Examples of Manuscript Components and Description of Electronic

SCEC 2013-2014 Annual Report:

Trimming the UCERF3 Hazard Logic Tree

Prepared for Southern California Earthquake Center Zumberge Hall of Science 169 University of Southern California 3651 Trousdale Parkway Los Angeles, CA 90089-0742

By Keith Porter and Charles Scawthorn SPA Risk LLC Denver CO, USA

19 February 2015

Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

Abstract. This report summarizes an effort to quantify the sensitivity of societal risk to branches in the logic tree of the Uniform California Earthquake Rupture Forecast version 3 (UCERF3, Field et al. 2014). It uses a deterministic sensitivity study sometimes called a tornado-diagram analysis to identify logic-tree branches that matter most to the probability distribution of expected annualized loss (EAL) to California woodframe single-family dwellings. The value of EAL varies between branches of UCERF3. Each branch of the logic tree has an associated marginal probability (in the Bayesian sense) or weight (in the language of frequentists), so EAL has a probability distribution. We identified the branches that matter most to that distribution: probability model, ground motion prediction equation (not an element of UCERF3, but important to EAL), total magnitude-5 rate, and choice of off-fault spatial seismicity probability density function. One can fix the other 5 parameters at arbitrary baseline values without a large bias in CDF of EAL. Varying only these 4 rather than 9 parameters, one can reduce by 99.6% the computational effort required to calculate the distribution of EAL without a substantial difference in the results.

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

Table of Contents 1 INTRODUCTION AND OBJECTIVES ...... 6

2 PROGRESS OF THE WORK ...... 9

2.1 COMPLETED WORK ...... 9 2.2 LIMITATIONS ...... 14

3 CONCLUSIONS ...... 15

4 REFERENCES CITED ...... 16

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

Index of Figures

Figure 1. Sensitivity of California statewide expected annualized loss to UCERF2 uncertainties (Porter et al. 2012) ...... 7

Figure 2. Portfolio of woodframe single family dwellings ...... 10

Figure 3. Vulnerability functions for woodframe single-family dwellings ...... 10

Figure 4. Cumulative distribution function of EAL from California woodframe single-family dwellings ...... 11

Figure 5. Tornado diagram of EAL for California woodframe dwellings ...... 13

Figure 6. EAL for California woodframe single-family dwellings using the trimmed UCERF3.3 logic tree (red line) versus the full tree (black line) ...... 14

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

Index of Tables

Table 1. UCERF3.3 branches (along with branches for the ground motion prediction equation)...... 8

Table 2. Sample of California portfolio ...... 9

Table 3. Layout of inventory table ...... 9

Table 4. Tornado diagram results for EAL in California woodframe dwellings ...... 12

Table 5. Tree-trimming of UCERF3.3 considering EAL in California woodframe dwellings ...... 13

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

1 INTRODUCTION AND OBJECTIVES

This report summarizes a study funded by the Southern California Earthquake Center. The goal of this project is to assess the sensitivity of selected measures of societal risk to selected branches in the hazard model, in particular to the branches of the Uniform California Earthquake Rupture Forecast version 3 (UCERF3, Field et al. 2014). This report was delayed to accommodate peer review of that publication.

By “measures of societal risk” is meant expected annualized loss—either economic or human deaths and injuries—or aspects of the loss-exceedance curve for the same measures. A loss- exceedance curve here means a relationship between loss and mean exceedance frequency. If one can identify a subset of branches that dominate uncertainty in risk, a future risk calculation can focus solely on those branches. The analyst can take the other branches at arbitrary, deterministic values, and thus reduce the computational effort of calculating risk. If the uncertainties that matter can be reduced by further scientific inquiry, one can prioritize future research.

