How good can hazard maps be & how good do they need to be Seth Stein, Earth & Planetary Sciences, Northwestern University Jerome Stein, Applied Mathematics, Brown University

How good can hazard maps be & how good do they need to be Seth Stein, Earth & Planetary Sciences, Northwestern University Jerome Stein, Applied Mathematics, Brown University What specifically are hazard maps designed to accomplish? How can we measure how well they do it? What is the best they can realistically do?

Insights from weather forecasting Different users want different things from forecasts Thus it’s important to have agreed criteria for "good" and "bad" forecasts of various types. "it is difficult to establish well-defined goals for any project designed to enhance forecasting performance without an unambiguous definition of what constitutes a good forecast." (Murphy, 1993)

Insights from weather forecasting There are fundamental limits to how accurate weather forecasts can be. If weather weren’t chaotic, every year storms would form at the same time & follow the same tracks (Lorenz, 1995) Ebeling and Stein, 2011

Insights from weather forecasting Communicating uncertainty is crucial and valuable Weather and climate forecasts should seek to to “routinely provide the nation with comprehensive, skillful, reliable, and useful information about the uncertainty of weather, water, and climate forecasts… Forecast uncertainty information can improve decisions and outcomes in various socioeconomic areas.” (Hirschberg et al., 2011)

Tohoku earthquake broke many segments 2011 Tohoku Earthquake 450 km long fault, M 9.1 (Aftershock map from USGS) J. Mori Expected Earthquake Sources 50 to 150 km segments M7.5 to 8.2 (Headquarters for Earthquake Research Promotion)

Tsunami runup approximately twice fault slip (Plafker, Okal & Synolakis 2004) M9 generates much larger tsunami Planning assumed maximum magnitude 8 Seawalls 5-10 m high CNN NYTStein & Okal, 2011

Assumed lack of M9s in record seemed consistent with model that M9s only occur where lithosphere younger than 80 Myr subducts faster than 50 mm/yr (Ruff and Kanamori, 1980) Disproved by Sumatra 2004 M9.3 and dataset reanalysis (Stein & Okal, 2007) Short record at most SZs didn’t include larger multisegment ruptures Stein & Okal, 2011

2008 Wenchuan earthquake (Mw 7.9) was not expected: map showed low hazard based on lack of recent earthquakes Didn’t use GPS data showing 1-2 mm/yr (~Wasatch) Earthquakes prior to the 2008 Wenchuan event Aftershocks of the Wenchuan event delineating the rupture zone Stein et al., 2012

Similar problems occur worldwide The earth often surprises us Do these reflect systemic problems with hazard mapping or simply low probability events (someone wins the lottery)?

Off Sanriku-oki North ~M8 0.2 to 10% Off Sanriku-oki Central ~M to 90% Off Fukushima ~M7.4 7% Off Ibaraki ~M6.7 – M7.2 90% Off Miyagi ~M7.5 > 90% Not a PSHA vs DSHA issue - issue is estimating poorly known model parameters Any map assuming no M9 would have done no better

Hazard maps are hard to get right: successfully predicting future shaking depends on accuracy of four assumptions over years Where will large earthquakes occur? When will they occur? How large will they be? How strong will their shaking be? Uncertainty & possible map failure result because these are often hard to assess, especially in plate interiors & other slowly deforming zones

GSHAP 1999 NUVEL-1 Argus, Gordon, DeMets & Stein, 1989 Swafford & Stein, 2007 Slow plate boundary Africa-Eurasia convergence rate varies smoothly (5 mm/yr)

Swafford & Stein, 2007 GSHAP 1999 NUVEL-1 Argus, Gordon, DeMets & Stein, 1989 M 6.4 M 6.3 Slow plate boundary Africa-Eurasia convergence rate varies smoothly (5 mm/yr)

Long record needed to see real hazard Swafford & Stein, M M 7.2

“Our glacial loading model suggests that earthquakes may occur anywhere along the rifted margin which has been glaciated.” Stein et al., Concentrated hazard bull's-eyes at historic earthquake sites 2005 Diffuse hazard along margin GSC Map depends greatly on assumptions & thus has large uncertainty

