Magnetotelluric Response Functions Research Articles

Analysis of Magnetotelluric Response Functions at the Cam Thuy Site, Northern Vietnam

During magnetotelluric studies in North Vietnam, an effect where the impedance phases Zxy and Zyx simultaneously left their quadrants was detected at one of the sounding sites, CTH, in the Cam Thuy District, Thanh Hoa Province. Investigation of this phenomenon has shown that it is accompanied by negative (bottom-up directed) components of the Umov–Poynting vector of electromagnetic energy flux density in the entire range of the recorded periods of magnetotelluric variations. This direction of the energy flux remains unchanged when the initial coordinate system is rotated by angles of 45° and 60°. Another characteristic feature of this phenomenon is an unusual behavior of the apparent resistivity curves: the apparent resistivity values increase starting from a period of 200 s up to a period of 104 s, and this increase occurs practically along the asymptote. The analysis of the polarization of the magnetotelluric field and the asymmetry parameter (skew angle) β of the phase tensor has shown that the section exhibits significant three-dimensional distortions as manifested by β values ranging from 20° to 90°.

An automatic preselection strategy for magnetotelluric single-site data processing based on linearity and polarization direction

The magnetotelluric response function can be severely disturbed by cultural electromagnetic noise. The preselection strategy is one of the effective ways to remove the influence of noise when calculating the response function. This study proposed three new parameters (the amplitude ratio predicted amplitude ratio and linear coherence (PLcoh) between the predicted and observed electric fields and the dispersion degree of the magnetic polarization direction (DDpol)) to detect noisy data, making the preselection strategy automatic. The first two were used to evaluate the linearity of binary linear regression to constrain incoherent noise, while the last was used to evaluate the magnetic polarization direction to constrain coherent noise. Finally, the technique is illustrated by applying it to two field datasets and comparing it with the previous studies. The results showed that these parameters can be used to effectively identify contaminated data, and a reliable response function can be obtained by using these parameters to extract high-quality data when intermittent noise contaminates field data.

A novel method for processing noisy magnetotelluric data based on independence of signal sources and continuity of response functions

Various schemes for processing magnetotelluric (MT) data have been reported aiming at suppressing the strong effect of artificial electromagnetic noises, especially coherent noise that is correlated between electric and magnetic time series. Many of the recent denoising schemes are based on decomposing MT data into the responses of the natural signals and noises. However, a steady differentiation of the natural signals from noises independent of any empirical choice of parameter setting is critically important. In addition, improper subtraction of values from the separated signals can lead to losing useful values of the natural signals or missing noise-affected values, which may result in failure in deriving the true MT responses. We propose a novel data-processing method that applies frequency-domain independent component analysis (FDICA) to both the local MT data and the reference magnetic data. Among the separated signals, the proposed method can quantitatively distinguish the natural signals from the noise-affected components by calculating the ratio of the cross-power spectrum with the reference data to the auto-power spectrum for each component. When determining which values to subtract from the separated signals, we introduce an evaluation index with respect to two characteristics of the MT response function: stationarity in the time domain and smoothness in the frequency domain. We conducted two types of experiments: with MT time series severely contaminated by synthetic coherent noises and with MT field data interfered with direct-current (DC) railways. Consequently, we confirmed the noise-suppression performance superiority of the proposed method over the conventional methods of MT data processing.

Analysis of sensitivity patterns for characteristics of magnetotelluric (MT) response functions in inversion

