Geodetic data inversion to estimate a strain-rate field by introducing sparse modeling

Abstract Many studies have estimated crustal deformation from observed geodetic data. So far, because most studies have applied a smoothness constraint, which includes the assumption of local uniformity of a strain-rate field, localized strain rates near fault zones have tended to be underestimated when we invert spatially sporadic GNSS data. To overcome this difficulty, we introduce sparse modeling into the estimation of a strain-rate field. Specifically, we impose a sparsity constraint as well as the smoothness constraint on strain rates as prior information, which are expressed by the L1-norm and the L2-norm of the second-order derivative of the velocity field, respectively. To investigate the validity and limitation of the proposed method, we conduct synthetic tests, in which we consider an anti-plane strain problem due to a steady slip on a buried strike-slip fault. As a result, we find: (1) regardless of the locking depth of the fault, the proposed method reproduces localized strain rates near the fault with almost equal or better accuracy than the L2 regularization method (i.e., only the smoothness constraint); (2) the advantage of the proposed method over the L2 regularization method is clearer when data coverage is worse (i.e., when fewer observation points are available); and (3) the proposed method can be applied when observation errors are small. Next, we apply the proposed method to the GNSS data across the Arima-Takatsuki fault zone, which is one of the most active strike-slip faults in Japan. As a result, the proposed method estimates about $$1.0\times {10}^{-8}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1.0</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> /yr faster strain rates near the fault zone than the L2 regularization method, which corresponds to a 20–30% greater strain-rate concentration. The faster strain rates result in the estimation of a shallower locking depth: 11 km by the proposed method, compared to 17 km by the L2 regularization method. The former is closer to the depth of D90, 12–14 km, above which 90% of earthquakes occur. Graphical Abstract