Paper II

Stress tensor inversion using detailed microearthquake informa-
tion and stability constraints: Application to Ölfus in southwest
Iceland

Introduction
In 1996 the Uppsala SIL group started its participation in the European Union project ambitiously referred to as PRENLAB, Earthquake Prediction Research in a Natural Laboratory. As my involvement in the Siljan project was coming to an end, and as one of the tasks of the Uppsala group in PRENLAB was stress estimation from microearthquake focal mechanisms, I was more than happy to join them. Ragnar Slunga had in 1988 developed a stress tensor inversion algorithm based on the Gephart and Forsyth [1984] formulation of the inverse problem, with a new approach to choosing the fault plane and a novel procedure for including uncertainties on the focal mechanism through the inclusion of acceptable focal mechanisms (Section 3.3). We continued development of the algorithm, I rewrote the code for increased efficiency and the inversion scheme was tested and put to work in the SIL environment.

Summary
The basis of stress tensor inversion from earthquake focal mechanisms was reviewed in Section 3.4. Our inversion scheme is based on the grid search method of Gephart and Forsyth [1984], although the final formulation is slightly different. We utilize the one-norm measure of misfit but only consider rotations in the fault plane, i.e. we use the pole rotation method in Gephart and Forsyth [1984] terminology. Errors in the focal mechanisms are accounted for in a manner that does not require the exact method of Gephart and Forsyth [1984]. In order not to under-estimate the size of the confidence limits, we follow Magee [1997] and assign both nodal plane normals to ${10}^{\circ}$ bins over the lower hemisphere and then use half the number of filled bins as our number of ``non-redundant'' data. This number is only used for the confidence limit calculation, we use all data for the inversion. In Paper III the issue of redundancy in the focal mechanism data is further investigated.

The selection of the fault plane from the two nodal planes was discussed in Section 3.4. In our inversion scheme we implemented three alternative techniques for the selection. We use the common slip angle criterion (SA), i.e. choosing the nodal plane with smallest misfit in the tested stress field, for comparison both with other inversion methods and with our other fault selection methods. The new selection criterion introduced in Paper II is based on the notion that the nodal plane that is most unstable in the tested stress field is the plane that slips. Based on a simple Mohr-Coulomb failure criterion we define the instability, $I$, of a nodal plane as

$$I = \tau - \mu\sigma_n$$ (76)

where $\tau$ is the magnitude of the shear stress on the nodal plane, $\sigma_n$ the normal stress on the plane and $\mu$ the coefficient of friction. In order to calculate the instability we need to know the the magnitudes of the principal stresses. Fortunately, we are only interested in which of the two nodal planes has the highest value of instability, and since we know from Section 2.6 that the relative position of the nodal planes in a Mohr diagram only depends on their orientation with respect to the stress axis, we can unambiguously decide which nodal plane is the most unstable for every value of the coefficient of friction. The coefficient of friction is usually not well known in the seismic zones and we see from the definition of instability that with varying $\mu$ the unstable nodal plane might become the stable nodal plane. We show that the value of $\mu$ at the point of cross-over, $\mu_x$, is independent of the magnitudes of the principal stresses, and can be determined from $R$ and the orientations of the nodal plane normals with respect to the principal stress axis and $R$. During the fault selection process in the inversion we calculate $\mu_x$ and if it is outside the range $0.4 \leq \mu \leq 1.5$, which covers friction coefficients for most rock types above 100 MPa [Byerlee, 1978], we select the most unstable nodal plane. If $\mu_x$ is within the range, we instead utilize the slip angle criterion. We want to emphasize that the instability (IS) criterion is only used to choose fault plane, it is not used as a misfit criterion. The IS criterion is only valid in areas where fractures in the crust have frictional properties independent of their orientation, in areas dominated by weak faults with a specific orientation the criterion is likely to fail.

As our third fault selection criterion we are able to include information from high accuracy relative relocations [Slunga et al., 1995] produced by the SIL system. If a group of relocated microearthquakes define a common fault plane, and the plane agrees with the events focal mechanisms, this common plane is used as the fault plane for the events.

