Fast computation of static stress changes on 2D faults from final slip distributions



[1] Computing the distribution of static stress changes on the fault plane of an earthquake, given the distribution of static displacements, is of great importance in earthquake dynamics. This study extends the approach developed by Andrews [1980], and compares it against existing analytical formulations. We present calculations for slip maps of past earthquakes and find that the stress-change results are accurate to about 1–2% of the maximum absolute stress change, while the computation time is greatly reduced. Our method therefore provides a reliable and fast alternative to other methods. In particular, its speed will make computation of large suites of models feasible, thus facilitating the construction of physically consistent source characterizations for strong motion simulations.

1. Introduction

[2] One of the fundamental quantities in earthquake source physics is the static stress change associated with earthquake faulting. The static stress change on the fault plane itself is linked to the dynamics of earthquake rupture and hence also to the associated energy release and seismic radiation. Its knowledge is therefore required in dynamic rupture modeling of past (and future) earthquakes [Peyrat et al., 2001]. Moreover, stress-drop distributions for simulated slip maps in scenario earthquakes for near-source strong-motion simulations allow to constrain the temporal rupture evolution [Guatteri et al., 2003] and the energy budget of earthquake rupture [Guatteri et al., 2004].

[3] Estimating the distributed stress changes on the fault plane directly from seismological data is cumbersome [Peyrat and Olsen, 2004], and usually it is inferred from imaged slip distributions [e.g., Bouchon, 1997]. Kinematic source inversions have revealed the complexity of earthquake rupture at all scales [Heaton, 1990; Somerville et al., 1999; Mai and Beroza, 2002]. Therefore, also static stress changes on the rupture plane are highly heterogeneous, exhibiting locally large stress drop in the region of high slip but also zones of stress increase.

[4] Okada [1992] derived analytical expressions to compute static stress changes for given final displacements in an elastic homogeneous half space. In contrast, finite difference methods (FDM) [e.g., Ide and Takeo, 1997; Day et al., 1998] and the discrete wave number method by Bouchon [1997] require knowledge of the entire slip-time history at each point of the fault, but these approaches return the complete spatio-temporal evolution of stress changes on a fault plane. However, computations are time consuming for all methods mentioned so far, particularly for large grid sizes.

[5] In this study, we propose a shortcut to calculating stress changes on a 2D fault plane, based on the work by Andrews [1980]. He presented a wave number representation relating the distribution of static slip to the associated collinear static stress change. We give a brief introduction of Andrews' method and our extensions to it and discuss its range of applicability. We then examine the method under various initial assumptions and evaluate its performance in terms of computational speed and accuracy, when compared to the analytical solutions obtained with Okada's formulations. This comparison is performed for synthetic slip distributions as well as for examples of inverted slip maps of past earthquakes.

2. Method

2.1. Description

[6] For a given slip distribution D(x) on a planar fault, Andrews [1974] presented the following expressions for the two components (parallel and perpendicular to the slip direction), of shear strain on the fault:

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Based on equation (1), Andrews [1978, 1980] developed a formulation which relates the slip-parallel shear-stress change to the slip distribution in the wave number domain:

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Here, written as functions of the wave number vector k, D is the static slip distribution, Δσ is the component of shear-stress change parallel to the slip direction and K is the static stiffness matrix given by

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with k and k being the wave numbers in the directions parallel and perpendicular to the slip direction, respectively, and λ and μ being the Lamé constants. Starting from equation (2) and proceeding in an analogous manner, we derive a similar expression for the component of shear-stress change perpendicular to the slip direction:

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where the slip-perpendicular static stiffness function is given by

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Calculating the static stress changes for a given slip distribution therefore only requires the Fourier transformation of slip into the wave number domain, application of equations (3) and (5) and the inverse transformation of Δσ(k) back into space domain. Varying rake angles are treated by splitting the total slip at each point in an along-strike and down-dip component and calculating parallel and perpendicular stress changes for each slip component separately. Then the resulting two distributions of the down-dip stress component are summed up, as well as the two distributions of the along-strike stress component. Especially for grids with a large number of points, Andrews' method reduces calculation time by several orders of magnitude compared to Okada's and Bouchon's method (Figure 1).

Figure 1.

Comparison of calculation times observed on a 1.7 GHz PC. They are proportional to N2 for Okada's and Bouchon's method and to N logN for Andrew's method. However, for Andrew's method also the proportionality factor is much smaller, so the total increase over the plotted range is only in the range of seconds, making the curve appear to be a horizontal line.

