1. Kenji MAEDA Meteorological Research Institute, JMA
Estimation of the Fault Constitutive Parameter Aσ and Stress Accumulation Rate from Seismicity Response to a Large Earthquake
Kenji MAEDA
Meteorological Research Institute, JMA
The title of my talk is “Estimation of the Fault Constitutive Parameter Aσ and Stress Accumulation Rate from Seismicity Response to a Large Earthquake”, and I would like to propose a method of estimating the fault constitutive parameter Aσand the background stressing rate, that is, stress accumulation rate by using the seismicity rate change induced by a large earthquake. The point of my talk is to calibrate the system parameter modeled by Dr.Dieterich’s theorem by using the stress step as a input to the system and the seismicity response as the output of the system.

2. Objectives
Estimate the fault constitutive parameter Aσ from the seismicity response to a large earthquake.
Estimate the stress accumulation rate from the seismicity response to a large earthquake.
Estimate the stress change from seismicity data.
As Dr. Dieterich shows earlier in his tutorial lecture, a seismicity might be connected to the stress change by the theory basing on the rate- and state- dependent constitutive friction law. And if one want to apply the theory to the actual seismicity data, it becomes very important to estimate the parameters relating to the theory properly.
So, today’s my talk is related to the estimation of the parameters concerning the Dr. Dieterich’s model.
And the objectives of my study are as follows. The first is to estimate the fault constitutive parameter Aσ from the seismicity response to a large earthquake, and the second is to estimate the stress accumulation rate from the seismicity response to a large earthquake, and the last is to estimate the stress change from seismicity data.
All the work of my study is basing on the Dieterich’s theoretical model of earthquake generation that is introduced by himself in today’s earlier tutorial lecture. I hope I understand his model correctly and my idea is useful.

3. Seismicity rate after the stress step
Dieterich(1994)
This equation shows the seismicity rate change as a function of time after a stress step is imposed on some area. This is one of the result of Dieterich (1994) model. In this equation, r represents the seismicity rate before the stress step, S’r and S’ represent stressing rate before and after the stress step, dS is modified Coulomb stress step, A and sigma are fault constitutive parameter and normal stress, respectively, and ta is the characteristic relaxation time, that is defined as A sigma divided by S’. t is measured from the time when the stress step is imposed on the area.
This expression seems a little complicated form, but characteristically this shows the time decaying pattern like an aftershock sequence that will be shown later. And if we focus on the earthquake rate just after the stress step, the expression becomes quite simple like R0=r exp (dS / A sigma) where R0 is the seismicity rate just after the stress step, and r is the background seismicity rate before the stress step, dS is the magnitude of the stress step at the site, and the A and sigma are the constitutive parameter and normal stress, and in this study we do not treat A sigma separately, but treat them as one parameter. So, we can see from this simple equation that if we know the stress step dS and the ratio of seismicity rate before and after the stress step, we can estimate the parameter A sigma.

4. Seismicity rate after the stress step
Dieterich(1994)
On the other hand, the previous equation can be approximated by the form like this by substituting K for a set of these parameters, c for these parameters, and r’ for r multiplied by the ratio of S dot to S dot r. This simple form is consist of background seismicity rate r dash plus well-known Omori’s formula of aftershock sequence. In the next slide, I will show you how well the top original form of equation is approximated by the bottom simplified equation. Anyway, we can see from this formulation that if the seismicity rate after a stress step is imposed changes in the form of Omori’s formula and we can define the parameter K and c, we can estimate the reference stressing rate S dot r using already estimated A sigma and background seismicity rate r before the stress step.

5. This slide shows the earthquake rate change as a function of time after the stress step. The solid lines are corresponding to the Dieterich’s original equation, and the broken lines are corresponding to the approximated equation basically expressed by the Omori’s formula. The different colors represent the different magnitude of stress steps normalized by A sigma.
From this figure we can see that the Dieterich’s formulation is generally well approximated by the simplified formulation based on Omori’s formula. However the background seismicity rate is a little overestimated by the simplified formulation and the difference is visible when the stress step is small.

6. Stress Step & Seismicity Rate Change
This slide schematically shows the method already mentioned of estimating the parameters A sigma and S dot r. The top figure shows the seismicity rate change expected by imposing a sudden stress step and the bottom figure shows the change of the stress. The seismicity rate goes up from a background rate r to R0 suddenly just after the stress step and then decays in the form following Omori’s formula. The parameter A sigma can be estimated by this equation using the observed R0 and r, and the value of stress step dS. The magnitude of dS will be given independently by the elasticity theory calculated for the source model of a large earthquake.
The background stressing rate before the stress step, that is, parameter S dot r can be estimated by the bottom formula using the estimated A sigma and the parameter of Omori’s formula K, and r.

7. When we select the target area to apply the method, we pay attention to
The fault model of a large earthquake is well determined.
The seismicity rate change caused by a stress step is clearly recognized.
The area is far enough from a large earthquake not to be effected by a slip inhomogeneity of the earthquake.
The rate change tend to follow Omori’s formula.