Plate Boundary Earthquakes Major fault loaded rapidly at constant rate Earthquakes spatially focused & temporally quasi-periodic Past is fair predictor Intraplate Earthquakes Tectonic loading collectively accommodated by a complex system of interacting faults Loading rate on a given fault is slow & may not be constant Earthquakes can cluster on a fault for a while then shift Past can be poor predictor Plate A Plate B Earthquakes at different time Stein, Liu & Wang 2009

during the period prior to the period instrumental events Earthquakes in North China Large events often pop up where there was little seismicity! Ordos Plateau Shanxi Graben Bohai Bay Beijing 1303 Hongtong M 8.0 Liu, Stein & Wang 2011 Weihi rift

during the period prior to the period instrumental events Earthquakes in North China Large events often pop up where there was little seismicity! Ordos Plateau Shanxi Graben Bohai Bay Beijing 1556 Huaxian M 8.3 Weihi rift Liu, Stein & Wang 2011

during the period prior to the period instrumental events Earthquakes in North China Large events often pop up where there was little seismicity! Ordos Plateau Shanxi Graben Bohai Bay Beijing 1668 Tancheng M 8.5 Weihi rift Liu, Stein & Wang 2011

during the period prior to the period instrumental events Earthquakes in North China Large events often pop up where there was little seismicity! Ordos Plateau Shanxi Graben Bohai Bay Beijing 1679 Sanhe M 8.0 Weihi rift Liu, Stein & Wang 2011

during the period prior to the period instrumental events Earthquakes in North China Large events often pop up where there was little seismicity! Ordos Plateau Shanxi Graben Bohai Bay Beijing 1966 Xingtai M Tangshan M Haicheng M 7.3 Weihi rift Liu, Stein & Wang 2011

No large (M>7) events ruptured the same fault segment twice in past 2000 years In past 200 years, quakes migrated from Shanxi Graben to N. China Plain Historical Instrumental Shanxi Graben Weihi rift

Aftershocks of 1976 Tangshan earthquake continue today Liu & Wang, 2012

Rate-state friction (Dieterich, 1994) predicts aftershock duration  1/loading rate Plate boundary faults quickly reloaded by steady plate motion after large earthquake Faults in continents reloaded much more slowly, so aftershocks continue much longer Stein & Liu, 2009 Often see long aftershock sequences in slowly deforming continental interiors Stein & Liu 2009

Tuttle (2009) Within plates or other low-deformation zones, present seismicity may be better indicator of locations of past events & future small events than future large ones Obermeier, (1998) Wabash: M~7 6 Kybp

Newman et al., % 275% Hazard maps involve assumptions about - Mmax of largest future events -Ground motion model -Timing of future earthquakes (time- independent or time- dependent) Since all have large uncertainties, wide range of plausible hazard models

154% %106 Hazard maps involve assumptions about - Mmax of largest future events -Ground motion model -Timing of future earthquakes (time- independent or time- dependent) Since all have large uncertainties, wide range of plausible hazard models

Stein et al, 2012 Stein et al., 2012 Uncertainty typically factor of 3-4 Often can’t be reduced much due to earthquake variability Hazard is essentially unknowable within broad range One can chose a particular value depending on preconception, but the uncertainty remains and only time will tell how good the choice was

Seismological assessment of hazard maps Various metrics could be used, e.g. compare maximum observed shaking in subregion i, x i to predicted maximum shaking p i Compute Hazard Map Error HME(p,x) =  i (x i - p i ) 2 /N and compare to error of reference map produced using a null hypothesis HME(r,x) =  i (x i - r i ) 2 /N using the skill score SS(p,r,x) = 1 - HME(p,x)/HME(r,x) Positive score if map does better than null

Some testing challenges 1)Short time record: can be worked around by aggregating regions. 2) Subjective nature of hazard mapping, resulting from need to chose faults, maximum magnitude, recurrence model, and ground motion model. This precludes the traditional method of developing a model from the first part of a time series and testing how well it does in the later part. That works if the model is "automatically" generated by some rules (e.g. least squares, etc). In the earthquake case, this can't be done easily because we know what happens in the later part of the series.