We found in prior research (Porter et al. 2012) that the three uncertainties that matter most to UCERF2 were the choice of probability model, ground motion prediction equation (not an element of UCERF2, but an important element of risk), and the magnitude-area relationship. We used a deterministic sensitivity sometimes called a tornado-diagram analysis to produce that knowledge (Howard 1988). The tornado diagram depicts which of two or more uncertain input parameters matter most to a quantity of interest. Let us denote the quantity of interest by y, the input uncertainties as x1, x2, etc., and the functional relationship between them as

y f x12, x ,... xin ,... x  (1)

Each xi is expressed either with a cumulative distribution function or merely reasonable low, typical and high values. Let us denote low, typical, and high values of xi by xilow, xityp, and xihigh, respectively. One first calculates a baseline value of y:

ybaseline = f(x1typ, x2typ, ... xityp, ... xntyp) (2)

Next, one quantifies the sensitivity of y to variability in each xi using a measure called swing:

yilow = f(x1typ, x2typ, ... xilow, ... xntyp) (3)

yihigh = f(x1high, x2typ, ... xihigh, ... xntyp) (4) www.sparisk.com 6

Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

swingi = | yihjgh - yilow | (5)

One then sorts the xi parameters in decreasing order of swing and plots the results in a horizontal bar chart. The horizontal axis measures y. The uppermost (top) horizontal bar in the chart measures yi with where i is the index for the input parameter with the largest swing. Its left end is the smaller of {yilow, yihigh}. Its right end is the larger of {yilow, yihigh}. The next horizontal bar measures yj where j is the index for the x-parameter with the 2nd-largest swing. Its left end is the smaller of {yjlow, yjhigh}. Its right end is the larger of {yjlow, yjhigh}. The result looks like a tornado in profile. A vertical line is drawn through ybaseline. The result can look like Figure 1, from our previous work on UCERF2. One can then model y as a function of the 2 or 3 or 4 x- parameters that matter most, fixing the rest to their baseline values.

Probability model Empirical BPT, 0.3 GMPE CB2008 AS2008 Magnitude-area relationship Ellsworth Hanks & Bakun A-fault solution type Unsegmented Segmented

Connect more B faults? True False Deformation model D2.6 D2.2 Fault-slip rates A priori Mo-rate bal Median and B-Faults b-value 0.0 0.8 weighted avg

0.5 1.0 1.5 2.0 Expected annualized loss, $ billion

Figure 1. Sensitivity of California statewide expected annualized loss to UCERF2 uncertainties (Porter et al. 2012)

The present research aims to perform a tornado-diagram analysis of UCERF3. The UCERF2 logic tree contained 480 logic tree branches. Adding 4 choices of ground motion prediction equation, the total number of branches was 1,920. UCERF3 contains 5,760 branches. With 5 choices of ground motion prediction equation, the total number of branches is 28,800. Despite the greater complexity of UCERF3, the concepts and procedures for trimming its logic tree are the same as in Porter et al. (2012). The uncertainties and their respective numbers of branches are shown in Table 1.

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

Table 1. UCERF3 branches (along with branches for the ground motion prediction equation)

Variable Branches Cumulative branches 1. ProbModel 4 4 2. GMPE 5 20 3. TotalMag5Rate 3 60 4. SpatialSeisPDF 2 120 5. DeformationModels 4 480 6. ScalingRelationships 5 2400 7. MaxMagOffFault 3 7200 8. SlipAlongRuptureModels 2 14400 9. FaultModels 2 28800

As with the tree trimming of UCERF2, the present analysis requires the following components.

1. An estimated portfolio of assets exposed to shaking—building occupants, building value, etc. The “portfolio” comprises an estimate, by relatively small geographic area such as a census tract, of the value of the assets (number of people or replacement cost of property) by asset class. An asset class might be defined in terms of model building type, height category, era of construction, structure type and possibly occupancy class.

2. A computer model capable of estimating the loss for a portfolio of assets exposed to seismic shaking. The model must be capable of estimating loss for each individual path along the hazard logic tree. To date, we have quantified loss in terms of expected annualized loss (EAL). We plan to quantify loss soon in terms of the portfolio loss exceedance curve (LEC).

