Parametric uncertainty quantification in the Rothermel model with ...

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International Journal of Wildland Fire 2015, 24, 307–316 http://dx.doi.org/10.1071/WF13097

Parametric uncertainty quantification in the Rothermel model with randomised quasi-Monte Carlo methods ¨ kten A Yaning Liu A,D,E, Edwin Jimenez B, M. Yousuff Hussaini A, Giray O C and Scott Goodrick A

Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA. Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, USA. C USDA Forest Service Center for Forest Disturbance Science, 320 Green Street, Athens, GA 30602, USA. D Present address: Earth Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA. E Corresponding author. Email: [email protected] B

Abstract. Rothermel’s wildland surface fire model is a popular model used in wildland fire management. The original model has a large number of parameters, making uncertainty quantification challenging. In this paper, we use variancebased global sensitivity analysis to reduce the number of model parameters, and apply randomised quasi-Monte Carlo methods to quantify parametric uncertainties for the reduced model. The Monte Carlo estimator used in these calculations is based on a control variate approach applied to the sensitivity derivative enhanced sampling. The chaparral fuel model, selected from Rothermel’s 11 original fuel models, is studied as an example. We obtain numerical results that improve the crude Monte Carlo sampling by factors as high as three orders of magnitude. Additional keywords: chaparral fuel model, fire propagation, global sensitivity analysis, variance reduction. Received 11 June 2013, accepted 24 September 2014, published online 7 April 2015

Introduction An important topic in wildland fire management is fire behaviour prediction. Fire behaviour has been modelled mathematically by a system of algebraic, integral or partial differential equations, which determine the values of output variables (model responses) for a given set of input data. Mathematical models cannot simulate all fire behaviours, but they quantify fire characteristics in a repeatable manner and facilitate comparisons of the distinct characteristics manifest in different scenarios. With the surge of numerical tools and rapid development of computational resources, fire behaviour models are growing both in number and complexity. Rothermel’s surface fire spread model (Rothermel 1972) is a widely used wildland fire model, particularly in North America. It was developed from the principle of conservation of energy, but completed through experimental tests. Values for input parameters of 11 fuel models for the fire-danger rating system are also specified in Rothermel (1972). Albini (1976) makes modifications that correct the formulae for oven-dry fuel loading and reaction intensity, in addition to reformulating certain equations so that they are more suitable for computer-based models. Moreover, two new fuel models are added, making the total number of standard fuel models 13. Wilson (1990) suggests using new empirical formulae coming from the analysis of Journal compilation Ó IAWF 2015

variance technique, based on which Catchpole and Catchpole (1991) further propose modifying the heat sink and damping moisture terms in the original Rothermel model. Various fire models have been integrated in software systems, such as FARSITE (Finney 2004) and BehavePlus (Andrews 2007), which serve a collection of purposes ranging from predicting fire behaviour and control of fire hazard, to developing fire models that are not included in the 13 standard fuel models. The underlying surface fire spread model adopted by these systems is the Rothermel model with modifications made by Albini (1976), owing to its wide applicability and robustness. However, the later proposed modifications, which are yet to be tested, are not incorporated. Simulations making use of a robust model are still subject to errors and uncertainty owing to the variability of the input parameters. In general, the errors and uncertainty can be classified into two categories (Oberkampf and Roy 2010): model (epistemic) uncertainty and parametric (aleatory) uncertainty. In this paper, we are primarily interested in the parametric uncertainty, which includes uncertainty in the input data for initial or boundary conditions that may spread temporally or spatially, affecting the model outputs, as well as error in the model parameters due to either measurement error or intrinsic variability in the parameters. In the Rothermel model, for example, www.publish.csiro.au/journals/ijwf

