Forecasting the Locations of Future Large Earthquakes: An Analysis and Verification
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DOI: 10.1007/s00024-010-0069-1
- Cite this article as:
- Shcherbakov, R., Turcotte, D.L., Rundle, J.B. et al. Pure Appl. Geophys. (2010) 167: 743. doi:10.1007/s00024-010-0069-1
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Abstract
The objective of this paper is to quantify the use of past seismicity to forecast the locations of future large earthquakes and introduce optimization methods for the model parameters. To achieve this the binary forecast approach is used where the surface of the Earth is divided into l° × l° cells. The cumulative Benioff strain of m ≥ m_{c} earthquakes that occurred during the training period, ΔT_{tr}, is used to retrospectively forecast the locations of large target earthquakes with magnitudes ≥m_{T} during the forecast period, ΔT_{for}. The success of a forecast is measured in terms of hit rates (fraction of earthquakes forecast) and false alarm rates (fraction of alarms that do not forecast earthquakes). This binary forecast approach is quantified using a receiver operating characteristic diagram and an error diagram. An optimal forecast can be obtained by taking the maximum value of Pierce’s skill score.
Keywords
Earthquake forecastingROC diagramerror diagramforecast verification1 Introduction
Forecasts of the locations of future major earthquakes play an important role in earthquake preparedness and determining earthquake insurance costs. Many such forecasts have been carried out, one example is the National Seismic Hazard Map for the United States (Frankelet al., 1996). This is a probabilistic estimate of the ground-shaking risk. A major input into this assessment is the observed rate of occurrence of past earthquakes. Kossobokovet al. (2000) utilized the rates of occurrence of m ≥ 4.0 earthquakes globally to forecast the expected rates of occurrence of larger earthquakes. Recently, the Regional Earthquake Likelihood Models (RELM) forecasts were published (Field, 2007). This represents the first-generation of forecasts based on a variety of approaches and methods. It also introduces a testing methodology to verify and compare these forecasts (Schorlemmeret al., 2007).
In this paper we have presented a method to construct a binary forecast of where future large earthquakes are likely to occur globally. We divide the Earth’s surface into l° × l° cells in order to construct the forecast and evaluate its success. The input for our forecast is a relative intensity (RI) map of m ≥ m_{c} earthquakes that occur in the cells during a specified training period, ΔT_{tr}. As a measure of RI we have utilized the cumulative Benioff strain in each cell during that period. The cumulative Benioff strain has been computed using the data from the CMT (Harvard) catalog (http://www.globalcmt.org). This catalog is complete from 1/1/1976 to 1/1/2008 to magnitude m_{c} = 5.5. The RI map is then converted to a binary forecast by introducing a threshold cumulative Benioff strain. Cells with Benioff strains greater than this threshold constitute alarm cells where future earthquakes are forecast to occur. A high threshold reduces the forecast areas but leads to more events that are not predicted. A low threshold reduces the failures to predict but increases the false alarms. We have introduced a standard optimization procedure for binary forecasts in order to select the optimal threshold.
In order to test our method we have performed a retrospective analysis. To define training and forecasting time periods we subdivide the catalog using a base year T_{b}, a forecasting year T_{f}, and an end year T_{e}. We have utilized seismicity during the training period from T_{b} = 1/1/1976 to T_{f} = 1/1/2004 (ΔT_{tr} ≡ T_{f} − T_{b} = 28 years) to make a forecast for the locations of future target earthquakes of a specified magnitude, m_{T} ≥ 7.2. The locations of earthquakes during the forecasting period T_{f} = 1/1/2004 to T_{e} = 1/1/2008 (ΔT_{for} ≡ T_{e} − T_{f} = 4 years) are then used to test the validity of the method. The success of the binary forecast has been evaluated using techniques developed in the atmospheric sciences (Mason, 2003). The evaluation is in terms of a hit rate (fraction of events that are successfully forecast) versus a false alarm rate (the fraction of forecast cells in which earthquakes do not occur). Both hit rate and false alarm rate are functions of the tuning parameter (in our case the threshold Benioff strain). Their parametric plot produces a receiver operating characteristic (ROC) diagram. This approach has been used by Hollidayet al. (2005) to test the success of forecasting algorithms in California. Similar analysis was also applied to the verification of the intermediate-term earthquake prediction algorithm M 8 by Minster and Williams (1999).
