Abstract
The purpose of using loss functions for earthquake prediction and disaster preparedness is to minimize expected costs when destructive earthquakes occur. This study focuses on developing a greater understanding of the interrelation between earthquake prediction characteristics and economic parameters. Equations for estimating economic losses and optimizing both earthquake prediction and disaster preparedness are presented in different dimensional forms to improve the understanding of parametric relationships between prediction and preparedness. The equations expand upon previously presented loss functions by explicitly considering loss as a function of both time and space and the cost parameters are clearly described to allow for practical application. Derivations reveal the close relationship between the loss function in terms of earthquake prediction characteristics and the benefit-cost analysis commonly used for disaster preparedness. An optimal preparedness scheme is presented based on a concept of unpreventable damage in extreme events and is shown to be a function of the level of damage prevented by taking action in response to an earthquake prediction. The formulations show that alarm durations are optimal relative to the type and time to implement different actions and the alarm area is optimal relative to the potential earthquake size and related geohazards. The presentation shows that earthquake prediction need not be constrained at a point in space to be useful for disaster preparedness and that mitigation activity is more economically feasible the smaller the area of prediction is with respect to the potential earthquake source size. Examples are used to show how loss functions can be utilized to determine if an algorithm may be useful to implement into practice and how earthquake prediction strategies can be implemented in coordination with other risk reduction strategies to make cost effective mitigations. Optimized earthquake prediction algorithms will greatly aid disaster managers and decision makers in their preparations once a prediction is made. The loss functions help to develop a greater understanding between earthquake prediction research and disaster preparedness implementation, allowing for future improvements in earthquake disaster prevention.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
(This is based on the Multihazard Mitigation Council criteria (2005) that showed how every dollar spent by the Federal Emergency Management Agency provides a future $4 benefit and leads to additional non-federally funded mitigation investments. A 100 % private to public investment is assumed, resulting in the factor of 8 used in calculations.)
Abbreviations
- A :
-
Number of alarms declared
- A f :
-
Number of false alarms
- A s :
-
Number of successfully predicted alarms
- N :
-
Number of earthquake events having M ≥ M o
- N m :
-
Number of earthquakes missed by alarms; number of failures to predict
- N I :
-
Number of earthquakes successfully identified by alarms
- f :
-
Rate of false alarms; f = A f/A
- n :
-
Rate of failures to predict; n = N m/N
- α:
-
Total space–time occupied by alarms; \( \alpha = a/(T \cdot R_{b} ) \)
- α j :
-
Space–time occupied by alarm j; \( \alpha_{j} = a_{j} /(T \cdot R_{b} ) \)
- a :
-
Total space–time covered by alarms
- a j :
-
Space–time covered by alarm j
- ρ:
-
Ratio of total space (regional area) occupied by alarms; ρ = ∆r/R b
- ρ j :
-
Ratio of total space (regional area) occupied by alarm j; ρ j = R a|j /R b
- τ:
-
Ratio of total time occupied by alarms; τ = ∆t/T
- τ j :
-
Ratio of total time occupied by alarm j; τ j = ∆ j /T
- ∆ j :
-
Time covered by alarm j; ∆-time covered by alarm
- ∆ j∩k :
-
Total duration of intersecting time between alarms j and k
- ∆t :
-
Total nonintersecting time covered by alarms; ∆t = Σ(∆ j − ∆ j∩k )
- ∆r :
-
Total nonintersecting space covered by alarms; ∆r = Σ(R a|j − R a|j∩k )
- \( \bar{p}_{R} \) :
-
Parameter indicating the efficiency by area of the predicted earthquake in alarm region causing damage
- \( \bar{p}_{R|j} \) :
-
Parameter indicating the efficiency by area of the predicted earthquake in alarm region causing damage for alarm j
- M :
-
Magnitude range
- M o :
-
Lower bound magnitude consistent with regional seismicity