The fault plane selection methods were tested using both synthetic data and geologic fault slip data. Using synthetic data with noise added and testing both single stress fields and mixed stress fields, the IS criterion performs generally slightly better than the SA criterion. We find, however, that the result of synthetic tests highly depends on the input parameters to the focal mechanism generation and, as such, are difficult to interpret. As a more appropriate test we converted the fault slip data of Angelier [1979] into focal mechanisms and inverted them for the stress tensor. The IS and SA criterion both yield similar stress estimates, in agreement with  Angelier [1979], but where the IS criterion picks all the correct fault planes, the SA criterion only selects 20 out of 38 fault planes correctly.

Figure: Stress tensor inversion results from Ölfus, Iceland. All inversions have been performed with the instability fault selection criterion. Grids are equal-area projections of the lower hemisphere. To the left are the resulting optimal stress tensors; $\sigma_{1}$, square, $\sigma_{2}$, diamond, and $\sigma_{3}$, triangle; together with the 10%, 68%, and 95% confidence limits for $\sigma_{1}$, red to yellow from 10% to 95%; and $\sigma_{3}$, purple to green from 10% to 95%. Deviation is the average angle between the directions of estimated shear stress and observed slip on the planes, misfit is deviation weighted with amplitude errors and $R$ is the optimum $R$ value. The black histogram on the perimeter shows the 95% confidence level for the direction of maximum horizontal stress. In the middle are histograms of the 10%, green, 68%, red, and 95%, blue, confidence limits in $R$, the black spike is the optimum value. To the right are Kamb contours of the fault plane normals chosen by the inversion. A) Inversion using only the optimal focal mechanism for each event. B) Inversion with all acceptable mechanisms for each event. C) Inversion with 41 events having predefined planes from relocation and using all acceptable mechanisms for the remaining 37 events.

The stress tensor inversion scheme takes advantage of the acceptable focal mechanisms produced by the SIL focal mechanisms algorithm (Section 3.3). The optimal focal mechanism for each event is accompanied by a number of acceptable mechanisms that satisfies the data only slightly less well. We test all these acceptable focal mechanisms for each event at every grid point in the stress inversion. Note that for each acceptable mechanism we apply the fault selection criterion to identify the proper nodal plane. The acceptable mechanisms have an associated amplitude error and these errors are used to weight the mechanisms relative to the optimal mechanism. The acceptable (including the optimal) mechanism which produce the lowest misfit is selected to represent the event in a particular grid point. When the optimal stress tensor has been identified, the inversion scheme has also selected the most appropriate focal mechanism for each event from the acceptable mechanisms.

The stress tensor inversion was applied to 78 microearthquakes from a small volume, 1.6 km N-S, 1.3 km E-W and 5.9 km deep, in Ölfus, southwest Iceland. 41 of the events could be assigned a ``correct'' fault plane from relative relocation. In Figure 17 we show the results of three different inversion using the instability fault selection criterion. Comparing Figures 17A and B, we see how the inclusion of acceptable focal mechanisms significantly reduces the misfit and deviation and also reduces the size of the confidence limits, making the resulting stress state more well constrained. The optimal stress states are approximately the same and the chosen fault planes have approximately the same orientations in both inversions. In Figure 17C we have included predefined fault planes for 41 of the 78 events. The first thing to note is that the chosen fault planes of all three inversions are very similar, i.e. the instability selection criterion is successful in picking the correct fault planes. The state of stress is less well constrained in Figure 17C and we see that the misfit has increased. All three inversions show a rather stable direction of maximum horizontal stress at approximately $\mathrm{N}{30}^{\circ}\mathrm{E}$.

Stress inversion with the instability selection criterion was compared to inversion using the slip angle criterion. The stress states estimated by using the two different criteria are similar but, when compared to the predefined planes, the slip angle criterion picks the wrong nodal plane for more than 50% of the events.

Concluding remarks
This study showed that a nodal plane selection criterion based on the stability of the planes in the stress field is significantly better at predicting the correct fault planes than a misfit based selection criterion. Including a range of acceptable focal mechanisms for each event constrains the stress estimate better than using a single mechanism per event. The range of mechanisms also lowers the misfit.

Testing the stability selection technique in an area with strongly anisotropic friction conditions, such as close to a large, well established fault, would be very interesting. Preliminary inversions of microearthquake from the Husavik fault region in northern Iceland, showed that the stability criterion very consistently picked the wrong fault planes, as compared to relative relocations. If this is a general pattern, the combination of stress tensor inversion and relative relocation could become useful as a means of assessing the stability of faults in different directions and identifying established faults at depth.