Erratum: An integral method to estimate the moment accumulation rate on the Creeping Section of the San Andreas Fault

Erratum: An integral method to estimate the moment accumulation rate on the Creeping Section of... Radar interferometry, Seismic cycle, Space geodetic surveys, Inverse theory, Continental tectonics: strike-slip and transform Erratum of the paper ‘An integral method to estimate the moment accumulation rate on the Creeping Section of the San Andreas Fault’, by Tong et al. published in Geophys. J. Int. (2015) 203, 48-62. Our recent publication (Tong et al. 2015) has a factor of two error in eq. (3) that relates the moment accumulation rate M per unit length of fault L to the integral of the surface residual velocity v(x) times the distance from the fault x for a 2-D dislocation in a uniform elastic half-space. The corrected formula is   \begin{equation}\frac{M}{L} = \mathop {\lim }\nolimits_{W \to \infty } \frac{{\mu \pi }}{{2W}}\int\nolimits_{{ - W}}^{W}{{x\nu \left( x \right)\,{\rm{d}}x}},\end{equation} (1) where W is the upper bound on the limit of integration. In practice, W should be much greater than the maximum locking depth of the fault Dm. This equation is conceptually important, although not practically important, because it shows that a slip rate inversion is not needed to estimate geodetic moment accumulation rate. The concept is used to provide formal bounds on moment accumulation rate directly from geodetic data. In addition to the use in our original publication, Maurer et al. (2017) reference this equation in their recent publication although it is not actually used for bounding the moment accumulation rate. This error affects two figures in our paper although the overall findings are unchanged. First, the values along the y-axis in Fig. 7 should be divided by 2. Second, the moment accumulation rate estimate (labelled profile a in Fig. 10) should be reduced by a factor of two. The reason these changes do not affect the overall findings of the paper is because we did not rely on this integral method for the moment accumulation rate estimation but only use the formula to explain why a slip rate inversion is not needed for this idealized example. To confirm that the new eq. (1) is correct, we begin with two well-known formulae. The first relates the moment accumulation rate per length of fault to the integral of the back-slip rate s(z) over depth:   \begin{equation}\frac{M}{L} = \mu \int\nolimits_{{ - {D_m}}}^{0}{{s\left( z \right)}}\,{\rm{d}}z.\end{equation} (2) The second relates the surface residual velocity to the convolution of the back-slip rate with the Green's function for a line dislocation (e.g. Segall 2010, eq. 2.41):   \begin{equation}\nu \left( x \right) = \frac{1}{\pi }\int\nolimits_{{ - {D_m}}}^{0}{{\frac{{s(z)x}}{{{x^2} + {z^2}}}}}{\rm{d}}z.\end{equation} (3) Inserting eq. (3) into eq. (1) and rearranging the order of integration, we obtain   \begin{equation}\frac{M}{L} = \mathop {\lim }\nolimits_{W \to \infty } \left[ {\frac{{\mu \pi }}{{2W}}\int\nolimits_{{ - W}}^{W}{{x\left( {\frac{1}{\pi }\int\nolimits_{{ - {D_m}}}^{0}{{\frac{{s\left( z \right)x}}{{{x^2} + {z^2}}}{\rm{d}}z}}} \right){\rm{d}}x}}} \right] = \mu \int\nolimits_{{ - {D_m}}}^{0}{{s\left( z \right)}}\left( {\mathop {\lim }\nolimits_{W \to \infty } \frac{1}{{2W}}\int\nolimits_{{ - W}}^{W}{{\frac{{{x^2}}}{{{x^2} + {z^2}}}{\rm{d}}x}}} \right){\rm{d}}z.\end{equation} (4) We perform the integral over x first and multiply by 2 after changing the limits because the integrand is symmetric about x = 0:   \begin{equation}\int\nolimits_{0}^{W}{{\frac{{{x^2}}}{{{x^2} + {z^2}}}{\rm{d}}x}} = \left. {\left( {x - z{\,{\tan }^{ - 1}}\frac{x}{z}} \right)} \right|_0^W.\end{equation} (5) In the limit as W → ∞ the final result is   \begin{equation}\mathop {\lim }\nolimits_{W \to \infty } \frac{1}{W}\left( {W - z{{\,\tan }^{ - 1}}\frac{W}{z}} \right) = 1,\quad{\rm{ for }}\,{z_{\max }} = {D_m} \ll W.\end{equation} (6) The moment accumulation rate is   \begin{equation}\frac{M}{L} = \mu \int\nolimits_{{ - {D_m}}}^{0}{{s\left( z \right){\rm{d}}z}},\end{equation} (7) which agrees with eq. (2). In addition to this check on eq. (1), one of the reviewers, Takeshi Sagiya, also confirmed our correction and determined where we went wrong in the original derivation. REFERENCES Maurer J., Segall P., Bradley A.M., 2017. Bounding the moment deficit rate on crustal faults using geodetic data: methods, J. geophys. Res. , 122( 8), 6811– 6835. https://doi.org/10.1002/2017JB014300 Google Scholar CrossRef Search ADS   Segall P., 2010. Earthquake and Volcano Deformation , Princeton University Press. Google Scholar CrossRef Search ADS   Tong X., Sandwell D.T., Smith-Konter B., 2015. An integral method to estimate the moment accumulation rate on the Creeping Section of the San Andreas Fault, Geophys. J. Int. , 203( 1), 48– 62. https://doi.org/10.1093/gji/ggv269 Google Scholar CrossRef Search ADS   © The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Geophysical Journal International Oxford University Press

Erratum: An integral method to estimate the moment accumulation rate on the Creeping Section of the San Andreas Fault