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fuel state can change in a short time scale, say hourly and daily. These changes are termed abrupt change and diurnal change respectively (Pyne et al. 1996). Changes in wind speed and direction have immediate impact on fire spread rate and reaction intensity, and changes in temperature and humidity directly impact the fuel moisture. The immediate consequence of the existence of parametric uncertainties is that the model outputs are also uncertain. Several methods exist in the literature that deal with the uncertainty quantification of fire models. Fujioka (2002) develops a method that studies two-dimensional fire spread modelling errors and utilises the correction factor to predict the next step. Bachmann and Allgo¨wer (2002) study the uncertainty propagation for the Rothermel model using the Taylor series method. Upadhyay and Ezekoye (2008) implement the quadrature method of moments, which shows its advantage over the Monte Carlo method. However, the method is for only one uncertain variable and it is not easy to extend to multivariate cases. In Jimenez et al. (2008), a sampling method named sensitivity derivative enhanced sampling is coupled with Monte Carlo simulation as well as stratified sampling to quantify uncertainties in the Rothermel model and substantial improvement is achieved. Finney et al. (2011) described a method for ensemble wildland fire simulation that focussed on uncertainty in the weather inputs (FSPro). Typical run times for the ensemble system were roughly 6 h on shared-memory computers using 16 or 32 processors. Ensemble size was found to be an important factor in reducing the variability of estimated burn probabilities as the larger ensemble size was better able to capture the rarest weather combinations that could have major consequences. Larger ensemble sizes require more computing resources, resulting in a trade-off between reducing the variability in the estimated burn probabilities and the length of time required for the simulation. The techniques described in the present paper seek to improve this trade-off by preconditioning the system by gaining knowledge of the parametric sensitivity of the model equations, which allows improvements over standard Monte Carlo methods. We extend the work of Jimenez et al. (2008) in several ways to quantify parametric uncertainties in the Rothermel model. We first use global sensitivity analysis (Sobol0 1993; Saltelli 2008) to identify the important input parameters of the Rothermel model, rank them in order of their importance, and use this information to reduce the dimensionality of the model. We then use randomised ¨ kten 2009) and an optimised quasi-Monte Carlo methods (O version of sensitivity derivative enhanced sampling used by Cao et al. (2003, 2004, 2006) to estimate the output variables. From the numerical results, we observe improvements by factors as high as three orders of magnitude over standard Monte Carlo simulation. Rothermel’s wildland fire model The main output variables of the Rothermel model are the rate of fire spread (ros in m s1), the direction of maximum spread (sdr in degrees), the effective wind speed (efw in m s1), and reaction intensity (ri in kW m2). The equation for ri is given by: ri ¼ G0  wn  heat  ZM  ZS

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where G0 is the optimum reaction velocity, wn is the net fuel loading, heat is the fuel low heat content, ZM is the moisture damping coefficient, and ZS is the mineral damping coefficient. The rate of fire spread ros is computed from the following equations: ros ¼

ri  x  ð1 þ Fc Þ rb  e  Qig

where x is the propagating flux ratio, rb is the ovendry bulk density, e is the effective heating number, Qig is the heat of preignition and Fc is formulated as: Fc ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½Fs þ Fw cosðyÞ2 þ ½Fw sinðyÞ2

Here y is the split angle between upslope direction and direction the wind is blowing to, Fs and Fw are slope and wind factors. The direction of maximum spread and effective wind speed are given by:  sdr ¼ arcsin

Fw sinðyÞ Fc



" #1=BðsÞ 1 Fc efw ¼   196:85 C ðsÞ b=b EðsÞ opt where s is the surface-area-to-volume ratio, B, C, E are some functions of s, and b is the packing ratio. The model inputs can generally be specified as 12 parameters, which are the oven-dry fuel loading wo, fuel depth d, surface-area-to-volume ratio s, fuel heat content heat, fuel particle density rp, fuel moisture content Mf, fuel total mineral content st, fuel effective mineral content se, wind speed at midflame height wsp, slope (vertical rise/horizontal run) slp, wind direction y, and fuel moisture of extinction mx. We can further classify wo as w0d1, w0d2, w0d3, w0lh, w0lw, s as svd1 , svd2 , svd3 , svlh , svlw and Mf as md1 , md2 , md3 , mlh , mlw . Here the subscripts d1, d2, d3, lh, lw denote the size classes used to categorise the different fuel moisture time-lag classes: dead fuel, 0–0.6 cm; dead fuel, 0.6–2.5 cm; dead fuel, 2.5– 7.5 cm; live herbaceous fuel; and live woody fuel. Therefore the total number of input parameters reaches 24. We note that ros and ri depend on all the 24 parameters, but sdr and efw on only 15 of them. As the values of the parameters cannot be accurately measured, parametric uncertainties exist in all the input parameters. We are interested in studying the uncertainties in the model outputs caused by the propagation of those in the input parameters. Our methodology is introduced in the next section. Numerical method Global sensitivity analysis Sensitivity analysis (SA) examines how uncertainties in the input variables of a model impact the output. An input parameter that gives rise to large uncertainty in the output is identified as important. By freezing the parameters that are not