3. A set of seismic vulnerability functions that relate (uncertain) loss to ground motion for each asset class in the portfolio. A mean seismic vulnerability function here refers to a relationship between the expected number of deaths or injuries (typically as a fraction of occupants, i.e., the mean damage factor) and a scalar or vector shaking intensity, whether shaking is measured in terms of damped elastic spectral acceleration response at some index period, or instrumental intensity, or some combination of these with one or more of magnitude, distance, site class, etc. For an LEC, the vulnerability functions must be probabilistic, i.e., providing an estimate of the probability distribution of loss of the asset class as a function of ground motion.

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

2 PROGRESS OF THE WORK 2.1 COMPLETED WORK Most of the components necessary for the work were completed in prior research for SCEC. In 2009-2010, the replacement cost of buildings, contents, and occupants was estimated and tabulated in a standard format. A sample of the database is shown in Table 2. The fields are defined in Table 2. Figure 2 illustrates the portfolio of single-family wood dwellings used in the present work. For the present work, we updated the values from prior work to 2013 replacement costs, accounting for population growth and inflation.

Previous work also produced an OpenSHA Portfolio EAL calculator. See Porter et al. (2012) for details. In the 2008-2009 work for USGS and SCEC, we developed seismic vulnerability functions for casualties and building losses. See Porter (2009a, 2009b, 2010) for details. Figure 3 illustrates the mean vulnerability functions used in the present work for woodframe single-family dwellings.

Table 2. Sample of California portfolio

ID Tract OccLabel SsType DesignLevel A Vb Vc PopDay PopNight PopCommute 349 06001400100 RES1 W1 MC 1482 249794 124905 247 1544 546 350 06001400100 RES3A C2L HC 2.00 184 92 0 2 1 351 06001400100 RES3A C2L LC 0.65 59 30 0 1 0 352 06001400100 RES3A C2L MC 1.94 178 90 0 2 1 353 06001400100 RES3A C3L MC 0.65 59 30 0 1 0

Table 3. Layout of inventory table

Field name Data type Description Comment ID Autonumber An index 1, 2, … Tract Text, 11 Census tract number e.g., 06001400100, from hzTract.Tract BldgSchemesId Text, 5 Identifies scheme to distribute from occupancy to In California, distinguishes L/M/H material, structure type and design level hazard OccLabel Text, 5 HAZUS-MH occupancy label RES1, RES2, … EDU2 SsType Text, 10 FEMA earthquake structure type W1, W2, … or MH DesignLevel Text, 2 HAZUS-MH seismic design level PC, LC, LS, MC, MS, HC, or HS A Double 1000 sq ft See Porter (2009a) for deriv ValYr MM/DD/YYYY Year in which dollar valuation is made Default = 1/1/2003 Vb Double Building replacement cost, $1000s See Porter (2009a) for deriv Vc Double Content replacement cost, $1000s See Porter (2009a) for deriv PopDay Long integer Daytime population, people at 2 PM See Porter (2009a) for deriv PopNight Long integer Nighttime population, people at 2 AM See Porter (2009a) for deriv PopCommute Long integer Commute-time population, people at 5 PM See Porter (2009a) for deriv

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

Replacement cost (2013 USD) per sq km < $1 M $1 - 10 M $10 - 100 M $100 - 1,000 M > $1,000 M

±

0 25 50 100 150 200 Miles

Figure 2. Portfolio of woodframe single family dwellings

1.00

0.75

W1 pre-code 0.50 single-family dwelling

Low code Mod code

Mean damage factor damage Mean 0.25

High code 0.00 0.00 0.50 1.00 1.50 S (1.0 sec, 5%), geomean, g a

Figure 3. Vulnerability functions for woodframe single-family dwellings

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

Milner updated the OpenSHA EAL calculator with UCERF3 and performed the calculations of y specified in Equations (2) through (4). We calculated the EAL for the portfolio of California woodframe single-family dwellings and each of the 28,800 branches discussed above. We also tested sensitivity to model of Vs30, choosing between Wills and Clahan (2006) and Wald and Allen (2007); results are similar enough to say that EAL is not strongly sensitive to Vs30 model. The following discussion illustrates results using Wald and Allen (2007).

