Paired t-test sharpens confidence intervals for subsurface resistivity models

A new statistical scheme replaces the F-test, used in previous magnetotelluric studies from 2006 to 2024, with a paired t-test for estimating confidence intervals of subsurface electrical resistivity models, arguing that the F-test is statistically inappropriate and tends to overestimate the intervals. Their replacement, a paired t-test applied to raw data misfits, yields confidence intervals that are systematically narrower and, they argue, statistically defensible.

Magnetotellurics and geomagnetic depth sounding infer electrical resistivity deep in the crust by measuring natural variations of electric and magnetic fields at the surface. The observed quantities are magnetotelluric impedances, which link measured electric fields to magnetic fields, and geomagnetic transfer functions, which relate vertical magnetic variations to horizontal ones. Regularized inversion of these data produces a three-dimensional resistivity model, but such a model is only useful if researchers can state how precisely a given feature, such as a magma reservoir, is resolved. Model uncertainty estimation has rarely been performed for resistivity models, the authors note, because the electromagnetic induction equation is highly nonlinear and because regularized inversion complicates the statistics.

The core of the new paper is an empirical demonstration that the statistical foundation of the old approach is flawed. Previous studies assessed uncertainty by modifying the optimal model, replacing a target region with a single constant resistivity, running forward modeling, and testing whether the chi-squared misfit changed significantly using an F-test. That test formally requires both chi-squared misfits in the ratio to follow chi-squared distributions, which in turn requires the underlying standardized residuals, called z misfits, to be normally distributed. Using data from 18 sites around Mount Azuma, an active volcano in northeastern Japan, the authors applied the Shapiro-Wilk test to the z misfit distributions. Only one site out of eighteen produced a p-value above the conventional 0.05 significance level, and the distributions showed large skewness and kurtosis. Even pooled across all sites, the misfits failed normality. The chi-squared misfits therefore do not follow chi-squared distributions, and the F-test is inappropriate.

The revised scheme keeps the forward-modeling strategy but changes both the statistic and the test. A target block, a contiguous group of model elements such as a low-resistivity reservoir zone, is assigned a single trial resistivity, and forward modeling computes the predicted impedances and transfer functions. Each standardized residual of the test model is paired with the corresponding residual of the optimal model, since both are evaluated at the same sites, the same data types and the same periods. The paired t-test then asks whether the mean of the paired differences differs significantly from zero, with degrees of freedom equal to the number of data minus one. Because the test statistic can be positive or negative, a two-tailed threshold is used, and cases in which the modified model actually improves the misfit are explicitly excluded from the significant set.

The authors applied the method to the Mount Azuma model, where the inversion of 3888 impedance and transfer function values, collected over periods from about 0.01 to 1024 seconds, had reached a normalized root mean squared misfit of 2.42. Three nested target blocks were tested, defined as regions where resistivity falls below 1, 3 or 5 ohm-meters, interpreted as the magma and hydrothermal reservoir. Testing site by site, the block below 3 ohm-meters yielded a 95 percent confidence interval of 1.0 to 2.6 ohm-meters and a 99 percent interval of 1.0 to 3.0 ohm-meters. The behavior of the other two blocks proved just as informative about the data as about the reservoir.

For the smallest block, below 1 ohm-meter, the 95 percent confidence interval split into two disconnected ranges, and the 99 percent interval could not be constrained at all between 0.01 and 100 ohm-meters. For the largest block, below 5 ohm-meters, the test statistic exceeded the threshold at every trial resistivity, collapsing the 95 percent interval to zero, while the 99 percent interval constrained resistivity to 2.3 to 3.2 ohm-meters. The authors interpret these outcomes as a diagnostic of dataset resolution: an unconstrained interval means the data cannot resolve the block at that scale, whereas a vanishing interval means the block is larger than the resolvable scale and cannot be represented by one number. By trying blocks of different sizes, a practitioner can locate the intrinsic resolution scale of the dataset.

