Accelerating Moment Release in Modified Stress Release Models of Regional SeismicitySteven C. Jaume´, Department of Geology, College of Charleston, Charleston, SC 29424, USA, Email: jaumes@cofc.edu AND

Mark S. Bebbington, IIS&T, Massey University, Private Bag 11222, Palmerston North, New Zealand, Email: m.bebbington.massey.ac.nz

Abstract: We show how the stress-release process, by making the distribution of assigned

magnitudes dependent on the stress, can produce earthquake sequences characterized by accelerating moment release (AMR). The magnitude distribution is a modified Gutenberg-Richter power law, which is equivalent to the square-root of energy released having a tapered Pareto distribution. The mean of this distribution is controlled by the location of the tail-off. In the limit as the tail-off point becomes large, so does the mean magnitude, corresponding to an “acceleration to criticality” of the system. Synthetic earthquake catalogs were produced by simulation of differing variants of the system. The factors examined were how the event rate and mean magnitude vary with the level of the process, and whether this underlying variable should itself correspond to strain or seismic moment. Those models where the stress drop due to an earthquakes is proportional to seismic moment produce AMR sequences, whereas the models with with stress drop proportional to Benioff strain do not. These results suggest the occurrence of AMR is strongly dependent upon how large earthquakes effect the dynamics of the fault system in which they are embedded. We have also demonstrated a means of simulating multiple AMR cycles and sequences, which may assist investigation of parameter estimation and hazard forecasting using AMR models.

1. The Stress Release Model (Vere-Jones, 1978): This is essentially a stochastic version of elastic rebound, with deterministic build-up of stress within a region and release (stochastically) through earthquakes. The state variable is the stress level, X(t) = X(0) + t - S(t), where is the tectonic loading rate, and S(t) = i:ti<t Si the accumulated stress release for events of `size’ Si at times t1,t2,…

The result is a point process with conditional intensity

(t) = (X(t)) = exp[ + (X(0) + t - S(t))].

The two obvious possibilities for the notional “stress” X(t) are Benioff strain i = 102.4+0.75Mi = Si, or seismic moment Ei=10 9.0+1.5Mi = Si.

2. The Tapered Pareto Distribution: To produce AMR through increasing event sizes, we need the mean magnitude to increase with the underlying variable. We will use the tapered Pareto distribution (Vere-Jones et al., 2001)

Pr (Y > y) = (y-y0)- exp[-(y-y0)/U], y > y0

for the stress drop Y, which is then converted to magnitude as M = m0 + 4/3 log10 Y. U = 102.4 + 0.75 is the ‘upper turning point’. We will allow the ‘equivalent magnitude’ to vary with X(t) as either

= u0 + u1eX or = u0 + u1 (1 + X+)u2

where the subscript ‘+’ denotes the positive part.

3. Simulations: We simulated 8 earthquake catalogs using different assumptions regarding (t), , and Si as follows:

A. (t) = exp[ + X(t)], = 7.0, Si = 102.4+0.75Mi

B. (t) = exp[ + X(t)], = 7.0, Si = 10 9.0+1.5Mi

C. (t) = constant, = 5 + 0.1eX(t), Si = 102.4+0.75Mi

D. (t) = constant, = 5 + 0.1eX(t), Si = 10 9.0+1.5Mi

E. (t) = exp[ + X(t)], = 5 + 0.1eX(t), Si = 102.4+0.75Mi

F. (t) = exp[ + X(t)], = 5 + 0.1eX(t), Si = 10 9.0+1.5Mi

G. (t) = exp[ + X(t)], = 5 +0.1(1 + X+)2, Si = 102.4+0.75Mi

H. (t) = exp[ + X(t)], = 5 +0.1(1 + X+)2, Si = 10 9.0+1.5Mi

Figure 3: Stress (red line) and event histories (vertical black lines) from CA models that show characteristics of self-organized (left) and intermittent (right)

criticality.

