Application of the fast and robust bootstrap method to the uncertainty analysis of the magnetotelluric transfer function

SUMMARY Because cultural noise can significantly distort the observed electromagnetic field data of the magnetotelluric measurements, robust estimators have been developed to estimate the magnetotelluric transfer function robustly against outlying data. Because the magnetotelluric transfer function is an estimate, it has uncertainty, and it is important to infer the uncertainty of the transfer function estimate accurately. However, it has been known that the parametric error estimation method is not appropriate for accurately inferring the standard errors of robust estimates of the magnetotelluric transfer function. The bootstrap method, which is the most popular non-parametric method, also has a disadvantage in that it can require a long computational time, particularly when it is applied to robust estimators. Although the fixed-weight approximation of the bootstrap method can significantly reduce the computational cost, it generally underestimates the standard error of the robust estimates. To overcome the limitations of the existing error estimation methods, we verified the effectiveness of the fast and robust bootstrap method by applying it to two robust remote reference estimators: the remote reference method using the M-estimator and the robust remote reference multivariate regression S-estimator. In the fast and robust bootstrap method, fixed-weight approximations of the estimates of the ordinary bootstrap method are corrected based on a linear approximation of the fixed-point equation corresponding to robust estimators. Using synthetic and real-world data sets, we confirmed that the fast and robust bootstrap method could provide standard errors comparable to those obtained by the ordinary bootstrap method in most cases, as fast as the fixed-weight bootstrap method. We also observed that the fast and robust bootstrap method could prevent the overestimation of the error by the ordinary bootstrap method, which stemmed from outlying data. Although the fast and robust method could not perfectly correct the fixed-weight approximations of transfer function estimates if their standard errors were large, its advantages are considered to outweigh this limitation. Thus, we conclude that the fast and robust bootstrap method is effective and practical for computing the standard error of the magnetotelluric transfer function.