Each branch has an associated Bayesian probability attached to it (a frequentist would refer to these probabilities as weights), so one can consider the EAL values as samples with varying probability. Sorting them in increasing value and calculating their cumulative probabilities, we created the cumulative distribution function shown in Figure 4. A Lilliefors (1967) goodness of fit test shows that one can reasonably treat the sample distribution as if it were ideally lognormally distributed, with median (in the real domain) $2.37 billion and logarithmic standard deviation 0.20. (With Wills and Clahan, the median is $2.08 billion with the same logarithmic standard deviation.)

1.00

0.75

0.50

0.25 Cumulative probability Cumulative

0.00 0.0 1.0 2.0 3.0 4.0 5.0 Expected anualized loss, $B

Figure 4. Cumulative distribution function of EAL from California woodframe single-family dwellings

Table 4 presents the results of the EAL sensitivity study calculations as shown in Equation (5). Sorting and plotting, we produced the tornado diagram shown in Figure 5. It shows that the four variables that matter most are the probability model, ground motion prediction equation,

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015 total magnitude-5 rate, and choice of off-fault spatial seismicity probability density function. Table 5 lists the logic-tree branches in order of decreasing swing, along with differences between the EAL CDF if one uses the full UCERF3.3 logic tree and the CDF if one uses only a subset of variables.

The 4th row of Table 5 shows for example that if one allows only the top 4 parameters to vary, the expected value of EAL differs from that of the full tree by 4% (an error of 5% or less is often considered by engineers to be small enough not to matter) and the coefficient of variation differs by 1%. Using the top 4 variables involves only 120 branches as opposed to 28,800. This means that essentially the same results are produced with 0.4% of the computational effort. Figure 6 shows the cumulative distribution function of EAL varying only the top 4 parameters. The difference between the red line (the CDF of EAL with the top 4 branches) and the black line (a lognormal CDF with median 2.37 billion and logarithmic standard deviation 0.20) reflects the 3.9% upward bias in the results using the trimmed tree. This error is small, for example compared with the difference between EAL using the Wald and Allen (2007) model of Vs30 and that of Wills and Clahan (2006).

Table 4. Tornado diagram results for EAL in California woodframe dwellings

Variable Low Baseline High Low EAL High EAL Swing $B $B $B 1. ProbModel Poisson Poisson Low $2.35 $3.45 $1.104 2. GMP BSSA2014 IDR2014 CY2014 $2.14 $2.74 $0.607 3. TotalMag5Rate 6.5 9.6 9.6 $1.74 $2.35 $0.606 4. SpatialSeisPDF U2 U3 U3 $2.11 $2.35 $0.242 5. DeformationModels NEOK GEOL ZENGBB $2.27 $2.37 $0.103 6. ScalingRelationships EllB EllBsqrtLen HB08 $2.31 $2.41 $0.099 7. MaxMagOffFault 7.9 7.3 7.3 $2.29 $2.35 $0.057 8. SlipAlongRuptureModels Tap Uni Uni $2.34 $2.35 $0.013 9. FaultModels FM3_2 FM3_1 FM3_1 $2.34 $2.35 $0.012

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

Prob model Poisson Low

GMP BSSA CY 2014

Total mag 5 rate 6.5 9.6

Spatial seis PDF U2 U3

Deformation models NEOK ZENGBB

Scaling relationships EllB HB08

Max mag off fault 7.9 7.3

Slip along rupture models Tapered Boxcar

Fault models FM3.2 FM3.1

$1.0 $2.0 $3.0 $4.0

Figure 5. Tornado diagram of EAL for California woodframe dwellings

Table 5. Tree-trimming of UCERF3.3 considering EAL in California woodframe dwellings

