Confidence limit of the magnetotelluric phase sensitive skew

Abstract

The rotationally invariant phase sensitive skew parameter, an indicator of dimensionality of conductivity structure, is a complicated non-linear function of the impedance tensor elements. In the presence of noise in the impedance data, skew can be significantly biased, leading to a false interpretation of dimensionality. Therefore, the probability function distribution of the skew parameter is derived to obtain its confidence limit, rather than treating a conventional linear propagation error. It is well known that the latter is only appropriate if the parameter is a function of independent random variables with small relative errors. The confidence limit is estimated by deriving its conditional probability function in terms of the tensor elements density function, using the Jacobi-matrix transformation of random variables, assuming the tensor elements to be normally distributed random variables. It is shown with synthetic and experimental data that the statistical confidence limit derived here truly reflects a probability range for the skew value. Bias of skew is seen to be significant with a small 5% of random Gaussian noise added to the tensor elements. Considering the 95% confidence limit instead of the measured skew itself results in a plausible approach to analyse dimensionality. The procedure developed here to estimate the confidence limit can also be extended to other functions of the tensor elements.