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[SGD01-P01] Why can a spatiotemporal functional model of postseismic deformations be expressed with common time constants?
Keywords:the 2011 Tohoku-Oki Earthquake, crustal deformation, postseismic deformation, GNSS time series, predicting model
Introduction
The time series of postseismic deformations of the 2011 Tohoku-Oki Earthquake observed at the GEONET stations can be approximated by a simple spatiotemporal function to predict the deformations over a wide area with high accuracy. The prediction can be done with sufficient accuracy even if the same time constants are used for all stations and components. This simplifies the functional form and its handling, which significantly enhances the convenience and applicability of the model. In this presentation, we discuss why the functional model can be expressed with common time constants.
Method and Results
Tobita (2016) showed that a mixed model of two logarithmic and one exponential functions, represented by the following equation, can be used to predict the postseismic deformation:
D(t) = a ln(1+ t/b) + c + d ln(1+ t/e) - f exp(– t/g) + Vt,
where D(t) is each component of the postseismic deformation at a station, t is days after the earthquake, ln is the natural logarithm, b, e, and g are the relaxation time constants common to all stations, and V is the steady-state velocity at each station. The time constants were determined for stations showing representative variations; for the other stations, the time constants were given in common, and the amplitude coefficients (a, d, f) were determined for each station and component by the least-squares method.
In addition to this equation, fluctuations that accumulate at a constant rate have been found since 2015, suggesting that a new steady slip at a plate interface has been occurring since 2015. We reduced the overall standard deviation by enhancing the functional model to incorporate this linear deformation (Fujiwara et al. 2022).
Spatial distribution of the coefficients (a, d, f) are nonrandom, and the distribution is spatially smooth (see Figure). In other words, the local effects near the observation station are sufficiently small, and afterslip at the plate interface and viscoelastic relaxation in the upper mantle are mainly observed on the ground (Fujiwara et al. 2022).
Here, the functions with the fitting period of 2.0-year and 3.9-year are compared, and the following results are obtained.
(1) The coefficients of space (a, d, f) and time (b, e, g) are strongly correlated with each other.
(2) When the time constants change, the distribution of the spatial coefficients (a, d, f) changes at a constant rate for all observation stations, absorbing the change in the total functional model. This is because the time constants have time periods of several days, tens of days, and thousands of days, and a small change in the time constants are replaced by a change in the distribution among the spatial coefficients.
(3) As the results, despite the differences in the original time constants for each observation station, the overall residuals can be made sufficiently small by changing the spatial coefficients.
Tobita (2016) reported that a single logarithmic function approximates the sum of other multiple logarithmic functions with different time constants, which explains why the whole can be approximated by a function with a uniform time constant, even if the individual observation stations comprise components with different time constants.
Conclusion
The postseismic deformations of the earthquake is a complex phenomenon with multiple relaxation time constants, and each phenomenon is systematically generated and propagated, and observed on the ground. In practical terms, it can be concisely expressed as a spatial function by combining three time functions with time constants ranging from several days to thousands of days, setting up a common time function.
Tobita M (2016) https://doi.org/10.1186/s40623-016-0422-4.
Fujiwara S, Tobita M, and Ozawa S (2022) https://doi.org/10.1186/s40623-021-01568-0.
The time series of postseismic deformations of the 2011 Tohoku-Oki Earthquake observed at the GEONET stations can be approximated by a simple spatiotemporal function to predict the deformations over a wide area with high accuracy. The prediction can be done with sufficient accuracy even if the same time constants are used for all stations and components. This simplifies the functional form and its handling, which significantly enhances the convenience and applicability of the model. In this presentation, we discuss why the functional model can be expressed with common time constants.
Method and Results
Tobita (2016) showed that a mixed model of two logarithmic and one exponential functions, represented by the following equation, can be used to predict the postseismic deformation:
D(t) = a ln(1+ t/b) + c + d ln(1+ t/e) - f exp(– t/g) + Vt,
where D(t) is each component of the postseismic deformation at a station, t is days after the earthquake, ln is the natural logarithm, b, e, and g are the relaxation time constants common to all stations, and V is the steady-state velocity at each station. The time constants were determined for stations showing representative variations; for the other stations, the time constants were given in common, and the amplitude coefficients (a, d, f) were determined for each station and component by the least-squares method.
In addition to this equation, fluctuations that accumulate at a constant rate have been found since 2015, suggesting that a new steady slip at a plate interface has been occurring since 2015. We reduced the overall standard deviation by enhancing the functional model to incorporate this linear deformation (Fujiwara et al. 2022).
Spatial distribution of the coefficients (a, d, f) are nonrandom, and the distribution is spatially smooth (see Figure). In other words, the local effects near the observation station are sufficiently small, and afterslip at the plate interface and viscoelastic relaxation in the upper mantle are mainly observed on the ground (Fujiwara et al. 2022).
Here, the functions with the fitting period of 2.0-year and 3.9-year are compared, and the following results are obtained.
(1) The coefficients of space (a, d, f) and time (b, e, g) are strongly correlated with each other.
(2) When the time constants change, the distribution of the spatial coefficients (a, d, f) changes at a constant rate for all observation stations, absorbing the change in the total functional model. This is because the time constants have time periods of several days, tens of days, and thousands of days, and a small change in the time constants are replaced by a change in the distribution among the spatial coefficients.
(3) As the results, despite the differences in the original time constants for each observation station, the overall residuals can be made sufficiently small by changing the spatial coefficients.
Tobita (2016) reported that a single logarithmic function approximates the sum of other multiple logarithmic functions with different time constants, which explains why the whole can be approximated by a function with a uniform time constant, even if the individual observation stations comprise components with different time constants.
Conclusion
The postseismic deformations of the earthquake is a complex phenomenon with multiple relaxation time constants, and each phenomenon is systematically generated and propagated, and observed on the ground. In practical terms, it can be concisely expressed as a spatial function by combining three time functions with time constants ranging from several days to thousands of days, setting up a common time function.
Tobita M (2016) https://doi.org/10.1186/s40623-016-0422-4.
Fujiwara S, Tobita M, and Ozawa S (2022) https://doi.org/10.1186/s40623-021-01568-0.