SUMMARYBecause the magnetotelluric (MT) method uses natural sources, the electric and magnetic fields recorded in the field acquisition are not directly used but usually converted into other MT response functions for interpretation such as inversion. Considering that inversion results are dependent on types of input data, it can be helpful to analyse different characteristics of MT response functions for inversion. In this study, we examine sensitivity patterns of MT response functions used commonly in MT inversion, which are the impedance tensor, apparent resistivity, phase, tipper, effective impedance and phase tensor; and investigate how their sensitivity patterns affect inversion results. We first describe overall tendencies of 3-D sensitivity patterns of the MT response functions, and then classify the MT response functions into six groups based on 2-D sensitivity patterns computed at the surface, which are briefly called ‘surface-sensitivity patterns’ in this study. The ’diagonal components of the impedance’ and ‘off-diagonal components of the phase tensor’, which have four petals-shaped surface-sensitivity patterns along the diagonal directions, belong to Group 1, and contribute to imaging 3-D subsurface structures from receivers installed evenly at the surface. Group 2 contains the ‘xy-components of the impedance, apparent resistivity and phase’ and ‘yy-component of the phase tensor’ whose surface-sensitivity patterns are linear in the y-axis. The ‘yx-components of the impedance, apparent resistivity and phase’ and ‘xx-component of the phase tensor’ that have strong linear surface-sensitivity patterns along the x-axis are classified into Group 3. The MT response functions of Groups 2 and 3 are useful for inversion of structures close to 2-D, whose strike extends along the y- and x-axes, respectively. Groups 4 and 5 include the ‘x- and y-components of tipper’ that possess linearly aligned two petals-shaped surface-sensitivity patterns in the x- and y-axes, respectively. The tipper can be helpful in imaging both 2-D and 3-D structures. The ‘effective impedance’ belongs to Group 6, whose surface-sensitivity patterns appear as a small circle. The surface-sensitivity patterns allow the effective impedance to have an advantage in interpretation of 1-D structures. By using several MT response functions for specific cases of 1-D, 2-D and 3-D interpretation of MT data, we investigate whether characteristics of the sensitivity patterns are reflected in modelling (simulating field data) and inversion results, and then suggest optimal MT response functions for those cases. In doing so, we show how to utilize the characteristics of the sensitivity patterns in inversion, and recommend the input MT response functions for inversion according to MT exploration situations. Our study provides basic information on similarities and differences of major MT response functions for inversion and insights on which MT response functions are suitable to increase the feasibility of MT inversion for different field situations based on the sensitivity patterns.

Augmented electrical resistivity tomography magnetotelluric response function

With the magnetotelluric (MT) method, conductive layers can mask deeper resistors. An analytical relationship is derived between electrical resistivity tomography (ERT) and MT data to calculate an augmented ERT-MT response function. The augmented ERT-MT response function adds additional high-frequency content in the MT response function. This allows us to model the near-surface conductive layer from the ERT data and resolve deeper resistive structures with the MT data. In general, for the 1D case, the augmented ERT-MT response function results in a 1D inverse model that demonstrates an improvement of up to approximately 91% in recovering the true resistivity model. The augmented ERT-MT response function also is tested on theoretical 2D models. Two electrical resistivity models are used for the 2D tests. For both 2D models, the resistive dike and conductive overburden that were initially concealed are recovered when using the augmented ERT-MT response function. The augmented ERT-MT response function also is applied to field data, allowing us to map a resistive dike below a conductive overburden that was absent in the standard MT inverse model. The recovered dike and fault zone in the augmented ERT-MT response function resistivity inverse model correlate with the same features observed in a magnetic susceptibility inverse model. The augmented ERT-MT response function technique improves the resolution of MT to recover resistors beneath a conductive layer for the theoretical and experimental cases.

Subsurface Structure around Mas Crater of Papandayan Volcano based on Magnetotelluric and Geomagnetic Data

Papandayan is a hydrothermally active volcano in Indonesia. We revealed the subsurface structure around the Mas Crater area of Papandayan based on the magnetotelluric (MT) and geomagnetic (GM) method. For the MT method, 14 sounding stations were deployed and two of them are located close to the active fumaroles. We estimated the MT response functions using a remote reference and then modeled the data with the aid of a 1-D robust inversion. The resistivity structure can generally be divided into three layers, namely a thin resistive surface layer, a middle conductive layer, and a more resistive basement. We interpreted the middle layer to be the hydrothermal zone or clay mineral. For the GM method, we measured the total intensity at 19 data points. The IGRF and diurnal variation were subtracted from the raw data. We then obtained the 2-D magnetic susceptibility model of magnetic field anomaly using an Occam inversion. The model shows a significantly low susceptibility structure beneath the fumaroles which might correlate with thermally demagnetized rocks.