Uncertainty quantification for Rothermel model

important at their nominal values, it is possible to reduce the dimension of the original model. In this way, uncertainty analysis can be carried out for the dimension-reduced model with greater efficiency. We next describe the variance-based Sobol0 sensitivity measures, which are a widely used global sensitivity analysis in applications (Sobol0 1993, 2001; Saltelli 2002; Liu and Owen 2006). It is worth noting that recently, Sobol0 and Kucherenko (2009) developed a new sensitivity measure, derivative-based global sensitivity measure (DGSM). For linear or quasi-linear models, DGSM agrees well with Sobol0 ’s variance-based measure and the computation can be cheaper. However, for non-linear models, DGSM can produce very different results and lead to false conclusions. Owing to the high non-linearity of the Rothermel model, we stick to the variance-based Sobol0 sensitivity measure. Consider a function f ðxÞ, defined on the k-dimensional unit hypercube ½0; 1k . Let u  f1; . . . ; kg be an index set and xu denote the juj-dimensional vector with elements xj for j 2 u. The ANOVA decomposition of f ðxÞ is: X fu ðxu Þ; f ðxÞ ¼ uf1;...;kg

where fu ðxu Þ is a functionRthat only depends on the variables in the index set u, and f; ¼ ½0;1k f ðxÞdx is a constant. The appealing feature of the ANOVA decomposition is that it can be used to express the total variance of the function f ðxÞ as a sum of variances of functions fu ðxu Þ that depend on smaller subsets of variables. Indeed, if we define variances as: Z Z fu ðxu Þ2 dxu ; s2 ¼ f ðxÞ2 dx  f;2 s2u ¼ ½0;1juj

½0;1k

then we have: s2 ¼

X

s2u

uf1;...;k g

The construction of functions fu and the proofs of these decompositions can be found in Sobol0 (1993). Sobol0 (1993) introduced two types of global sensitivity indices (GSI): Su ¼

1 X 2  1 X 2 sv ; Su ¼ 2 s 2 s vu s v\ u6¼; v

for u  f1; . . . ; kg. S u is a normalised sum of all variances whose index sets are subsets of u, and Su is a normalised sum of all variances whose index sets have non-empty intersections with u. Note that S u  Su , and hence they can be used as lower and upper bounds of the sensitivity measures on the parameters xu . Sobol0 (1993) first proposed Monte Carlo algorithms to compute GSI and Saltelli (2002) further improved the efficiency of the algorithms. Algorithms to improve the accuracy of estimating small sensitivity indices are discussed in Sobol0 and Myshetskaya (2007). For a k-dimensional model, the number of GSI can be as many as 2k , which can make the computation of all GSI very expensive. For this reason, in the literature

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typically the indices with respect to singletons, S fig and Sfig for i 2 f1; . . . ; kg, are computed. If Sfig is relatively small, then the corresponding parameter can be frozen at its nominal value. Monte Carlo methods The Monte Carlo (MC) method is a popular numerical tool used in many applications. In performing uncertainty quantification, one often needs to compute the expectation of a random variable f ðX Þ: Z f ðxÞpX ðxÞdx; E½ f ðX Þ ¼ Rk

where X 2 Rk is a random vector of k input parameters and pX ðxÞ is its probability density function (PDF). The MC method estimates this integral by sums of the form: ¼ YMC f