- M max :
-
Maximum magnitude anticipated in a region
- O :
-
Total set of vulnerable objects
- \( O_{i} (*) \) :
-
Subset of vulnerable objects; \( (*) \)-index identifying the type of vulnerable object
- R a :
-
Region (area) of alarm
- R a|j :
-
Region (area) of alarm j
- R a|j∩k :
-
Region (area) of intersecting space between alarms j and k
- R b :
-
Region (area) under study for predicting earthquakes
- R d :
-
Region (area) of expected damage
- R d|k :
-
Region (area) of expected damage for earthquake k
- R do :
-
The region (area) having an outer boundary of potential loss or damage resulting directly from an earthquake
- R do|k :
-
The region (area) of potential loss for earthquake k
- T :
-
Time period for applying a prediction algorithm
- T r :
-
Recovery time period
- Λs :
-
Source zone size
- X :
-
Dimension between R a and R do
- C a :
-
Average cost of actions for all alarms; C a = ΣC a|j /A = ∆C a/A
- C a|ij :
-
Cost of implementing actions to mitigate potential damage to object O i for alarm j
- C a|gj :
-
Cost of implementing general actions for alarm j
- \( C_{{{\text{a - }}R_{\text{d}} }} \) :
-
Costs C a for uniformly distributed population density and objects in R d
- ∆C a :
-
Total cost of implementing actions in all alarms
- C s :
-
Average cost of switching from non-alarm to alarm and back again; C s = ΣC s|j /A
- C s|j :
-
Total cost of switching from non-alarm to alarm j and back again
- C st|j :
-
Cost incurred when switching to an alarm j
- C so|j :
-
Cost incurred when switching out of an alarm j
- \( C_{{{\text{s - }}R_{\text{d}} }} \) :
-
Costs C s for uniformly distributed population density and objects in R d
- C m|j :
-
Cost of implementing and maintaining preparations in response to an alarm j per unit of time; C m|j = C a|j /∆ j
- C m :
-
Average cost of implementing and maintaining preparations in response to all alarms per unit of time; C m = ΣC m|j /A
- \( \hat{C}_{{{\text{m}}|j}} \) :
-
Cost of implementing and maintaining preparations in response to an alarm j per unit of space; \( \hat{C}_{{{\text{m}}|j}} = C_{{{\text{a|}}j}} /R_{{{\text{a}}|j}} \)
- \( \hat{C}_{\text{m}} \) :
-
Average cost of implementing and maintaining preparations in response to all alarms per unit of alarm area; \( \hat{C}_{\text{m}} = \Upsigma \hat{C}_{{{\text{m}}|j}} /A \)
- \( \hat{C}_{{{\text{m}}|j - R_{\text{d}} }} \) :
-
Costs \( \hat{C}_{{{\text{m}}|j}} \) for uniformly distributed population density and objects in R d
- \( \dot{C}_{{{\text{m}}|j}} \) :
-
Cost of implementing and maintaining preparations in response to an alarm j per unit of space–time; \( \dot{C}_{{{\text{m}}|j}} = C_{{{\text{a}}|j}} /(\Updelta_{j} R_{{{\text{a}}|j}} ) \)
- \( \dot{C}_{\text{m}} \) :
-
Average cost of implementing and maintaining preparations in response to all alarms per unit of alarm space–time; \( \dot{C}_{\text{m}} = \Upsigma \dot{C}_{{{\text{m}}|j}} /A \)
- D :
-
Average cost of damage resulting from a failure to predict or not preparing in response to an alarm; D = ΣD k /N
- D ik :
-
Damage to object i in earthquake k
- D e :
-
Losses resulting from disruption of the regional economic system
- D ek :
-
Losses resulting from disruption of the regional economic system in earthquake k
- P :
-
Average cost of damage prevented for all successful alarms; P = ΣP j /A s
- P v :
-
Damage prevented for successfully predicted alarm v
- P ij :
-
Cost of prevented damage that may be achieved by making investment C a|ij in response to alarm j
- P gj :
-
Cost of prevented damage that may be achieved by making investment C a|gj in response to alarm j
- P ik :
-
Damage prevented to object i in earthquake k
- Ωd :
-
Loss per unit area from a failure to predict or not preparing in response to an alarm over regional area R d; Ωd = D/R d
- Ωs :
-
Loss per unit area for switching to and from the alarm applied to region R do; Ωs = C s/R do
- Ωa :
-
Loss per unit space–time for implementing actions maintaining the alarm applied to the region R do; Ωa = C a/R do
- Ωp :
-
Gain per unit area resulting from actions implemented in region R d; Ωp = P/R d
- i :
-
Index identifying vulnerable objects
- j :
-
Index identifying number of alarms
- k :
-
Index identifying number of earthquakes (or number of failures to predict)
- v :