Method for obtaining response functions from noisy magnetotelluric data using frequency-domain independent component analysis

The magnetotelluric (MT) method has been used for visualizing subsurface resistivity structures and more recently for monitoring resistivity changes. However, electromagnetic data often include cultural noise, which can cause errors in the estimation of MT response functions and subsurface resistivity structure analysis. Frequency-domain independent component analysis (FDICA) offers advantages for MT data processing particularly because this method can extract hidden components in the observed data. These components can be decomposed into natural MT signals and cultural noise so that the noise effect in the recovered MT data is reduced. FDICA is applied to MT data acquired at the Kakioka Magnetic Observatory in Japan. The apparent resistivity and phase curves are obtained with small estimated errors between periods of 7 and 12,000 s, although the length of the time-series data is limited. The curves are smoother than those obtained using a conventional method. Various types of synthetic noise are added to the time series at Kakioka to test the noise-reduction performance of FDICA for MT data with high noise contamination. The results demonstrate that FDICA can be used to estimate MT response functions with high accuracy even under conditions in which more than half of the time-series data are contaminated by noise.

Comparison of stable maximum likelihood estimator with traditional robust estimator in magnetotelluric impedance estimation

The newly developed stable maximum likelihood estimator (MLE) has been considered to be superior to the traditional robust estimation for Magnetotelluric (MT) transfer functions, based on the examination of field data. However, due to the unknown MT response function for real data, it is difficult to judge the validity of any comparisons between the performance of the two estimators. This can be realized by comparing the performances on synthetic data, as different ideal error distributions can be modeled (e.g., noise contaminated magnetic fields can be simulated by adding desired noise into the assumed clean version). In this paper, we generate synthetic MT data with desired error distributions (e.g. Gaussian and Cauchy distributions) based on the estimated MT impedance and Fourier transformed measured magnetic data (as the input) from one MT site located in southeast part of Tibet Plateau collected under the SinoProbe project. The comparison is demonstrated in terms of the apparent resistivity and phase, and their estimated standard deviations. The performance of the two methods and their variants are tested based on generated synthetic data with various noise patterns. Synthetic tests indicate that their performance in terms of handling noisy data is generally comparable. The bias in the estimation is more sensitive to noise in the input. Then they are applied to process real data from two MT sites in the same region, indicating their solutions are generally similar.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ МАГНИТОТЕЛЛУРИЧЕСКОГО ПОЛЯ В ОДНОМЕРНЫХ КУСОЧНО-ГРАДИЕНТНЫХ СРЕДАХ

The solution of forward problems of magnetotelluric sounding for a case of one-dimensional piecewise gradient environments is received. Two problems are considered: the problem of calculating the impedance on the Earth’s surface for a gradient model of the upper half-space (Earth's atmosphere) and the problem of determining the magnetotelluric impedance for layered, piecewise-gradient models of the geological medium, which allow mathematical modeling of the magnetotelluric response functions and study the interaction of electromagnetic field and gradient models of lithosphere media. Such models extend the possibilities of development of the theory and practice of magnetotelluric research, first of all, when solving inverse problems, which is associated with the advantage of analytical solutions over numerical solutions of inverse problems of geophysics in terms of sharp reduction of their dimensionality. The main focus of the work is on the study of geodynamic processes in passive electromagnetic monitoring, the foundations of which are laid in the article (Alexandrov, Rybin, Zabinyakova, 2018).

Direct Method Solution of 3-D Magnetotelluric Modeling Using Vector Finite Element Method

As exploration is forced into more difficult areas with complex three-dimensional (3-D) structural environments, the importance of 3-D magnetotelluric (MT) modeling is essential for the correct interpretation of MT data. To reduce the complexity of the calculations introduced by 3-D models, iterative forward calculation of MT response functions is used as basis for inversion of 3-D MT data. This paper proposes an alternative procedure for making reliable 3-D MT modeling codes for forward calculation that is not only effective but also accurate. This is accomplished by using a direct method to solve the linear systems arising from the discretization process in the vector finite element approach. The vector finite element method is known for its capability of overcoming difficulties in modeling caused by possible jumps of normal components across discontinuities of material properties. Meanwhile, by using a direct method rather than an iterative method, the process of solving the linear equations does not suffer from slow convergence. Here, we present a comparison between our modeling codes and codes based on a different approach. In the resulting 3-D MT responses it was found that the proposed method has high accuracy.