N 1 X f ðxi Þ  E½f ðX Þ; N i¼1

where xi , i ¼ 1; . . . ; N , are independent and identically distributed (i.i.d.) samples drawn from the joint PDF pX ðxÞ using a pseudorandom number generator. The convergence of the estimator to the true value E½f  is guaranteed by the Strong Law of Large Numbers and error can be estimated by the Central Limit Theorem (CLT) (see Billingsley 1995 for details). depends on the The error of the (crude) MC estimator YMC f sample size, N, and the variance of f ðX Þ. Consequently, for a fixed sample size, the error can be improved if the variance of f ðX Þ can be made smaller. Techniques that make this possible are known as the variance reduction techniques (see Rubinstein 2007). Next, we will discuss a variance reduction technique that has proved to be very useful in applications to Rothermel’s model. Optimised sensitivity derivative enhanced sampling Sensitivity derivative enhanced sampling (SDES), developed by Cao et al. (2003, 2004, 2006), has been used successfully in a variety of problems such as Burgers’ equation, optimal control problems, and aircraft wing structure design. Jimenez et al. (2008) implemented SDES to quantify uncertainty in Rothermel’s model and showed that an order of magnitude reduction can be obtained in the estimate of first moments. For simplicity, we explain SDES for models with a single uncertain parameter. The method can be easily extended to cases where multiple uncertain parameters exist. The scalar model with a single source of parametric uncertainty is described by Y ¼ f ðX Þ, where f : R ! R is a smooth function. Consider J ðnÞ ðxÞ, the n-th order Taylor expansion of f ðxÞ about the mean of X , denoted by E½X : 0

J ðnÞ ð xÞ ¼ f ðE½ X Þ þ f ðE½ X Þðx  E½ X Þ 1 þ f ð2Þ ðE½ X Þðx  E½ X Þ2 2 1 þ    þ f ðnÞ ðE½ X Þðx  E½ X Þn n!

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The n-th order SDES for E½ f  is defined as: N   1X f ðxi Þ  J ðnÞ ðxi Þ N i¼1 N    1X f ðxi Þ  J ðnÞ ðxi Þ  E½J ðnÞ ðX Þ ¼ N i¼1

YSDES ¼ E½J ðnÞ ð X Þ þ f ;n

ð1Þ

Jimenez et al. (2013) give sufficient conditions that show when the variance of SDES is smaller than the variance of the crude MC method; however, these conditions are not practical to use. In this paper, we will view the SDES method in the context of a well-known variance reduction technique, the control variates method, and slightly modify Eqn 1 as: ¼ YOSDES f ;n

N    1X f ðxi Þ  b J ðnÞ ðxi Þ  E½J ðnÞ ðX Þ : ð2Þ N i¼1 

Here the constant b is estimated from the simulation data. N  P

b ¼ i¼1

  J ðnÞ ðxi Þ  J f ðxi Þ  f N P

ðJ ðnÞ ðx

 2 iÞ  J Þ

;

i¼1

P P where J ¼ N1 Ni¼1 J ðnÞ ðxi Þ, and f ¼ N1 Ni¼1 f ðxi Þ. We name this method as the n-th order optimised SDES (OSDES). From the theory of control variates, we know that the variance of OSDES is always less than or equal to the variance of the crude MC method, and the way b is chosen is optimal, ignoring the error in its estimation. The SDES method, however, takes b to be 1, and this may actually increase its error compared with crude MC. A more rigorous investigation of OSDES can be found in Liu et al. (2013a). Quasi-Monte Carlo and randomised quasi-Monte Carlo methods Quasi-Monte Carlo (QMC) methods are often described as the deterministic version of MC. The main difference between these methods is the way they simulate R the underlying model. Both methods estimate the integral f ðxÞdx using sums of the form PN 1 i¼1 f ðxi Þ. In MC, the numbers xi come from a pseudorandom N sequence, whereas in QMC, they come from a ‘low-discrepancy’ sequence. Intuitively, whereas a pseudorandom sequence tries to mimic the behaviour of a true random sample of numbers in some domain, the low-discrepancy sequence does not attempt to look random, but tries to divide its domain as evenly as possible. MC emphasises ‘randomness’, whereas QMC emphasises ‘evenness’. The popularity of QMC follows from the fact that it often provides significantly lower error than   MC. The QMC error has the convergence rate O

ðlog N Þk N

, where k is the

dimension of the integrand f . The MC convergence rate, however, is OðN 0:5 Þ, which is asymptotically larger than the QMC rate. Some commonly used low-discrepancy sequences are Halton, Sobol0 , Faure and Niederreiter sequences. We refer the