[7] A general problem in calculating stress changes from slip arises if non-zero slip occurs at the edges of the fault plane. Non-zero slip at the fault boundaries constitutes a discontinuity in slip that will always lead to unrealistically high values of stress change at the edges. In our tests with synthetic slip maps we avoid this problem by using slip distributions which decrease to zero at all boundaries. In the case of inverted slip distributions for past earthquakes we reduce the influence of non-zero slip at the boundaries with the following procedure: The numerical grid is extended outward across the physical fault edges by 10 grid points, over which slip is tapered to zero. For both methods, computations are performed for the expanded slip distribution, but we only retain the inner part of the stress change distribution, corresponding to the original fault dimensions.

2.2. Range of Validity

[8] Strictly speaking, equations (3) and (5) are valid for stress changes on a 2D fault plane embedded in a homogeneous full space, and hence several limitations apply.

[9] First of all, the shape of the fault to be modeled is restricted to a single 2D fault plane, thus excluding more complex fault geometries, e.g., multiple segments. In contrast, Okada's code allows for modeling arbitrary displacement and observation point positions. This is not a serious limitation, however, since for many seismological applications the assumption of a 2D fault plane constitutes a good approximation of the true, complex fault geometry.

[10] Secondly, both Andrews' and Okada's formulation assume a homogeneous medium. We can approximately account for variations in shear wave velocity and density by scaling the calculated stress change with rigidity μ. However, this can not take into account variations in Poisson's ratio ν.

[11] Finally, being defined for a full space, Andrews' method lacks the effect of the free surface, whereas Okada's formulations are valid for a half space. The resulting differences in the stress changes depend on the actual slip distribution, with the effect being stronger if large slip occurs at shallower depth. For the special case of strike slip on a vertical fault, the free-surface effect can be approximated by including a mirror image of the slip distribution above the free surface [Steketee, 1958], resulting in very small stress differences of 1–2% even for events with very large surface slip (e.g., Landers). However, the free-surface effect is generally alleviated by the fact that material strength usually decreases towards the free surface (rigidity often varies by a factor of about 2 between the uppermost and lowermost part of the fault plane, see next section for details). If depth-dependent rigidity is taken into account, the absolute stress change values are decreased in the uppermost low-strength part, also decreasing the differences due to the missing free-surface effect.

3. Example Calculations

3.1. Synthetic Slip Distributions

[12] As a first test, we assume a simple strike-slip fault embedded in a full space. To emulate the full space in the calculations with Okada's method, we position the fault top at 200 km depth. The fault plane is vertically dipping and has extensions of 20 km × 20 km, discretized with a spatial sampling of 0.2 km. Rigidity is set to μ = 3.3 · 1010 N/m2 and a Poisson ratio of ν = 0.25 is assumed. Slip on the fault plane is modeled as d(x, z) = dmax · exp(−r(x, z)2/a2) with dmax = 1 m, a = 2500 m and r(x, z) being the distance to the center of the fault plane at (10,10) km. This yields a mean slip of 0.048 m and a magnitude of roughly MW = 5.8.

[13] The static stress changes in the slip-parallel component range from a maximum stress drop of 13.6 MPa to a maximum stress increase of 1.5 MPa. The absolute differences in both shear-stress components between the results of Andrews' and Okada's method are very small with maximum values of 0.05 MPa, equal to about 0.36% of the maximum stress drop.

[14] We repeat the calculation with Okada's method for the same slip distribution with the top of the fault plane coinciding with the free surface. In this case the influence of the free surface results in a difference of about 0.036 MPa (≈0.26%) at the surface. If we switch from strike slip to dip slip with the same distribution, we observe slightly higher differences of 0.12 MPa at the free surface, equal to 0.9% of the maximum stress drop. For source models with most of the slip happening in a depth of around 10 kilometers, the effect of the free surface is therefore negligible.

3.2. Past Earthquakes

[15] In the next section we evaluate how Andrews' method performs for past earthquakes. We use published slip distributions obtained from inversion of strong-motion and/or teleseismic data. The slip models are interpolated onto a grid with 0.2 km spacing using a spline interpolation. In general the choice of the interpolation method is not trivial, but is of minor importance here, as we are interested only in the relative differences between the methods.

3.2.1. Morgan Hill

[16] The 1984, Morgan Hill, California, earthquake provides an example of a pure strike-slip event on a buried fault. We use the slip model from Beroza and Spudich [1988] with a 10 × 30 km fault plane dipping down from 2.5 km depth at an angle of 85°. The calculated stress changes are scaled with the depth-dependent rigidity model (Figure 2d), derived from the velocity and density profiles specified by Beroza and Spudich [1988]. The differences between the static stress changes calculated with Okada's and Andrews' method have mean and maximum absolute values of 0.09 MPa and 1.82 MPa, respectively, corresponding to 0.1% and 1.9% of the maximum stress drop of 95 MPa.