A time-frequency analysis method to obtain stable estimates of magnetotelluric response function based on Hilbert-Huang transform

The time-frequency analysis method represents signal as a function of time and frequency, and it is considered a powerful tool for handling arbitrary non-stationary time series by using instantaneous frequency and instantaneous amplitude. It also provides a possible alternative to the analysis of the non-stationary magnetotelluric (MT) signal. Based on the Hilbert-Huang transform (HHT), a time-frequency analysis method is proposed to obtain stable estimates of the magnetotelluric response function. In contrast to conventional methods, the response function estimation is performed in the time-frequency domain using instantaneous spectra rather than in the frequency domain, which allows for imaging the response parameter content as a function of time and frequency. The theory of the method is presented and the mathematical model and calculation procedure, which are used to estimate response function based on HHT time-frequency spectrum, are discussed. To evaluate the results, response function estimates are compared with estimates from a standard MT data processing method based on the Fourier transform. All results show that apparent resistivities and phases, which are calculated from the HHT time-frequency method, are generally more stable and reliable than those determined from the simple Fourier analysis. The proposed method overcomes the drawbacks of the traditional Fourier methods, and the resulting parameter minimises the estimation bias caused by the non-stationary characteristics of the MT data.

Estimation of the Magnetotelluric Response Function: The Path from Robust Estimation to a Stable Maximum Likelihood Estimator

The robust statistical model of a Gaussian core contaminated by outlying data in use since the 1980s, and which underlies modern estimation of the magnetotelluric (MT) response function, is re-examined from first principles. The residuals from robust estimators applied to MT data are shown to be systematically long-tailed compared to a distribution based on the Gaussian and hence inconsistent with the robust model. Instead, MT data are pervasively described by the stable distribution family for which the Gaussian is an end member, but whose remaining distributions have algebraic rather than exponential tails. The validity of the stable model is rigorously demonstrated using a permutation test. A maximum likelihood estimator (MLE), including the use of a remote reference, that exploits the stable nature of MT data is formulated, and its two-stage implementation, in which stable parameters are first fit to the residuals, and then the MT responses are solved for, with iteration between them, is described. The MLE is inherently robust, but differs from a conventional robust estimator because it is based on a statistical model derived from the data rather than being ad hoc. Finally, the covariance matrices obtained from MT data are pervasively improper as a result of weak non-stationarity, and the Cramer–Rao lower bound for the improper covariance matrix is derived, resulting in reliable second-order statistics for MT responses. The stable MLE was applied to an exemplar broadband data set from northwest Namibia. The stable MLE is shown to be consistent with the statistical model underlying linear regression and hence is unconditionally unbiased, in contrast to the robust model. The MLE is compared to conventional robust remote reference and two-stage estimators, establishing that the standard errors of the former are systematically smaller than for either of the latter, and that the standardized differences between them exhibit excursions that are both too frequent and too large to be described by Gaussian statistics. These excursions are more prevalent when the tail thickness parameter of the stable distribution is small, and are attributed to rising bias in the robust estimator that is also consistent with the Berry–Esseen theorem that defines the rate of convergence to the central limit theorem value. An explanation for weak non-stationarity of MT data is proposed, and several extensions to the present work are described.

New application of wavelets in magnetotelluric data processing: reducing impedance bias

Magnetotelluric (MT) data consist of the sum of several types of natural sources including transient and quasiperiodic signals and noise sources (instrumental, anthropogenic) whose nature has to be taken into account in MT data processing. Most processing techniques are based on a Fourier transform of MT time series, and robust statistics at a fixed frequency are used to compute the MT response functions, but only a few take into account the nature of the sources. Moreover, to reduce the influence of noise in the inversion of the response functions, one often sets up another MT station called a remote station. However, even careful setup of this remote station cannot prevent its failure in some cases. Here, we propose the use of the continuous wavelet transform on magnetotelluric time series to reduce the influence of noise even for single site processing. We use two different types of wavelets, Cauchy and Morlet, according to the shape of observed geomagnetic events. We show that by using wavelet coefficients at clearly identified geomagnetic events, we are able to recover the unbiased response function obtained through robust remote processing algorithms. This makes it possible to process even single station sites and increase the confidence in data interpretation.