Figure 4: Stress (red line) and event histories (vertical black lines) from stress release

models that show characteristics of SOC (left) and intermittent (right) criticality.

Figure 1: Example of a simulated earthquake sequence (black circles) fit by the power-law time to failure equation (solid line). C = 0.49 for this example.

Figure 2: Normalized event size distributions from stress release model simulations. Left: If only is controlled by X(t) the tapered Pareto distribution is recovered (cases A and B). Right: However if is controlled by X(t) the event size distributions are a mix over U and show more large events (Cases C and D). (Solid circles – Si = 102.4+0.75Mi; Open circles- Si = 10 9.0+1.5Mi)

4. Continued:

where is the cumulative Benioff strain at time t, tf is the time of the mainshock event, and A, B, and m are constants to be determined. t was restricted to be between 0.1 and 10.0 of the mean mainshock interevent time and m was restricted to be between 0.0 and 0.8 to insure a minimum amount of curvature in the fit. The quality of the fit was evaluated using the method of Bowman et al. (1998):

C = power-law RMS/linear RMS

The A, B, m and t that minimized C was taken to be the best fit. Bowman et al. (1998) found for 8 mainshocks in California preceded by AMR, all had C < 0.7, which we will take as our definition of a ‘good fit’. Figure 1 is an example of a fit to an AMR sequence from case B (see section 3).

5. Results: We evaluate the degree to which each simulated catalog generates AMR sequences by determining the percentage of mainshock events for which C < 0.7. We also conducted the same search on a randomized version of each simulated catalog, where each event was randomly reassigned to a new time within the same time interval. Results are in Table 1.

Table 1: Percentage of Mainshock Events Preceded by AMR in Simulated and Randomized Earthquake Catalogs. The first value corresponds to the analysis of the 10+ largest events and the second value to the 25+ largest events.

Simulation % mainshock AMR % mainshock AMR (random)

Case A 27/24 58/69

Case B 92/85 58/50

Case C 0/30 56/50

Case D 78/78 60/67

Case E 8/22 64/58

Case F 92/93 38/52

Case G 27/19 53/56

Case H 100/96 54/67

In addition we examined the magnitude-frequency distributions of the resulting simulated earthquake catalogs (Figure 2). While the cases where = constant = 7.0 reproduced the tapered Pareto distribution as expected (Fig. 2 left), those cases where was allowed to vary with X(t) produced very different distributions (Fig. 2 right). See Vere-Jones et al. (2001) for a more detailed discussion of the variety of distributions.

6. Comparison With Cellular Automaton Models: It is instructive to compare our results with those obtained by Jaume´ et al. (2000), who analyzed the output of two cellular automaton (CA) models of seismicity for AMR sequences. They found that a CA model showing characteristics of self-organized criticality (SOC) had < 10% mainshocks preceded by AMR, whereas a CA model showing Intermittent Criticality had > 90% mainshocks preceded by AMR. Figures 3 and 4 compare stress and event histories of the CA (Fig. 3) and our SRM (Fig 4.) models.

4. Search For Presence of Accelerating Moment Release: We took each of the 8 simulated earthquake catalogs and searched for the presence of accelerating seismic moment/energy release (AMR) before the largest 10+ and 25+ events (mainshocks) in each catalog. Following empirical practice, we fitted the cumulative Benioff strain prior to the mainshocks with a power-law time to failure equation of the form (Bufe and Varnes, 1993):

(t) = A + B(tf – t)m

7. Conclusions: Our results suggest that it is the form of the earthquake stress drop that controls the presence or absence of AMR in stress release-type models. Large earthquakes strongly effect the dynamics of the system in simulations with stress drop scaled as seismic moment and AMR occurs during an “acceleration to criticality” following a previous large event. Comparison with CA model results reinforce this conclusion, in that stress and event histories of SRM models that produce many AMR sequences are similar to corresponding CA models.

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