8. dCFF by the 2003 Tokachi-oki Eq.
0.07~0.16MPa
Now, I will show you the actual case of estimating the parameters. The 2003 Tokachi-oki earthquake whose magnitude is 8.0 occurred in the sea area of Hokkaido, northern part of Japan, and caused a clear activation of seismicity in some regions in inland Hokkaido. This slide shows the distribution of Coulomb stress change caused by the Tokachi-oki earthquake as well as its source area. The source fault model we adopted here is the single fault plane model estimated by GSI (2004) using geodetic data. As for the receiver faults we assume vertical right-lateral faults with the strike direction of N70°E which represent the typical fault mechanism around the regions expressed as A to D. From this figure we can see the stress increased area in the middle of Hokkaido, and the seismicity in that area is actually stimulated after the earthquake that will be shown in the following slide. The magnitude of increased Coulomb stress around the region A to D is estimated about 0.07 to 0.16 MPa.

9. Seismicity in Hokkaido
This figure shows the epicentral distribution of earthquakes with magnitude greater than or equal to 0.5 shallower than 15 km before and after the Tokachi-oki earthquake totally in two and half years period. The large blue star indicates the epicenter of the Tokachi-oki earthquake. The space time distribution in the area enclosed by rectangle will be shown in the next slide.

10. Space-Time Distribution of Eqs.
2003 Tokachi-oki Eq. (M8.0) A D C B
This is the space-time distribution and the horizontal axis is the time axis. The vertical arrow at the bottom is the occurrence time of Tokachi-oki earthquake. It is clearly seen that around the region marked A to D the number of earthquake is greatly increased just after the Tokachi-oki earthquake. So, we have decided to focus on these regions.

11. Investigated Regions A B C D
The regions we selected to apply the method are displayed here as the enclosed polygons with the mark of A to D.

12. Region A (No decluster)
2003 Tokachi-oki Eq. (M8.0)
Cumulative
Space-Time
Depth-Time
The variation of seismicity in the region A, for example, is shown here. We can see the activated seismicity after the Tokachi-oki earthquake. And we can also notice that many small clusters are included in the activity. By analyzing the activity after the Tokachi-oki event it is found that this activated activity is not well fitted to the Omori’s formula that is requested to apply the previously mentioned method, but is well expressed by the modified Omori’s formula with the parameter of p=1.8. This is probably because small clusters are activated by the unknown local stress changes and the total activity does not represent the pure stress step response to the Tokachi-oki earthquake. So, we decided to apply the declustering process to the original data in order to remove the activity induced by probable unknown local stress change.

13. Declustering Algorithm
Time
Space
Time
Space
Cluster 1
Cluster 2
Connection Time
Connection Distance
The method of declustering we adopted is very simple and displayed here schematically. The earthquakes located each other within some parameterized connection distance and connection time are classified into the same cluster and this process is repeated successively. This connection process is performed only forwardly in time, therefore in the example shown here the earthquakes are classified into two clusters. After the clustering is performed, each cluster is represented by one largest earthquake in the cluster.

14. Region A (Decluster 1km,1day)
2003 Tokachi-oki Eq. (M8.0)
Cumulative
Space-Time
Depth-Time
This slide shows the declustered seismicity in the region A, for example, obtained by the connecting parameters of 1 km in epicentral distance and 1 day in time. It is clear that the clustering activity is significantly removed. Then the declustered seismicity is well approximated by Omori’s formula, so we can estimate the parameters Aσ and S dot r by applying the method earlier mentioned. It should be noted here that when we use the declustered data to apply the method, the reference seismicity rate r before the stress step is reduced in proportion to the reduction rate for the declustered data after the stress step.

15. Region C (No Decluster)
2003 Tokachi-oki Eq. (M8.0)
Cumulative
Space-Time
Depth-Time
This is another example of the region C. In this case the original activity does not decay but increase in the earlier time interval after the Tokachi-oki earthquake and later the rate is decreasing. Therefore this activity cannot be modeled either the Omori’s formula nor the modified Omori’s formula.

16. Region C (Decluster 3km,7day)
2003 Tokachi-oki Eq. (M8.0)
Cumulative
Space-Time
Depth-Time
After the declustering process is applied to the previously shown data in the region C, the clustering activity is greatly reduced and the activity becomes to be modeled by the Omori’s formula.

17. Fitness of (Modi-)Omori Form.
These figures show examples of how the Omori’s or the modified Omori’s formula fits the data. The left hand side is for the original data, that is, non-declustered data of the region A. In this case the modified Omori’s formula is used to fit the data because the fitness of the Omori’s formula to the data is much worse than that of the modified Omori’s formula. However, even for the modified Omori’s formula, the result of the statistical test shows that the model is rejected with 95 % reliability. On the other hand, the right hand side is for the declustered data of the region A. In this case, the Omori’s formula is used to fit the data and the fitness is so good that the statistical test indicates the model cannot be rejected with 95 % reliability.