3) New maps made after a large earthquake that earlier maps missed are problem for counting statistics. Frankel et al, 2010 Before 2010 Haiti M7After 2010 Haiti M7 4X

4) Overparameterized model (overfit data): Given a trend with scatter, fitting a higher order polynomial can give a better fit to the past data but a worse fit to future data Analogously, a seismic hazard map fit to details of past earthquakes could be a worse predictor of future ones than a smoothed map How much detail is useful? Linear fit Quadratic fit

Consider map as means, not end Assess map’s success in terms of contribution to mitigation Even uncertain or poor maps may do some good Societal assessment of hazard maps

Goal: Assess the hazard in order to develop cost-effective mitigation policy Mitigation diverts resources from present needs - education, health, environment, economic growth, etc - in hope of reducing possible future losses Mitigation policies typically made - without considering uncertainties in hazard estimate (map) - without analysis of costs & benefits Challenge: how much is enough?

Societally optimal level of mitigation minimizes total cost = sum of mitigation cost + expected loss Expected loss = ∑ (loss in i th expected event x assumed probability of that event) Compared to optimum Less mitigation decreases construction costs but increases expected loss and thus total cost More mitigation gives less expected loss but higher total cost Stein & Stein, 2012 For earthquake, mitigation level is construction code Loss depends on earthquake & mitigation level Optimum

Consider marginal costs C’(n) & benefits Q’(n) (derivatives) Optimum is where marginal curves are equal, n* More mitigation costs more But reduces loss Stein & Stein, 2012 Benefit (loss reduction ) cost Less mitigation costs less But increases loss

Loss estimate scenarios based on hazard model Estimate loss as function of magnitude, ground shaking model, recurrence rate, and mitigation level This case Current mitigation fatalities ~ $100B damage Examine range of parameters & use to find optimum / php

Present Value of Future Losses Average loss over T years is L T Interest rate i PVFL = L T  t 1/(1+i) t = L T D T D T = 1/(1+i) + 1/(1+i) /(1+i) T = ((1+i) T -1 ) / (i(1+i) T ) ≈ 1/i for T large For interest rate i=0.05, D T = 15.4 for 30 years, and 19.8 for 100 years. For long enough times, the limit as T becomes infinite is D T = 1 / I, so if i = 0.05, D = 20. This is essentially the same as the value for 100 years.

Even without uncertainty, mitigation rarely will be optimal for societal reasons,but can still do some good Net benefit when mitigation lowers total cost below that of no mitigation Net loss when mitigation raises total cost above that of no mitigation

Within range, inaccurate hazard maps produce nonoptimal mitigation, raising cost, but still do some good (net benefit)

A hazard map that greatly overestimates the hazard can do net harm Resulting overmitigation cost can cause net loss

Epistemic uncertainty gives range of hazard estimates Select mitigation level between optima

Including risk aversion & aleatory uncertainty Consider marginal costs C’(n) & benefits Q’(n) (derivatives) Aleatory uncertainty in hazard model & mitigation efficiency causes uncertainty in expected loss. We are risk averse, so add risk term R(n) proportional to uncertainty in loss, yielding higher mitigation level n** cost Benefit (loss reduction ) Stein & Stein, 2012

Summary Limitations in our knowledge about earthquakes, notably space-time variability, limit how accurately hazard maps can be made Although uncertain maps likely produce nonoptimal mitigation, they still do some good if they’re not too bad Testing maps & quantifying uncertainties will help some Need to recognize & accept uncertainties

Haiti 2001 hazard map M7 earthquake shaking much greater than predicted for next 500 years

Epistemic uncertainty gives range of hazard estimates Select mitigation level between optima