We have also quantified our approach using the variation of the error diagram introduced by Molchan (1991), Molchan and Keilis-Borok (2008). The comparison of the error diagram approach and the maximum likelihood method for evaluation of earthquake forecasting methods was given by Kagan (2007). The connection between the entropy score and log-likelihood estimates with applications to several seismicity models is given in Harte and Vere-Jones (2005). The analysis of the error diagram approach applied to California seismicity was performed by Zechar and Jordan (2008). Kossobokov and Shebalin (2003) and Zaliapinet al. (2003) have applied the error diagram to test a number of earthquake predictions algorithms.
2 Relative Intensity of Global Seismicity
The proposed analysis is based on the RI of past seismicity where large target earthquakes with magnitudes greater than m_{T} ≥ 7.2 are forecasted. The moment magnitude m of earthquakes has been calculated from the scalar moment according to the formula \(m={\frac{2}{3}}\hbox{log}_{10}(M_0)-10.7,\) where M_{0} is a scalar moment reported in the CMT catalog. The grid size plays an important role in this analysis. In this study the Earth’s surface has been subdivided into 2° × 2° cells
The normalized RI in a cell is given by B_{xy}/B_{max} where B_{xy} is the cumulative Benioff strain in the active cell under consideration and B_{max} is the Benioff strain for the most active cell. The cumulative Benioff strain is also normalized by the actual area of the corresponding cell measured in square kilometers.
3 Forecast Verification
The success of a forecast is based on maximizing the fraction of the earthquakes that occur in alarm cells and minimizing the fraction of the alarm cells in which no earthquake occurs. This analysis is carried out by utilizing a contingency table approach. Each cell in which an earthquake is forecast to occur constitutes an alarm cell. Earthquakes occur in some of these alarm cells but also in cells where alarms have not been issued. As a result of this test each cell falls in one of four categories: a, alarm and earthquake (successful forecast); b, alarm but not earthquake (false alarm); c, no alarm but an earthquake (failure to predict); d, no alarm and no earthquake (successful forecast of non-occurrence). In addition a is the number of target earthquakes that occurred in alarm cells, b is the number of alarm cells with no earthquakes in them, c is the number of target earthquakes that occurred outside alarm cells, and d is the number of cells with no target earthquakes and which are not alarm cells.
As a specific example we consider the forecast given in Fig. 1. We retrospectively test this forecast in terms of the locations of m_{T} ≥ 7.2 target earthquakes that occurred during the forecast period T_{f} = 1/1/2004 to T_{e} = 1/1/2008. The total number of 2° × 2° cells in which a m ≥ 5.5 occurred during the entire time period (active cells) was 1,503. During the forecast period a + c = 34 earthquakes with m_{T} ≥ 7.2 occurred. Of these a = 23 occurred in alarm cells (were forecast) and c = 11 occurred in the other cells (were not forecast).
Contingency table for earthquakes with m_{T} ≥ 7.2 that occurred during the forecast period T_{f} = 1/1/2004 to T_{e} = 1/1/2008 (ΔT_{for} = 4 years)
Forecast | Observed | ||
---|---|---|---|
Yes | No | Total | |
Yes | a = 23 | b = 371 | a + b = 394 |
No | c = 11 | d = 1098 | c + d = 1109 |
Total | a + c = 34 | b + d = 1469 | a + b + c + d = 1503 |
The maximum value of \(\hbox{PSS}_{\rm ROC}^{\rm max}=\hbox{PSS}_{\rm ROC}\) from an ROC diagram can be used to find an optimizing value of the tuning parameter. For large earthquakes m_{T} ≥ 7.2 the maximum value of Pierce’s skill score is \(\hbox{PSS}_{\rm ROC}^{\rm max}=0.424\) corresponding to the threshold value B/B_{max} = 0.0417. This corresponds to H = 0.676 and F = 0.253 and was the basis for selecting this threshold value for the example given in Fig. 1. The value of the hit rate H = 0.676 (67.6%) gives the probability of having a large earthquake in the alarm cells during the forecast period. The threshold values obtained from retrospective forecasts can be applied a priori in subsequent prospective forecasts. This can be done by analyzing a particular region of study where the magnitude of completeness of a catalog is obtained and the optimal threshold and cell size values were determined from the retrospective analysis.