Study on the deep conductive geologic structure by magnetovariational method

In the magnetotelluric method, along with the magnetotelluric response functions originating from linear relations between components of the electric and magnetic fields we can determine the magnetovariational response functions derived from linear relations between components of the magnetic field Hx, Hy, Hz. This consideration may significantly enhance the capabilities of the magnetotellurics, since at low frequencies the magnetic field becomes free of nearsurface distortions and shines a nondeceptive light on the deep geoelectric structures. The components of the magnetic field are represented by Wiese–Parkinson matrix 𝑊 . From the matrix, we use the transformations to construct Vozoff tipper 𝑉 , tipper phase ψ, and tipper ellipticity of magnetic field H   to study the electrical heterogeneity. We obtain more information than previous methods because V gives two parameters: direction and amplitude, |V| > |ReW| and |V| > |ImW|; therefore data interpretation has many advantages over previous methods. The results allow us to give meaningful conclusions about the geology, such as mapping some deep conductive geologic structures of the crust.

Electrical resistivity structure under the western Cosmonauts Sea at the continental margin of East Antarctica inferred via a marine magnetotelluric experiment

The western Cosmonauts Sea, off the coast of East Antarctica, was a site of rifting of the Gondwana supercontinent and subsequent early seafloor spreading. To improve our understanding of the breakup of Gondwana, we conducted a marine magnetotelluric experiment to determine the electrical resistivity structure within the uppermost several hundred kilometers beneath the western Cosmonauts Sea. Magnetotelluric response functions at two sites, obtained after considering possible influences of non-plane magnetic field sources, suggest that these responses include distortions by topographic variations and conductive anomalies around the observation sites. Three-dimensional forward modeling confirmed that these distortions due to topographic variations and a thin (∼2-km thick) conductive layer immediately under the sites (mostly sediments) are severe. Furthermore, three-dimensional forward modeling to investigate the resistivity structure at deeper depths revealed an upper resistive layer (≥300 Ω-m), with a thickness of <100 km, and an underlying conductive half-space (∼10 Ω-m). The upper resistive layer and the underlying conductive structure most likely represent dry and water/melt-rich oceanic upper mantle, respectively. The upper resistive layer may be thinner than anticipated under the old seafloor of the study area (likely >90 Ma), and may suggest a conductive anomaly in the upper mantle produced by mantle convection and/or upwelling.

Magnetotelluric data, stable distributions and impropriety: an existential combination

The robust statistical model of a Gaussian core contaminated by outlying data that underlies robust estimation of the magnetotelluric (MT) response function has been re-examined. The residuals from robust estimators are systematically long tailed compared to a distribution based on the Gaussian, and hence are inconsistent with the robust model. Instead, MT data are pervasively described by the alpha stable distribution family whose variance and sometimes mean are undefined. A maximum likelihood estimator (MLE) that exploits the stable nature of MT data is formulated, and its two-stage implementation in which stable parameters are first fit to the data and then the MT responses are solved for is described. The MLE is shown to be inherently robust, but differs from the conventional robust estimator because it is based on a model derived from the data, while robust estimators are ad hoc, being based on the robust model that is inconsistent with actual data. Propriety versus impropriety of the complex MT response was investigated, and a likelihood ratio test for propriety and its null distribution was established. The Cramér-Rao lower bounds for the covariance matrix of proper and improper MT responses were specified. The MLE was applied to exemplar long period and broad-band data sets from South Africa. Both are shown to be significantly stably distributed using the Kolmogorov–Smirnov goodness of fit and Ansari-Bradley non-parametric dispersion tests. Impropriety of the MT responses at both sites is pervasive, hence the improper Cramér-Rao bound was used to estimate the MLE covariance. The MLE is shown to be nearly unbiased and well described by a Gaussian distribution based on bootstrap simulation. The MLE was compared to a conventional robust estimator, establishing that the standard errors of the former are systematically smaller than for the latter and that the standardized differences between them exhibit excursions that are both too frequent and too large to be described by a Gaussian model. This is ascribed to pervasive bias of the robust estimator that is to some degree obscured by their systematically large confidence bounds. Finally, a series of topics for further investigation is proposed.

Magnetotelluric Response Function Estimation Based on Hilbert–Huang Transform

Magnetotelluric (MT) data series are non-stationary random signals that do not meet the basic requirements of conventional methods based on the Fourier transform. To minimize the estimation bias errors brought about by the non-stationary characteristics of MT data, a new method, based on the Hilbert–Huang transform (HHT), is proposed for the first time for estimating the MT response functions from a time series of electromagnetic field variations. With the HHT method, the amplitude of data series are expressed as a function of frequency and time and then response functions are estimated statistically from the time–frequency spectrum. Mathematical model and calculation processes are introduced and some simulated data are analyzed to verify the correctness of the method. Finally, the measured MT data is facilitated by applying the HHT to assess the ability of HHT method to quantify meaningful geologic information.

