Comment on “Systematic survey of high‐resolution b value imaging along Californian faults: Inference on asperities” by T. Tormann et al.

Abstract

Tormann et al. [2014] propose a distance exponential weighted (DEW) b value mapping approach as an improvement to previous methods of constant radius and nearest neighborhood. To test the performance of their proposed method the authors introduce a score function:where N is the total number of grid nodes, n is the number of nonempty nodes, and b true and b est are the true and estimated b values, respectively. This score function is applied on a semisynthetic earthquake catalog to make inference on the parameters of the method. In this comment we argue that the proposed methodology cannot be applied to seismic analysis since it requires a priori knowledge of the spatial b value distribution, which it aims to reveal.The score function given in equation (1) seeks to minimize the absolute difference between the generating (true) and estimated b values. Like any statistical parameter, the estimation error of the b value decreases with increasing sample size. However, since the b value is a measure of slope on a log-log plot, for any given sample size, the estimation error is not only asymmetric but also its amplitude is dependent on the b value itself.To make a pedagogical analogy; conducting b value analysis by limiting the sample size (Tormann et al. use 150) is similar to measuring temperature with a peculiarly short thermometer. For small sample sizes, such a thermometer would measure lower temperatures more precisely than higher ones, to the extent that it would be impossible to distinguish between values on the upper end. Only when the sample size, and hence the resolution, is increased does the thermometer become reliable across the whole measurement range. For an illustration in Figure 1a, we show the confidence intervals for 6 b values; accordingly, we divide our thermometer into 6 intervals. The length of each interval is scaled with 1/σ N as a proxy for resolution, where σ N is the standard deviation for N number of samples. The analogous thermometers obtained for N = 150, 300, and 1500 are shown in Figure 1b.Intersecting the confidence interval curves of different b values, one can derive that the crudest measurement in the range [0.6-1.5] would require a resolution of Δb = 0.3 that is achieved with at least~170 samples. An analysis in this constrained setting can correspond only to a mere classification of b values as low (0.70 ± 0.10), medium (1.00 ± 0.13), or high (1.30 ± 0.17). Such a graphical inference is, however, an oversimplification because determining if two Gutenberg-Richter (GR) distributions are significantly different (for a given p value) would require nested-hypothesis tests (see supporting information).Compared to the previous sampling techniques, the proposed DEW method features two additional parameters; number of maximum events (N max ) and coefficient of exponential magnitude weight decay with distance (λ). The remaining parameters, common with the previous approaches, are set to the following values: minimum number of events (N min = 50), maximum search radius (R max = 7.5), and m...

Reply to Comment by Kamer on “Systematic survey of highresolution b value imaging along Californian faults: Inference on asperities”

“…Kamer [] suggests that the DEW sampling parameter choice ( λ ) is ambiguous and dependent on the setup of the synthetic test. That is true, of course, and we acknowledge it in our paper (p. 6), but argue based on a sensitivity analysis that our choice of 0.7 is robust and that results do not critically depend on this choice.…”

“…Kamer [] claims that limiting the maximum radius biases the resolved structures and finds that a sampling radius of 40 km together with a distance decay of λ = 0.01 gives inconsistently different results compared with our chosen parameter combination. It is obvious that the resolution of any sampling method, i.e., minimum identifiable structure size, is limited by the sampling parameters selected.…”

“…Imagine doing a tomographic study along this section but proposing sampling volumes as large as the study area—hardly a study that would pass peer review. Figure b in Kamer [] that, obviously, with this parameter choice, also the recovery of any synthetic cross section with known significant input anomalies would look as homogeneous as for the real Parkfield magnitudes shown in his Figure 3. Already a decade ago, Schorlemmer et al .…”

“…But we initially like to point out that it is largely not pertinent to our publication in the first place, because the objective of the Tormann et al . [] study was not to derive an automated and self‐optimizing way to map b values as Kamer [] implies. Our goal was targeted toward imaging potential asperities, low b value structures along faults, of sizes capable of accommodating M 6+ events, i.e., structures of ~20–40 km length.…”

“…Kamer [] addresses three main aspects to underline his impression that our approach would not be applicable to seismicity analysis: (1) the well‐known and often published sample size and absolute value‐dependent uncertainty of b values; (2) the obvious and previously published finding that large sampling radii (he suggests R max = 40 km and negligible distance decay λ = 0.01) recover a homogeneous b value distribution, “inconsistent” with high‐resolution sampling (e.g., R max = 7.5 km and λ = 0.7 in our study), and thus significant differences in derived probabilities; and (3) the claim that choice of parameters would favor and even artificially create b value anomalies.…”