18. Obtaining Stress Change from Seismicity Rate Change
Dieterich(2000)
By the way, once the parameters of A sigma and the reference stressing rate S dot r are obtained, we can calculate the stress change from the seismicity rate variation. This slide shows the method also indicated by Dieterich (2000) basing on the previously mentioned rate- and state- dependent seismicity model. The differential stress can be calculated step by step by using this formula, and the state variable appearing in the formula is defined like this.

19. Declustered Seismicity Change and Estimated Stress
This slide shows an example of the summary of the estimated parameters as well as the related seismicity change and stress variation in the region A. In the top figure, the dark blue line represents the cumulative number of declustered earthquakes, and the light blue line is the cumulative function of the Omori’s formula estimated by the maximum likelihood method, and the green line indicates the estimated rate function of the Omori’s formula. When we estimate the parameters of the Omori’s formula, we assume the background earthquake rate after the stress step, that is represented by the parameter B, is 0. This assumption is corresponding to assume that the stressing rate after the stress step is 0. The estimated parameters of Omori’s formula, A sigma, and r are listed here.
In the bottom figure, the red line is the solution obtained by applying the method mentioned earlier, and the estimated value of background stressing rate, S dot r, is MPa/year. The magnitude of the stress step caused by the Tokachi-oki earthquake is about 0.07 MPa, which is given by the source model and elasticity theory as mentioned earlier. The other lines are also displayed in order to show how the stress variation is affected by the value of S dot r for the same seismicity rate change shown in the top figure.. We can see if S dot r is larger than the estimated value, the stressing rate after the stress step increases, and if S dot r is smaller than the estimated value, the stressing rate after the stress step decreases.

20. Horizontal Movement (1/1/03~2/28/05）
Teshio
Shintoku2
Now, in order to obtain an approximate image of the stress variation before and after the Tokachi-oki earthquake independently from the method mentioned so far, I will show you the geodetic data. This slide shows the horizontal movement observed by the GPS array deployed by Geographical Survey Institute for the period including the mainshock. The station named Shintoku2 is located near the region A. So, the variation of the baseline length between the station of Teshio and Shintoku2 will be shown in the next slide.
GPS data from GSI

21. Baseline Length and Estimated Stress
基線長変化
GPS data from GSI
Baseline Length Change Between Teshio and Shintoku2
Estimated Stress Change in region A
The top figure shows the baseline length variation between Teshio and Shintoku2. Of course this is not a stress but a relative displacement between two stations. However, we might imagine the general feature of variation of the stress change. We can see a large step at the time of Tokachi-oki earthquake. Though small after-slip can be seen just after the mainshock, there seems no large variation after the stress step. In the bottom, the estimated stress change obtained earlier is also shown as a reference. The red line is our estimated result that is based on assuming no stressing rate change after the stress step.

22. Estimated Values Region Decluster Aσ r A None 0.016 (0.004) 0.16
(MPa)
(MPa/y) r
(/day) A
None
0.016
(0.004)
0.16
1km, 1day
0.0028
0.028
2km, 2days
0.020
0.0023
0.019 B
0.017
(0.006)
0.041
0.0054
0.024
0.0048
0.014
This and next slides are the summary of the results of the estimated parameters for each regions and different declustering parameters.
(The values in parentheses are those estimated by applying the modified Omori’s formula instead of the Omori’s formula to the seismicity data.)
The values of Aσ obtained for four regions have relatively small variation ranging from 0.02 to 0.03 MPa, but those of S dot r vary widely ranging with the order of 2 such as from about to 0.01 MPa/year. On the other hand, it is found that the values of declustering parameters do not much affect the result of estimated values.
(One possible reason for the wide ranging values of S dot r is that the seismicity caused by the unknown local stress could not be removed by the declustering algorithm we adopted here.)

23. Estimated Values Region Decluster Aσ r C None -- 0.024 0.0002 0.0005
(MPa)
(MPa/y) r
(/day) C
None --
2km, 5days
0.024
0.0002
0.0005
3km, 7days
0.0004
5km, 7days
0.0003 D
0.031
(0.02)
0.026
1km, 1day
0.032
0.012
0.018
2km, 2days
0.034
0.011
0.015

24. Non-declustered Seismicity Change and Estimated Stress
Finally, in order to estimate the total stress loaded to the region we applied the method mentioned earlier to the non-declustered seismicity data by using the same parameter values estimated so far. The top figure of this slide represents the seismicity rate change and cumulative number in the region A based on the original non-declustered data. We use the combination of three functions of the modified Omori’s formula to fit the data. And corresponding stress change estimated by previously mentioned method is shown in the bottom. The first stress step shown in the bottom figure is caused by the Tokachi-oki earthquake and the following step-like changes are interpreted as caused by unknown local stress changes.

25. Summary
The fault constitutive parameter Aσ is estimated from the declustered seismicity that is activated by a large earthquake.
The reference stress accumulation rate is also estimated by assuming no large stress change followed the mainshock.
The time variation of stress is estimated from the seismicity rate change by using estimated Aσ and stress accumulation rate .