Upper mantle electrical resistivity structure beneath the central Mariana subduction system

This paper reports on a magnetotelluric (MT) survey across the central Mariana subduction system, providing a comprehensive electrical resistivity image of the upper mantle to address issues of mantle dynamics in the mantle wedge and beneath the slow back‐arc spreading ridge. After calculation of MT response functions and their correction for topographic distortion, two‐dimensional electrical resistivity structures were generated using an inversion algorithm with a smoothness constraint and with additional restrictions imposed by the subducting slab. The resultant isotropic electrical resistivity structure contains several key features. There is an uppermost resistive layer with a thickness of up to 150 km beneath the Pacific Ocean Basin, 80–100 km beneath the Mariana Trough, and 60 km beneath the Parece Vela Basin along with a conductive mantle beneath the resistive layer. A resistive region down to 60 km depth and a conductive region at greater depth are inferred beneath the volcanic arc in the mantle wedge. There is no evidence for a conductive feature beneath the back‐arc spreading center. Sensitivity tests were applied to these features through inversion of synthetic data. The uppermost resistive layer is the cool, dry residual from the plate accretion process. Its thickness beneath the Pacific Ocean Basin is controlled mainly by temperature, whereas the roughly constant thickness beneath the Mariana Trough and beneath the Parece Vela Basin regardless of seafloor age is controlled by composition. The conductive mantle beneath the uppermost resistive layer requires hydration of olivine and/or melting of the mantle. The resistive region beneath the volcanic arc down to 60 km suggests that fluids such as melt or free water are not well connected or are highly three‐dimensional and of limited size. In contrast, the conductive region beneath the volcanic arc below 60 km depth reflects melting and hydration driven by water release from the subducting slab. The resistive region beneath the back‐arc spreading center can be explained by dry mantle with typical temperatures, suggesting that any melt present is either poorly connected or distributed discontinuously along the strike of the ridge. Evidence for electrical anisotropy in the central Mariana upper mantle is weak.

A discussion of bias in magnetotelluric responses

Magnetic-field noise biases magnitudes of magnetotelluric response functions such as resistivity, impedance, and the induction vector (tipper). This is particularly severe in the so-called dead band between about 1- and [Formula: see text] periods. The standard approach to remove these biases is to use a magnetic remote reference. In cases where this is unavailable or does not work because noise is correlated at the local and remote sites, it is possible to extract unbiased estimates using an inversion algorithm that predicts the magnitude from the phase. However, an important condition for this to be successful is that phase estimates are unbiased. Phase can be biased if parallel components of the electric and magnetic fields are correlated. This arises when the structure is 3D or when a 2D response is estimated in coordinates not parallel to strike. Improvements in electric field measurements make electric field references a viable option for estimating low-bias responses up to periods approaching [Formula: see text].

The statistical distribution of magnetotelluric apparent resistivity and phase

SUMMARY The marginal distributions for the magnetotelluric (MT) magnitude squared response function (and hence apparent resistivity) and phase are derived from the bivariate complex normal distribution that describes the distribution of response function estimates when the Gauss‐ Markov theorem is satisfied and the regression random errors are normally distributed. The distribution of the magnitude squared response function is shown to be non-central chi-squared with 2 degrees of freedom, with the non-centrality parameter given by the squared magnitude of the true MT response. The standard estimate for the magnitude squared response function is biased, with the bias proportional to the variance and hence important when the uncertainty is large. The distribution reduces to the exponential when the expected value of the MT response function is zero. The distribution for the phase is also obtained in closed form. It reduces to the uniform distribution when the squared magnitude of the true MT response function is zero or its variance is very large. The phase distribution is symmetric and becomes increasingly concentrated as the variance decreases, although it is shorter-tailed than the Gaussian. The standard estimate for phase is unbiased. Confidence limits are derived from the distributions for magnitude squared response function and phase. Using a data set taken from the 2003 Kaapvaal transect, it is shown that the bias in the apparent resistivity is small and that confidence intervals obtained using the non-parametric delta method are very close to the true values obtained from the distributions. Thus, it appears that the computationally simple delta approximation provides accurate estimates for the confidence intervals, provided that the MT response function is obtained using an estimator that bounds the influence of extreme data.