Optimizing differential travel-time measurements with dynamic time warping

Abstract

Precise differential travel-time measurement is essential for earthquake relative locating. The waveform cross-correlation (WCC) technique is widely regarded as the most effective method for calculating the differential travel-time of seismic phases. However, for earthquake pairs with large magnitude differences, substantial biases can arise due to disparities in the duration of the initial pulse, potentially leading to significant mislocations, particularly for mainshocks. To overcome this limitation, we propose to use the dynamic time warping (DTW) algorithm to optimize differential travel-time calculation. Using high-quality earthquake waveform data from the San Andreas Fault (2012 −2019), we systematically compared the performance of DTW and WCC, respectively. Our results demonstrate that DTW substantially improves differential travel-time measurements, especially in cases involving large magnitude differences. In addition, we tested the robustness of DTW using noisy seismic data, demonstrating its superior resilience to noise.

1. Introduction

High-precision earthquake locations play a vital role in revealing the underlying source process, resolving fault zone complexities, and effectively assessing seismic risks [1 −7], with accurate measurement of differential travel-times (DTT) being a key prerequisite for precise relocation. Currently, waveform cross-correlation (WCC) method is considered the most reliable approach for determining the DTT between earthquake pairs. The use of WCC measurements can significantly reduce errors in calculating the DTT, typically achieving millisecond-level precision for similar events, thereby greatly improving the precision of relative location [8 −12]. Nevertheless, a recent study [13] found that WCC is prone to introducing substantial errors in DTT calculation when the magnitude difference between an earthquake pair is large due to the unequal first pulse durations. Such errors may lead to considerable mislocations, including a marked displacement of the mainshock relative to the associated earthquake cluster [13]. To obtain more reliable seismic location results, researchers sometimes exclude earthquakes with large magnitudes (e.g., mainshocks) from the dataset when calculating the DTT with WCC, relocating only the remaining events [14, 15]. While this approach may mitigate the impact of magnitude differences to some extent, it could also compromise the comprehensiveness and precision of the relative location results, hindering subsequent interpretation and analysis.

The dynamic time warping (DTW) algorithm is a powerful tool for analyzing the similarity of time series. Originally developed for speech recognition [16, 17], it has since been applied in numerous fields, including handwritten signature matching [18], computer vision and animation [19], data mining [20], time series clustering [21], and healthcare [22], among others. Moreover, extensive research in seismology has confirmed the feasibility of DTW applications [23−25]. Compared to WCC, DTW offers distinct advantages in assessing time series similarity. Specifically, WCC requires the compared time series to be of equal length and relies on strict point-to-point correspondence to calculate similarity (Figure 1(a)). In contrast, the DTW algorithm is more flexible and achieves optimal alignment between time series through an elastic matching mechanism, allowing non-linear scaling along the time axis and minimizing mismatches caused by time shifts and distortions (Figure 1(b)). This flexible one-to-many or many-to-one mapping enables DTW to capture more complex and intrinsic similarities between time series without requiring them to be of equal length, as it can effectively handle both equal and unequal-length inputs. KUMAR et al [24] demonstrated the superior performance of the DTW algorithm in measuring time series similarity, showing that it offers higher precision than WCC and recommending it as a viable alternative for specific applications such as earthquake detection and waveform clustering.