-
Index identifying number of successful alarms
- g :
-
Index identifying number of general actions implemented
- λ:
-
Rate of target events; \( \lambda = N/T \)
- ν :
-
Rate of alarm transitions, number of alarm transition sets (on and off) per unit of time; \( \nu = A/T \)
- β:
-
Density of target events; \( \beta = N/R_{\text{b}} \)
- χ:
-
Alarm density, alarms per unit of area; \( \chi = A/R_{\text{b}} \)
- μ:
-
Alarms per unit of alarm time; \( \mu = A/\Updelta t;1/\mu \) is the average time per alarm
- ψ:
-
Alarms per unit of alarm area; \( \psi = A/\Updelta r;1/\psi \) is the average area per alarm
- ω:
-
Rate density of target events; \( \omega = N/(TR_{\text{b}} ) \)
- ξ:
-
Alarms per unit of alarm space–time; \( \xi = A/a;1/\xi \) is the average space–time per alarm
- \( \zeta \) :
-
Rate density of alarm transitions, number of alarm transition sets (on and off) per unit of space–time; \( \zeta = A/(TR_{\text{b}} ) \)
- r(t):
-
Conditional rate of the predicted event
- r j (t):
-
Conditional rate of the predicted event j
- r 0 :
-
Threshold value for calling alarm
- r 0|j :
-
Threshold value for calling alarm j
- L :
-
Total loss
- δL :
-
Incremental loss
- G :
-
Total gain
- δG :
-
Incremental gain
- γ:
-
Loss per unit time
- η:
-
Loss per unit area
- θ:
-
Loss per unit space–time
References
Allen, C. R. (Chairman), W. Edwards, W. J. Hall, L. Knopoff, C. B. Raleigh, C. H. Savit, M. N. Toksoz, and R. H. Turner, 1976, “Predicting Earthquakes: A Scientific and Technical Evaluation—With Implications for Society,” Panel on Earthquake Prediction of the Committee on Seismology, Assembly of Mathematical and Physical Sciences, National Research Council, U.S. National Academy of Sciences, Washington, D.C.
Console, R., D. Pantosti, and G. D’Addezio, 2002, “Probabilistic Approach to Earthquake Prediction,” Annals of Geophysics, Vol. 45, No. 6, pp. 723–731.
Davis, C. A., and J.P. Bardet, 2011, “Lifelines in Megacities: Future Directions of Lifeline Systems for Sustainable Megacities,” Chapter 3 in “Geotechnics and Earthquake Geotechnics towards Global Sustainability”, S. Iai, Ed., Springer, The Netherlands.
Davis, C., V. Keilis-Borok, G. Molchan, P. Shebalin, P. Lahr, and C. Plumb, 2010, “Earthquake Prediction and Disaster Preparedness: Interactive Analysis,” ASCE Natural Hazards Review, Vol. 11, No. 4, pp 173–184.
Davis, C., V. Keilis-Borok, K. Goss, V. Kossobokov, and A. Soloveiv, 2012, “Advance Prediction of the March 11, 2011 Great East Japan Earthquake: A Missed Opportunity for Disaster Preparedness,” International Journal of Disaster Risk Reduction, Vol. 1, http://dx.doi.org/10.1016/j.ijdrr.2012.03.001.
Ellis, S. P., 1985, “An Optimal Statistical Decision Rule for Calling Earthquake Alerts,” Earthquake Prediction Research, 3, pp. 1–10.
International Commission on Earthquake Forecasting for Civil Protection (ICEF), 2011, Report submitted to the Department of Civil Protection, Rome, Italy, May 30, 2011, T. H. Jordan Chair, Published in Annals of Geophysics, 54, 4, 2011; doi:10.4401/ag-5350.
Jones, L.M., R. Bernknopf, D. Cox, J. Goltz, K. Hudnut, D. Mileti, S. Perry, D. Ponti, K. Porter, M. Reichle, H. Seligson, K. Shoaf, J. Treiman, and A. Wein, 2008. “The ShakeOut Scenario,” U.S. Geological Survey OFR 2008-1150 and California Geological Survey Preliminary Report 25.
Jordan, T. H., Y.-T. Chen, P. Gasparini, R. Madariaga, I. Main, W. Marzocchi, G. Papadopoulos, G. Sobolev, K. Yamaoka, J. Zschau, 2009, “Operational Earthquake Forecasting: State of Knowledge and Guidelines of Implementation. Findings and Recommendations of the International Commission on Earthquake Forecasting for Civil Protection, released by the Dipartimento della Protezione Civile, Rome, Italy, 2 October, 2009. Available at http://www.prevententionweb.net/english/professional/publications/v.php?id=11788.
Jordan, T. H., and L. M. Jones, 2010, “Operational Earthquake Forecasting: Some Thoughts on Why and How,” Seismological Research Letters, Seismological Society of America, Vol. 81, No. 4, pp. 571–574.
Kanamori, H., 1977, “The Energy Release in Great Earthquakes,” Journal of Geophysical Research, 82, 2981–2987.
Keilis-Borok, V. I., 2002, “Earthquake Prediction: State-of-the-Art and Emerging Possibilities,” Annu. Rev. Earth Planet. Sci., 30, pp. 1–33.