reader to Niederreiter (1992) for a comprehensive survey of MC and QMC methods. One drawback of QMC lies in the difficulty of estimating the actual error of a particular estimate obtained by a low-discrepancy sequence. As a remedy, the randomised quasi-Monte Carlo (RQMC) methods are often used when error estimation is desired. Essentially, an RQMC method enables generating independent ‘copies’ of a low-discrepancy sequence. Each copy gives an estimate, and independent estimates can be used to assess the statistical accuracy, by, for example, computing the sample standard deviations and constructing confidence intervals. In this paper, we will use the RQMC method known as ‘random-start Halton sequences’. For details on this method as ¨ kten and Eastman well as a survey of RQMC methods, see O ¨ (2004) and Okten (2009). Results and discussion In this section, we present the main numerical results of uncertainty quantification for the Rothermel model. Along the lines of our discussion in the previous section, the Rothermel model can be mathematically represented as Y ¼ f ðX Þ, where Y ¼ fros; sdr; efw; rig are the four outputs we are interested in, and f ðX Þ ¼ ffros ðX Þ; fsdr ðX Þ; fefw ðX Þ; fri ðX Þg are the algebraic equations of the model that map X , the set of input parameters, to the model responses Y . For ros and ri, X is a 24-dimensional vector of parameters composed of all of those listed in Table 1, and for sdr and efw, X has 15 elements out of the 24 including d, rp , slp, svd1 , svd2 , svd3 , svlh , svlw , y, w0d1 , w0d2 , w0d3 , w0lh , w0lw , and wsp. We consider the chaparral fuel model originally found in Rothermel (1972). The parameter values are also summarised

Table 1. Chaparral fuel model parameters Parameter

Symbol

Fuel bed depth Low heat content 1-h fuel moisture 10-h fuel moisture 100-h fuel moisture Live herbaceous fuel moisture Live woody fuel moisture Moisture of extinction Particle density Effective mineral content Slope Total mineral content 1-h surface area/volume ratio 10-h surface area/volume ratio 100-h surface area/volume ratio Live herb surface area/volume ratio Live woody surface area/volume ratio Direction of wind vector (from upslope) 1-h fuel load 10-h fuel load 100-h fuel load Live herbaceous fuel load Live woody fuel load Midflame wind speed

d heat md1 md2 md3 mlh mlw mx rp se slp st svd1 svd2 svd3 svlh svlw y w0d1 w0d2 w0d3 w0lh w0lw wsp

Value

Units

1.83 18 622.0 8.0 8.0 8.0 150.0 150.0 20 512.5 1.0 14.04 5.55 6562.0 358.0 98.0 4921.0 4921.0 45 1.12 0.90 0.45 0 1.12 2.3

m kJ kg1 % % % % % % kg m3 % 8 % m2 m3 m2 m3 m2 m3 m2 m3 m2 m3 8 kg m2 kg m2 kg m2 kg m2 kg m2 m s1

Uncertainty quantification for Rothermel model

in Table 1. The application of our methodology is independent of the choice of the fuel model. The global SA of some other standard fuel models is studied by Liu et al. (2013b). We assume uncertainties exist in all input parameters. Despite the fact that FARSITE keeps some parameters, such as heat and st , fixed for all the 13 standard fuel types, uncertainties in these parameters can still result from errors in experimental measurements and other factors. For the chaparral fuel model, we assign to each parameter a uniform distribution with the mean listed in Table 1 and the standard deviation 5% of the mean. The 5% coefficient of variation (defined as the ratio of the standard deviation to the mean) is adopted in order for both dead and living fuel damping moistures not to exceed their extinction moistures. This allows us to study the propagation of fire on both categories of fuel, which is more important for fire stations and departments. Although the real world variance of each input parameter may be very different from what is assumed here, the goal of a sensitivity analysis is to better understand how uncertainty in model inputs propagates through the model, thus determining which input parameters are most important to improving model performance. Global SA is performed for each output variable before uncertainty quantification, for the purpose of reducing the model complexity by identifying the insignificant parameters that can be set to their mean values. Each sensitivity index is computed with 104 samples. From our numerical calculations, we observed that Sobol0 indices Sfig and S fig converge fairly fast. We compared the index values simulated with 104 samples and those with 106, and obtained agreements to two decimal places. The fast computation of Sobol0 indices further validates our employment of global SA. Fig. 1a shows that seven parameters among the 24, which are wsp, d, svd1 , heat, mlw , w0d1 and w0lw , can be tagged as significant. In fact, as S fig and Sfig offer the lower and upper bounds of the sensitivity measures, their sums over the seven parameters constitute the range of their importance measure without incurring additional computation (results for S fig are not listed). Similarly, it can be shown that the seven significant parameters for ros altogether contribute 95.0–96.9% of the total output variance. Fig. 1b and 1c demonstrate that the topographyand wind-related parameters, as predicted, are dominant factors that influence the outputs sdr and efw. All the three variables y, slp and wsp, are labelled significant for the output sdr, with y alone accounting for 95.8% and all three 99.7% of the total variance. As for efw, the parameter wsp explains 98.7% of the output variation on its own and 99.8% combined with slp. As far as ri is concerned (Fig. 1d), 10 variables can be identified as significant. In descending order of importance, these variables are heat, w0d1 , w0lw , svd1 , mx , se , d, rp , mlw and md1 . More than 98.3% of the total variance of ri is attributed to this group of parameters. To support the results of global SA, Fig. 2 qualitatively contrasts for each output the histogram of the full model with the dimension-reduced model. All parameters follow the uniform distributions described earlier. However, the reduced models only incorporate the significant parameters. The other parameters are fixed at their means. As a result, the dimensions of the models for ros and ri drop from 24 to 7 and 10 respectively, and sdr and efw from 15 to 3 and 2. Fig. 2 shows that the histograms for the