From the analysis of the error diagram curve the maximum value of this skill score is \(\hbox{SS}_{\rm ED}^{\rm max}=1-0.324-0.262=0.414\) corresponding to the threshold value B/B_{max} = 0.0417. It is worthwhile to note that the threshold values corresponding to the maximum values of the skill scores for both ROC and error diagrams are the same. This is due to the fact that the values of F and τ are very close to each other for corresponding threshold values as they are both dominated by large b + d values compared to small a + c. As a result the error diagram curve is almost a mirror image of the ROC curve due to the fact that ν = 1 − H. In general this is not always the case.
In the above analysis we have used a uniform prior distribution for the seismically active cells and assumed a null hypothesis where alarm cells are redistributed randomly. To reject this null hypothesis in favor of the method presented here we have also included in Fig. 2a and b the results of 150,300 Monte-Carlo simulations in which alarm cells have been redistributed randomly and the same target earthquakes were used to construct ROC and error diagrams. The confidence intervals were evaluated using the exact analytical result for the binomial distribution given in Zechar and Jordan (2008).
4 Discussion
The primary purpose of this paper is to quantify the extent to which locations of past smaller earthquakes can be used to forecast the locations of future large earthquakes. Our study has been carried out globally. The surface of the Earth has been divided into a grid of l° × l° cells with l = 2°. The objective is to forecast the locations of future large target earthquakes with magnitudes greater than m_{T} based on the past seismicity. Another important aspect of this work is to introduce different optimization procedures to find the best forecast and optimal values for model parameters. It is also important to note that in the presented analysis the measure of seismicity based on cumulative Benioff strain does not perform better than the predictor based on a long-run rate of occurrence of earthquakes.
The sensitivity of the method to the model parameters is an important issue. The cell size l = 2° has been chosen based on the analysis of the areas under the ROC curves constructed for different cell sizes and different forecasting magnitudes. The analysis shows that the areas tend to increase (better forecast) with increasing cell sizes (Fig. 3b). Bigger cell sizes cover a larger Earth’s surface and increase the total success rate. However, bigger cell sizes are less practical in forecasting as they include larger geographical regions and decrease the value of the forecast. In this analysis we assume all earthquakes are point events. Cell size can be an important factor if forecast earthquakes are defined through their rupture areas which can span several cell sizes.
Finally, the dependence of the method on the lower magnitude cutoff, m_{c}, shows a strong trend in the performance increase of the method with decreasing m_{c}. However, the completeness of the catalog prevents the use of events smaller than m < 5.5. The full analysis of the dependence of the method on the above parameters will be presented elsewhere. By varying the threshold B/B_{max} the method generates a spectrum of forecasts. The use of the Pierce’s skill score is probably the most accepted optimization tool, but several others have been proposed (Mason, 2003; Harte and Vere-Jones, 2005). An actual optimization should also be based on other factors such as available resources, population densities, quality of construction, and other risk factors.
Acknowledgments
This work has been supported by NSERC Discovery grant (RS), by NSF grant ATM-0327558 (DLT), by NASA grant NGT5 (JRH), by a US Department of Energy grant to UC Davis DEFG03-14499 (JRH, JBR), and by the NSERC and Benfield/ICLR Chair in Earthquake Hazard Assessment (KFT).
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