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The Royal Astronomical Society
Copyright
© The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society.
ISSN
0956-540X
eISSN
1365-246X
D.O.I.
10.1093/gji/ggy043
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Abstract

Radar interferometry, Seismic cycle, Space geodetic surveys, Inverse theory, Continental tectonics: strike-slip and transform Erratum of the paper ‘An integral method to estimate the moment accumulation rate on the Creeping Section of the San Andreas Fault’, by Tong et al. published in Geophys. J. Int. (2015) 203, 48-62. Our recent publication (Tong et al. 2015) has a factor of two error in eq. (3) that relates the moment accumulation rate M per unit length of fault L to the integral of the surface residual velocity v(x) times the distance from the fault x for a 2-D dislocation in a uniform elastic half-space. The corrected formula is   \begin{equation}\frac{M}{L} = \mathop {\lim }\nolimits_{W \to \infty } \frac{{\mu \pi }}{{2W}}\int\nolimits_{{ - W}}^{W}{{x\nu \left( x \right)\,{\rm{d}}x}},\end{equation} (1) where W is the upper bound on the limit of integration. In practice, W should be much greater than the maximum locking depth of the fault Dm. This equation is conceptually important, although not practically important, because it shows that a slip rate inversion is not needed to estimate geodetic moment accumulation rate. The concept is used to provide formal bounds on moment accumulation rate directly from geodetic data. In addition to the use in our original publication, Maurer et al. (2017) reference this equation in their recent publication although it is not actually used for bounding the moment accumulation rate. This error affects two figures in our paper although the overall findings are unchanged. First, the values along the y-axis in Fig. 7 should be divided by 2. Second, the moment accumulation rate estimate (labelled profile a in Fig. 10) should be reduced by a factor of two. The reason these changes do not affect the overall findings of the paper is because we did not rely on this integral method for the moment accumulation rate estimation but only use the formula to explain why a slip rate inversion is not needed for this idealized example. To confirm that the new eq. (1) is correct, we begin with two well-known formulae. The first relates the moment accumulation rate per length of fault to the integral of the back-slip rate s(z) over depth:   \begin{equation}\frac{M}{L} = \mu \int\nolimits_{{ - {D_m}}}^{0}{{s\left( z \right)}}\,{\rm{d}}z.\end{equation} (2) The second relates the surface residual velocity to the convolution of the back-slip rate with the Green's function for a line dislocation (e.g. Segall 2010, eq. 2.41):   \begin{equation}\nu \left( x \right) = \frac{1}{\pi }\int\nolimits_{{ - {D_m}}}^{0}{{\frac{{s(z)x}}{{{x^2} + {z^2}}}}}{\rm{d}}z.\end{equation} (3) Inserting eq. (3) into eq. (1) and rearranging the order of integration, we obtain   \begin{equation}\frac{M}{L} = \mathop {\lim }\nolimits_{W \to \infty } \left[ {\frac{{\mu \pi }}{{2W}}\int\nolimits_{{ - W}}^{W}{{x\left( {\frac{1}{\pi }\int\nolimits_{{ - {D_m}}}^{0}{{\frac{{s\left( z \right)x}}{{{x^2} + {z^2}}}{\rm{d}}z}}} \right){\rm{d}}x}}} \right] = \mu \int\nolimits_{{ - {D_m}}}^{0}{{s\left( z \right)}}\left( {\mathop {\lim }\nolimits_{W \to \infty } \frac{1}{{2W}}\int\nolimits_{{ - W}}^{W}{{\frac{{{x^2}}}{{{x^2} + {z^2}}}{\rm{d}}x}}} \right){\rm{d}}z.\end{equation} (4) We perform the integral over x first and multiply by 2 after changing the limits because the integrand is symmetric about x = 0:   \begin{equation}\int\nolimits_{0}^{W}{{\frac{{{x^2}}}{{{x^2} + {z^2}}}{\rm{d}}x}} = \left. {\left( {x - z{\,{\tan }^{ - 1}}\frac{x}{z}} \right)} \right|_0^W.\end{equation} (5) In the limit as W → ∞ the final result is   \begin{equation}\mathop {\lim }\nolimits_{W \to \infty } \frac{1}{W}\left( {W - z{{\,\tan }^{ - 1}}\frac{W}{z}} \right) = 1,\quad{\rm{ for }}\,{z_{\max }} = {D_m} \ll W.\end{equation} (6) The moment accumulation rate is   \begin{equation}\frac{M}{L} = \mu \int\nolimits_{{ - {D_m}}}^{0}{{s\left( z \right){\rm{d}}z}},\end{equation} (7) which agrees with eq. (2). In addition to this check on eq. (1), one of the reviewers, Takeshi Sagiya, also confirmed our correction and determined where we went wrong in the original derivation. REFERENCES Maurer J., Segall P., Bradley A.M., 2017. Bounding the moment deficit rate on crustal faults using geodetic data: methods, J. geophys. Res. , 122( 8), 6811– 6835. https://doi.org/10.1002/2017JB014300 Google Scholar CrossRef Search ADS   Segall P., 2010. Earthquake and Volcano Deformation , Princeton University Press. Google Scholar CrossRef Search ADS   Tong X., Sandwell D.T., Smith-Konter B., 2015. An integral method to estimate the moment accumulation rate on the Creeping Section of the San Andreas Fault, Geophys. J. Int. , 203( 1), 48– 62. https://doi.org/10.1093/gji/ggv269 Google Scholar CrossRef Search ADS   © The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society.

Journal

Geophysical Journal InternationalOxford University Press

Published: May 1, 2018

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