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reduced models are visually indistinguishable from those for the full models. Propagating uncertainties simply in the significant parameters improves computational efficiency to a great extent and at the same time reflects accurately the output uncertainties. Uncertainty quantifications are done just for the reduced models. We consider estimating the first and second moments of each output using a crude MC estimator, MC coupled with firstorder OSDES (MCþOSDES), RQMC estimator RQMC, and RQMC coupled with first-order OSDES (RQMCþOSDES). It is observed that b  1 in all cases, so that the OSDES only has a slight advantage over the original SDES. To take advantage of the first-order OSDES, we need the following generalised version of Eqn 2 with n ¼ 1: ¼ b f ðmX Þ þ YOSDES f ;1

N   1X f ðxi Þ  b J ð1Þ ðxi Þ N i¼1

where mX ¼ fE½X 1 ; . . . ; E½X k g is the collection of means of the k uncertain parameters fXi gki¼1 ; and J ð1Þ ðxÞ is the first order multivariate Taylor expansion J ð1Þ ðxÞ ¼ f ðmX Þ þ rf ðmX Þðx  mX Þ: Bachmann (2001) presents a list of analytical formulae for the first-order partial derivatives of the model output variables as well as some intermediate variables with respect to the input parameters. The partial derivatives in our numerical simulations are computed with the AUTO_DERIV software written in Fortran 95 by Stamatiadis and Farantos (2010), which is an enhanced version based on the Fortran 90 module by Stamatiadis et al. (2000). Our results show that the additional computational expense to compute the partial derivatives is negligible, especially considering the efficiencies gained. Fig. 3 plots the errors in the estimates of the first moment of the model outputs. Because the exact values are not known analytically, we approximate them numerically using large MC samples. For ros, sdr and ri, the numerically exact values are computed as the average of 320 repetitions of independent crude MC estimates using 1.6 105 samples each. The sample sizes in Fig. 3 are chosen as 25 ; 26 ; . . . ; 211 . Picking 25 as the smallest sample size is consistent with Godines and Manteufel (2002), who empirically find that the mean estimator is unbiased and normally distributed for sample sizes larger than 30. The ‘exact’ moments of efw are approximated using Gauss–Legendre quadrature. Table 2 lists all the numerically exact values for the first and second moments. For each sample size, 100 independent simulations of each estimator are realised and the root mean square error (RMSE) is computed as:

RMSENY

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 100  u 1 X 2 ¼t YNi  ^I 100 i¼1

where YNi denotes the ith realisation of the estimator with N samples and ^I is the numerically exact value. In all the plots of Fig. 3, the errors of MC (including MC and MCþOSDES) and RQMC (RQMC and RQMCþOSDES)

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Y. Liu et al.