The comparison with the F-test is stark. Applied to the same data, the F-test failed to constrain the blocks below 1 and 3 ohm-meters at either confidence level, and for the block below 5 ohm-meters it produced intervals of 0.053 to 20 ohm-meters at 95 percent and 0.013 to 50 ohm-meters at 99 percent. The F-test thus tends to overestimate confidence intervals, sometimes dramatically, because it is being asked a question its assumptions cannot answer. When the paired t-test was run on all data pooled across sites rather than site by site, the intervals widened, since improvements at some sites can cancel deteriorations at others; the authors argue that site-by-site assessment extracts more of each dataset’s character and generally gives narrower intervals.

The tighter intervals change physical interpretations of the volcano. The earlier study of Ichiki and colleagues, working from a 67 percent F-test interval of roughly 0.3 to 5 ohm-meters, concluded that the shallow reservoir required a water-saturated dacitic melt fraction of about 22 percent by volume, with conductive hydrothermal fluids needed to explain the upper bound. Under the new 99 percent interval of 1.0 to 3.0 ohm-meters, neither bound of the shallow reservoir can be explained by water-saturated dacitic melt, implying an even larger share of conductive hydrothermal fluid than previously thought. For the deep portion, the permissible water-saturated andesitic melt fraction narrows to 5 to 29 percent, tightening the range consistent with seismic S-wave velocity constraints. The estimated resistivity lower bound of the silica sealing layer above the reservoir drops from 5.0 to 3.0 ohm-meters, corresponding to a porosity of about 5 percent with 0.1 ohm-meter fluid, which coincides with the percolation threshold for effusive eruption and makes the hydrological picture more self-consistent.

The authors also compared their intervals with the model covariance matrix, an alternative uncertainty measure extended to nonlinear electromagnetic inversion by Thiel and colleagues in 2009. Expressed as relative error, twice the square root of the covariance diagonal ranged from about 0.85 to 11 percent across the target blocks, far narrower than the forward-modeling intervals. The comparison suggests the covariance-based measure can be excessively narrow, because it reflects only local behavior around the optimal model. Notably, the volumetric mean of the logarithmic conductivities of elements in each block fell near the midpoint of the paired t-test interval, supporting the idea that electromagnetic sensitivity is proportional to logarithmic conductivity rather than conductivity itself. A block whose interval matches the standard deviation of its elements’ logarithmic conductivities appears to be sized appropriately to the data resolution.

The method has acknowledged limitations. It does not account for the regularization effect on uncertainty, confidence intervals can split into disconnected zones with ambiguous interpretation, and the degrees of freedom treatment ignores the number of model parameters, which could become problematic in underdetermined inversions. The authors flag comparisons with Bayesian, Markov chain Monte Carlo approaches and computationally efficient treatments of effective degrees of freedom as future work. Even so, the message for geophysicists is immediate: confidence intervals reported from F-tests on magnetotelluric inversions deserve re-examination, and the paired t-test offers a statistically grounded, computationally practical alternative that any group with a forward-modeling code can adopt.

Subject of Research: Statistical confidence interval estimation for subsurface electrical resistivity models using magnetotelluric data and forward modeling

Article Title: Statistical confidence interval inference for subsurface electrical resistivity structure from magnetotelluric impedances and geomagnetic transfer functions, assisted by forward modeling

Article References: Ichiki, M., & Siripunvaraporn, W. (2026). Statistical confidence interval inference for subsurface electrical resistivity structure from magnetotelluric impedances and geomagnetic transfer functions, assisted by forward modeling. Earth, Planets and Space, 78(1), Article 216. https://doi.org/10.1186/s40623-026-02531-7

Image Credits: AI Generated

DOI: 10.1186/s40623-026-02531-7

Keywords: magnetotellurics, geomagnetic depth sounding, electrical resistivity, confidence interval, paired t-test, F-test, forward modeling, model uncertainty, Mount Azuma, magma reservoir, regularized inversion, Shapiro-Wilk test