Variable Branches Cumulative % of total EAL COV 1. ProbModel 4 4 0.01% 21.0% -37% 2. GMP 5 20 0.07% 30.4% 3% 3. TotalMag5Rate 3 60 0.2% 10.2% 2% 4. SpatialSeisPDF 2 120 0.4% 3.9% 1% 5. DeformationModels 4 480 1.7% 6. ScalingRelationships 5 2400 8.3% 7. MaxMagOffFault 3 7200 25% 8. SlipAlongRuptureModels 2 14400 50% 9. FaultModels 2 28800 100%

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

1.00

0.75

0.50

0.25 Cumulative probability Cumulative

0.00 0.0 1.0 2.0 3.0 4.0 5.0 Expected anualized loss, $B

Figure 6. EAL for California woodframe single-family dwellings using the trimmed UCERF3.3 logic tree (red line) versus the full tree (black line)

2.2 LIMITATIONS We have not repeated the calculations with all buildings or with fatalities. Results would likely be similar for a portfolio with all buildings (that was the case with UCERF2) and with fatalities. We do not plan to examine a geographically concentrated portfolio. If we were to do so, the top parameters could be different, especially for portfolios near faults that are the focus of certain branches. We have not examined sensitivity of the loss exceedance curve to UCERF3.3 branches, which is the subject of work planned using 2014-2015 funds.

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

3 CONCLUSIONS Tornado-diagram analysis was used to identify the branches of the UCERF3.3 logic tree that matter most to expected annualized loss (EAL) of a statewide portfolio of woodframe buildings. We found that four parameters matter most to the cumulative distribution function of EAL: probability model, ground motion prediction equation, total magnitude-5 rate, and choice of off- fault spatial seismicity probability density function. One can fix the other 5 parameters at arbitrary baseline values without a large bias in CDF of EAL. Varying only these 4 rather than 9 parameters, the calculation of EAL can be reduced by 99.6% without a substantial difference in the results. We are documenting these results in a manuscript for submission as a technical note to Earthquake Spectra (since the previous work was published in SRL, which might demur from publishing work that is so similar to our 2012 article).

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Trimming the UCERF 3.3 logic tree, 2013-2014 annual report 19 Feb 2015

4 REFERENCES CITED Field, E.H., Arrowsmith, R.J., Biasi, G.P., Bird, P., Dawson, T.E., Felzer, K.R., Jackson, D.D., Johnson, K.M., Jordon, T.H., Madden, C., Michael A.J., Milner, K.R., Page, M.T., Parsons, T., Powers, P.M., Shaw, B.E., Thatcher, W.R., Weldon, R.J., & Zeng, Y., 2014. Uniform California Earthquake Rupture Forecast, version 3 (UCERF3)—the time‐independent model. Bulletin of the Seismological Society of America, 104(3), 1122-1180. Howard, R.A., 1988. Decision analysis: practice and promise. Management Science 34 (6), 679- 695. Lilliefors, H., 1967. On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown, Journal of the American Statistical Association, 62 (318). (June 1967), 399-402 Porter, K.A., 2009a. Cracking an open safe: HAZUS vulnerability functions in terms of structure-independent spectral acceleration. Earthquake Spectra, 25 (2), 361-378 Porter, K.A., 2009b. Cracking an open safe: more HAZUS vulnerability functions in terms of structure-independent spectral acceleration. Earthquake Spectra, 25 (3), 607-618. Porter, K.A., 2010. Cracking an open safe: uncertainty in HAZUS-based seismic vulnerability functions. Earthquake Spectra, 26 (3), 893-900 http://www.sparisk.com/pubs/Porter-2010-Safecrack-COV.pdf Porter, K.A., E.H. Field, and K. Milner, 2012. Trimming the UCERF2 hazard logic tree. Seismological Research Letters, 83 (5), 815-828 Wald, D.J. and T.I. Allen, 2007. Topographic slope as a proxy for seismic site conditions and amplification. Bulletin of the Seismological Society of America, 97, 1379-1395 Wills, C.J. and K.B. Clahan, 2006. Developing a map of geologically defined site-conditions categories for California. Bulletin of the Seismological Society of America, 96 (4A), pp. 1483-1501

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