(d )

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

wsp

w0lh

w0lw

w0d3

d ρp

d Heat md1 md 2 md 3 mlh mlw mx ρp se slp st svd 1 svd 2 svd 3 svlh svlw θ w0d1 w0d2 w0d3 w0lh w0lw wsp

0.6 0.5 0.4 0.3 0.2 0.1

d Heat md1 md 2 md 3 mlh mlw mx ρp se slp st svd 1 svd 2 svd 3 svlh svlw θ w0d1 w0d2 w0d3 w0lh w0lw wsp

wsp

w0lh

w0lw

w0d3

w0d2

θ w0d1

svlh

svlw

svd 3

svd 2

0 slp svd 1

(c)

d ρp

S{i }

0

w0d2

0.05

θ w0d1

0.10

svlh

0.15

svlw

0.20

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 svd 3

(b) 0.25

svd 2

(a)

slp svd 1

312

Parameters Fig. 1. Sobol0 indices Sfig for the Rothermel model with chaparral fuel type. Considered outputs are ros, sdr, efw and ri. The full models for ros and ri contain 24 input parameters, and sdr and efw involve 15 parameters. The indices are obtained by Monte Carlo method with 104 samples.

converge to zero with rates of approximately OðN 0:5 Þ and OðN 1 Þ respectively. Results of least-square fitting reveal that the convergence rates of MC methods are in the range of ½OðN 0:49 Þ; OðN 0:52 Þ, whereas the range is ½OðN 0:98 Þ; OðN 1:02 Þ for RQMC methods. Fig. 3a and 3d exhibit that OSDES improves both MC and RQMC by an order of magnitude. It is equivalent to saying that 100 and 10 times more samples are necessary for estimates of MC and RQMC respectively to be of the same accuracy as MCþOSDES and RQMCþOSDES. More drastic acceleration gains can be observed from Fig. 3b and 3c, where two orders of magnitude additional accuracy is produced by OSDES. Thus, coupled with OSDES, MC and RQMC can achieve the same accuracy as crude MC and RQMC by using 104 and 102 times fewer samples. It is also noticed that the errors of RQMC are larger than or comparable with MCþOSDES for small sample sizes. However, RQMC eventually outperforms MCþOSDES at some point as sample sizes get larger. When coupled with SDES, RQMC provides higher accuracy than any other estimator even at small sample sizes. Table 3 lists the improvement ratios that reflect the levels of improvement over crude MC by the other sampling techniques. The improvement ratio (IR) of a sampling method Y to crude MC with N samples is defined as: IRNY ¼

RMSENMC : RMSENY

The improvement ratios for second moments (shown in Table 4) are smaller compared with first moments, partly

because the non-linearity is increased for estimating second moments. Using RQMCþOSDES with 211 samples, the convergence gain, which reaches 7004, is largest when the first moment of efw is estimated. Alternatively, three more digits compared with the crude MC can be achieved that are in agreement with the numerically exact value. The smallest gain happens for estimating the second moment of ri, which is still as high as 40. Fig. 4 shows the 95% level of confidence intervals (CI) that demonstrate the reliability of the estimators considered. CIs can be constructed based on the CLT Theorem. The 95% CI for a function f estimated with sample size N is given by: 2f 2f s s f þ 1:96 pffiffiffiffi f  1:96 pffiffiffiffi ; m m N N

! ð3Þ

f and s 2f are the sample mean and variance of f. The where m meaning of a CI with confidence level 95% is that the proportion of the independent CIs that are formed by the same method (Eqn 3) containing the true value mf is ,95%. Standard QMC is not able to provide CI and therefore it is difficult to assess the errors intrinsic in the estimates. RQMC makes it possible by facilitating the construction of CIs with multiple estimates obtained from randomised copies of a low-discrepancy sequence. In all cases, MC has the largest CIs in length, indicating its inaccuracy in the estimates of first and second moments. MC paired with OSDES or standard RQMC gives narrower CIs by improving the estimation accuracy. The CIs of

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Fig. 2. Comparisons of histograms for the full and dimension-reduced models. The full models consider variations of all the parameters, whereas the dimension-reduced models freeze the insignificant parameters that are detected by global sensitivity analysis (SA) and uncertainties are only assigned to the significant ones. The boxes are histograms and solid lines are probability density functions (PDFs). Each histogram is constructed using 3.2 104 samples.

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N Fig. 3. Root mean square errors (RMSEs) in first-moment estimates for the outputs in the reduced models. (a) ros. Uncertain parameters are wsp, d, svd1 , heat, mlw , w0d1 and w0lw . (b) sdr. Uncertain parameters are y, slp and wsp. (c) efw. Uncertain parameters are wsp and slp. (d) ri. Uncertain parameters are heat, w0d1 , w0lw , svd1 , mx, se, d, rp , mlw and md1 .