Figure 2.

Morgan Hill earthquake. (a) Slip distribution interpolated from the model by Beroza and Spudich [1988]. (b) Static stress drop distribution calculated with Andrews' method. (c) Difference between the two calculated stress-drop distributions, ΔσOkada − ΔσAndrews. The regularity of the pattern is due to the interpolation of the slip map. (d) Depth dependent rigidity model used to scale the stress changes. Horizontal lines mark the fault extensions. (e) Cumulative distribution function of the stress-drop difference displayed in c).

3.2.2. Northridge

[17] The 1994 Northridge earthquake constitutes an example of a blind thrust event on a shallow dipping fault (extending from 5 to 21 km depth with a dip of 40°). We use the slip model obtained by Hartzell et al. [1996] (source size 25 × 20 km), treating the total slip amplitude to be pure dip slip. A homogeneous medium is assumed, because the whole fault plane is located within a single layer of the velocity model used by Hartzell et al. [1996]. Figure 3 displays the stress changes calculated with Andrews' method and the difference to the Okada results. The most pronounced differences are observed in the upper part and can be attributed to the influence of the free surface. However, the differences remain small (0.1 MPa mean and 0.5 MPa maximum absolute value, equal to 0.3% and 1.3% of the maximum stress drop of 37 MPa).

Figure 3.

Northridge earthquake. (a) Slip distribution, modified from Hartzell et al. [1996]. The total slip amplitude is treated as being pure dip slip. (b) Down-dip component of static stress drop calculated with Andrews' method. (c) Difference in static stress drop calculated with both methods ΔσOkada − ΔσAndrews. The regularity in the pattern arises from the interpolation of the slip distribution. (d) Cumulative distribution function of the stress-drop difference displayed in c).

3.2.3. Loma Prieta

[18] The 1989 Loma Prieta earthquake finally is a case in which significant amounts of both dip slip and strike slip have occurred. We use the inverted slip distribution from Beroza [1991] with a source size of 14 × 40 km, extending from about 5 to 18 km depth with a 70° dip angle (Figure 4). The observed stress drop differences are again in the same range (0.3 MPa mean and 1.5 MPa maximum absolute value, equal to 0.4% and 2.1% of the maximum stress drop of 72 MPa).

Figure 4.

Loma Prieta Earthquake. (a) and (b) Strike-slip and dip-slip distribution, interpolated from the model by Beroza [1991]. (c) and (d) Along-strike and down-dip component of static stress drop calculated with Andrews' method. (e) and (f) Difference in along-strike and down-dip component of static stress drop, respectively, calculated as ΔσOkada − ΔσAndrews. (g) and (h) Cumulative distribution functions of the differences displayed in e) and f), respectively. (i) Depth dependent rigidity model used to scale the stress changes. It corresponds to the velocity and density model specified by Beroza [1991] for the region SW of the fault. Horizontal lines mark the fault extensions.

4. Conclusions

[19] We have expanded the approach by Andrews [1980] for computing static stress changes on a two-dimensional fault plane from a given slip distribution, in order to account for both slip-parallel and slip-perpendicular stress changes. Computations with this method are fast, and especially for large grid sizes the increase in computation speed compared to other methods is enormous.

[20] However, it has to be kept in mind that the formulation is valid only for stress changes on a fault embedded in a homogeneous full space. Under these assumptions, the calculated stress changes show little or no deviation (i.e., ≈0.3% of the maximum stress change) from results obtained with the analytical formulations derived by Okada [1992]. To evaluate the performance of Andrews' method for realistic model setups, we applied it to published slip maps from past earthquakes. We found, that the errors generally remain negligible (max. ≈ 2%, mean ≈ 0.3%).

[21] In particular, the accuracy is considered sufficient for the purpose of constraining dynamic rupture models and constructing physically consistent source characterizations for strong-motion simulations [Guatteri et al., 2003]. Especially in the latter field of research, the speed of the method will provide an important advantage, allowing fast calculations for large sets of slip models.


[22] We thank M. Bouchon for providing his code to compute stress changes for finite-source models. We thank Joe Andrews and one anonymous reviewer for valuable suggestions. Comments by G. Hillers, D. Schorlemmer, S. Wiemer and J. Wössner also helped to improve the manuscript. This is contribution number 1356 of the Institute of Geophysics, ETH Zürich.