are required. The major limitation on the successful implementation of the numerical Hilbert transform is limited bandwidth measurements which introduce significant noise constraints. Frequency domain measurements rely on the existence of frequency dependent responses of materials for the detection of anomalies. The response function is nominally the resistivity (conductivity) and experimental determination of this response function involves two measurements per frequency, the in-phase and quadrature components.

There are several possible applications of the prediction of in-phase response from quadrature data including 1) base level, calibration and phase checks on electromagnetic and induced polarisation systems 2) prediction and validation of noise levels in in-phase from quadrature measurements and vice versa and 3) interpolation and extrapolation of sparsely sampled data enforcing causality and better frequency-domain - time-domain transformations. This could be used for example to speed up forward and inverse modeling without loss of accuracy. In practice, using recalibrated Resolve HFEM data, in-phase data points can only be predicted using a scaled Hilbert transform with a standard deviation between 20 ppm (central frequencies) and 100 ppm (outer frequencies. This error is too large to be used reliably in any of the potential applications listed above.