Table 2. Numerically exact first and second moments of the model outputs Results for efw are computed using a 2-D Gauss–Legendre quadrature. All the others are simulated by crude Monte Carlo using 320 sets of independent samples, each with sample size of 1.6 105 Output variables ros sdr efw ri

First moment

Second moment

1

0.111 m2 s22

1:71 103 2 2 5.84 m s 4.38 106 kW2 m4

0.331 m s 41.38 2.41 m s1 2.09 103 kW m2

Table 3. Improvement ratios for first-moment estimates MC, Monte Carlo; OSDES, optimised sensitivity derivative enhanced sampling; RQMC, randomised quasi-Monte Carlo. Simulations are based on 100 sets of samples, each of sample size 211 Output variables ros sdr efw ri

MCþOSDES

RQMC

RQMCþOSDES

11 59 250 10

10 24 21 10

114 1004 7404 42

Table 4. Improvement ratios for second-moment estimates Simulations are based on 100 sets of samples, each of sample size 211 Output variables ros sdr efw ri

MCþOSDES

RQMC

RQMCþOSDES

7 28 41 9

17 24 21 10

105 692 1296 40

RQMCþOSDES are the smallest in length, manifesting that RQMC coupled with OSDES always gives rise to the most accurate estimates. Table 5 summarises the computational times for estimating E[ros] that are averaged over 100 sets of samples, each of sample size 211. We measured the computational costs with a single-core on a machine whose CPU has 2.26-GHz clock speed and 4 GB memory. The programming language was Fortran 95. Recall that the improvement ratio of RQMCþOSDES over MC is approximately 105. Table 5 simply shows that the additional computational time incurred is only 9.5%. The additional expense is due to the generation of RSHS as well as the computation of sensitivity derivatives with AUTO_DERIV. The additional computational cost can be regarded as marginal compared with the huge convergence gain.

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Fig. 4. Confidence intervals for first moments constructed with 100 sets of samples, each of sample size 211. Confidence intervals that are too small to read are magnified as shown by the smaller figures. The horizontal lines represent numerically exact values.

Table 5. Average computational times (in seconds) for estimating E[ros] using MC, MC1OSDES, RQMC, and RQMC1OSDES Sample size 11

2

MC

MCþOSDES

RQMC

RQMCþOSDES

1.2969

1.3426

1.4043

1.4200

Conclusions Effective wildland fire management requires fast prediction of potential or ongoing fire. Mathematical models built for predicting fire behaviour are based on several input fire environment parameters, which are inevitably subject to uncertainties. Predictions based on a single model simulation can thus be misleading and potentially harmful. However, multiple model simulations that incorporate the uncertainty in the model parameters enable the use of descriptive statistics to predict the behaviour of fire. Confidence intervals can be built for output variables, helping fire managers make better predictions and thus better management decisions. However, for complicated models with many parameters, simulation can be computationally expensive. In this paper, we proposed using global sensitivity analysis to reduce model complexity, and use OSDES, a control variate MC approach, together with random-start Halton sequences, an RQMC

method, to simulate the reduced model. Our proposed method improves standard MC simulation error by factors as high as three orders of magnitude when applied to the parametric uncertainty quantification of the Rothermel model at a computational overhead of less than 10%. This makes our proposed method significantly more efficient than the crude MC sampling. When applied to the ensemble wildland fire simulation of Finney et al. (2011), these methods could be used to yield either reduced variability in estimated burn probability by using larger ensemble sizes for similar computational time or produce similar results with significantly lower computational time. It remains an important problem to study the parametric uncertainties associated with spatiotemporal evolutionary fire growth models, as well as the interactive impacts of fire propagation and uncertainty quantifications. Such problems will be addressed in our future work. References Albini FA (1976) Estimating wildfire behavior and effects. USDA Forest Service, Intermountain Forest and Experiment Station, General Technical Report INT-30. (Ogden, UT) Andrews PL (2007) BehavePlus fire modeling system: past, present and future. In ‘Proceedings of 7th Symposium on Fire and Forest Meteorology’. 23–25 October 2007, Bar Harbor, ME. (American Meteorological Society: Boston, MA)

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