Simple and Fast Generalized - M (GM) Estimator and Its Application to Real Data Set

It is now evident that some robust methods such as MM-estimator do not address the concept of bounded influence function, which means that their estimates still be affected by outliers in the X directions or high leverage points (HLPs), even though they have high efficiency and high breakdown point (BDP). The Generalized M(GM) estimator, such as the GM6 estimator is put forward with the main aim of making a bound for the influence of HLPs by some weight function. The limitation of GM6 is that it gives lower weight to both bad leverage points (BLPs) and good leverage points (GLPs) which make its efficiency decreases when more GLPs are present in a data set. Moreover, the GM6 takes longer computational time. In this paper, we develop a new version of GM-estimator which is based on simple and fast algorithm. The attractive feature of this method is that it only downs weights BLPs and vertical outliers (VOs) and increases its efficiency. The merit of our proposed GM estimator is studied by simulation study and well-known aircraft data set.

Comparison of three robust impedance estimators for the MT method

The initial magnetotelluric (MT) response function estimator is based on the least-square theory; it can be severely disturbed by the cultural noise. The different robust procedures have been developed and improve the performance of response function estimation dramatically. It is hard to say which method is better or not. In a specific situation, a different approach has different performance. Therefore, it is important to know the different property of them. Between the robust procedures, the robust M-estimator gives a small weight to reject the outlier based on the residual between the output (electric filed) of the least-squares estimator and the observed data. M-estimator can reduce the influence of unusual data in the electric field (outliers) but are not sensitive to exceptional input (magnetic field) data, which are termed leverage points. The bounded influence (BI) estimator combines the standard robust M-estimator with leverage weighting based on the hat matrix diagonal element's statistics, a standard statistical measure of leverage point. Chave (2004) also creates an open-source code, Bounded Influence Remote Reference Processing (BIRRP), and it is widely used in the MT community. But the leverage point corresponds to the large variation of the magnetic field. It may be an energetic signal or active noise. The performance of the M-estimator and the BI-estimator was dramatically different in the two situations. On the other hand, the repeat median algorithm can protect against unusual data (outlier and leverage point) maximum. We researched the difference property of bound influence (BI-estimator), maximum likelihood (M-estimator), and repeat median (RM-estimator) signal site MT respond function estimator. Three independent code (BIRRP code, robust M-estimator code, and RM-estimator code) are used to compare them. At last, two typical field data are used, making the difference between the bound influence estimator and robust M-estimator transparent. We found that when the leverage point is the energetic signal, the M-estimator will perform better than the BI-estimator. When the leverage point is the active noise, the BI-estimator will work much better than the M-estimator. Finally, we also investigated the ability of the three estimators at a single site.

Source-dependent bias of sferics in magnetotelluric responses

The predominant signals of audio-frequency magnetotellurics (AMT) are called sferics, and they are generated by global lightning activity. When sferic signals are small or infrequent, measurement noise in electric and magnetic fields causes errors in estimated apparent resistivity and phase curves, leading to great model uncertainty. To reduce bias in apparent resistivity and phase, we use a global propagation model to link sferic signals in time series AMT data with commercially available lightning source information including strike time, location, and peak current. We then investigate relationships between lightning strike location, peak current, and the quality of the estimated apparent resistivity and phase curves using the bounded influence remote reference processing code. We use two empirical approaches to preprocessing time-series AMT data before estimation of apparent resistivity and phase: stitching and stacking (averaging). We find that for single-site AMT data, bias can be reduced by processing sferics from the closest and most powerful lightning strikes and omitting the lower amplitude signal-deficient segments in between. We hypothesized that bias can be further reduced by stacking sferics on the assumptions that lightning dipole moments are log-normally distributed whereas the superposed noise is normally distributed. Due to interference between dissimilar sferic waveforms, we tested a hybrid stitching-stacking approached based on clustering sferics using a wavelet-based waveform similarity algorithm. Our results indicate that the best approach to reduce bias was to stitch the closest and highest amplitude data.