Kossobokov, V. G., 2011, “Are Mega Earthquakes Predictable?” Pleiades Publishing, Ltd, Atmospheric and Oceanic Physics, Vol. 46, No. 8, pp. 951–961.
Kossobokov, V. and, P. Shebalin, 2003, “Earthquake Prediction,” Chapter 4 in “Nonlinear Dynamics of the Lithosphere and Earthquake Prediction,” V.I. Keilis-Borok and A.A. Soloviev, eds., Springer, Heidelberg, pp. 141–207.
Marzocchi, W., and G. Woo, 2007, “Probabilistic Eruption Forecasting and the Call for an Evacuation,” Geophysical Research Letters, Vol. 34, L22310, 4 pp, doi:10.1029/2007GL031922.
Molchan, G. M., 1990, “Strategies in Strong Motion Prediction,” Physics of the Earth and Planetary Interiors, 61, pp. 84–98.
Molchan, G.M., 1991, “The Structure of Optimal Strategies in Earthquake Prediction,” Tectonophysics 193, pp. 255–265.
Molchan, G. M., 1994, “Models of the Optimization of Earthquake Prediction,” D. K. Chowdhury, ed, Computational Seismology and Geodynamics, Vol. 2, pp. 1–10, Am. Geophys. Union, Washington, D. C.
Molchan, G. M., 1997, “Earthquake Prediction as a Decision Making Problem,” Pure and Applied Geophysics, 149, pp. 233–247.
Molchan, G. M., 2003, “Earthquake Prediction Strategies: A Theoretical Analysis,” Chapter 5 in “Nonlinear Dynamics of the Lithosphere and Earthquake Prediction,” V.I. Keilis-Borok and A.A. Soloviev, eds., Springer, Heidelberg, pp. 209–237.
Molchan, G. M., 2010, “Space-Time Earthquake Prediction: The Error Diagrams,” Pure and Applied Geophysics, published on line 10, March, 2010: http://springerlink.com/content/y14x42201r13747m/.
Molchan, G. M., and Y. Y. Kagan, 1992, “Earthquake Prediction and its Optimization,” Journal of Geophysical Research, 97, pp 4823–4838.
Molchan, G. M., and V. Keilis-Borok, 2008, “Earthquake Prediction: Probabilistic Aspect,” Geophys J Int, 173(3):1012–1017.
Murphy, A. H., 1977, “The Value of Climatological, Categorical and Probabilistic Forecasts in the Cost-Loss Ratio Situation,” Mon. Wea. Rev., 105, 803–816.
Quiroga, S., and E. Cerdá, 2009, “Optimal Crop Protection against Climate Risk in a Dynamic Cost-Loss Decision-Making Model,” International Association of Agricultural Economists Conference, Beijing, China, August 16-22, paper 598 KB, available at http://ageconsearch.umn.edu/bitstream/51702/2/166_Beijing_2009_Quiroga_Cerda.pdf, Sept. 25, 2010.
Thompson, J.C., 1952, “On the Operational Deficiencies in Categorical Weather Forecasts,” Bull. Amer. Meteor. Soc., 33, 223–226.
van Stiphout, T., S. Wiemer, and W. Marzocchi, 2010, “Are short-term evacuations warranted? Case of the 2009 L’Aquila earthquake,” Geophysical Research Letters, Vol. 37, L06306, 5 pp., 2010, doi:10.1029/2009GL042352.
Vere-Jones, D., 1995, “Forecasting Earthquakes and Earthquake Risk,” International Journal of Forecasting, 11, pp. 503–538.
Wein, A., and A. Rose, 2008, “Regional Economic Consequences,” Chapter 7 in Jones et al., “The ShakeOut Scenario,” U.S. Geological Survey OFR 2008-1150 and California Geological Survey Preliminary Report 25.
Wells, D., and K. Coppersmith, 1994, “New Empirical Relationships among Magnitude, Rupture Length, Rupture Width, Rupture Area, and Surface Displacement, Bull. Seism, Soc. Am., 84, No.4, pp. 974–1002.
Acknowledgments
The author is grateful to Professor George Molchan for his kind review and comments and especially to Professor Vladimir Keilis-Borok for the encouragement to perform studies in the areas discussed in this manuscript, enlightening discussions, and helpful comments. Valuable comments from anonymous reviews are acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Davis, C.A. Loss Functions for Temporal and Spatial Optimizing of Earthquake Prediction and Disaster Preparedness. Pure Appl. Geophys. 169, 1989–2010 (2012). https://doi.org/10.1007/s00024-012-0502-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00024-012-0502-8