Statistical Analysis of Initial-condition Constraints and Parametric Sensitivity

Transcription

1 Statistical Analysis of Initial-condition Constraints and Parametric Sensitivity Aimé Fournier, Jan CU Denver Adam U Utah SIAM Central States Section 2017 Meeting 2017/10/1

2 Introduction Experimental burns (e.g., FASMEE) require high-dimensional uncertainty quantification: Identifying initial conditions rigorously typical conditioned on suitability for burns Sensitivity analysis of measurements to each parameter Identifying locations where optimal sensor placement could improve parameters settings, thus improving coupled fire-weather simulations 2

3 Introduction... Heat flux (colors) Fire area in contour 3

4 Introduction... 4

5 Introduction... Wind velocity (speed colors) Volume rendered smoke Red line is burner 5

6 Typical initial conditions State space presents as all meteorological, chemical and fire variables at all 3D grid points. Full uncertainty of such a high-dimensional problem (even obtaining mean states) is not feasible. Instead, long histories of measurements at selected weather stations are analyzed, to identify initial conditions that are somehow rigorously close to the long-term mean. 6

7 Typical initial conditions... The data comprise: temperature T in (, ); relative humidity 0 < 1; horizontal wind speed s 0; wind direction 0 d < 360 ; and wind gust g > 0. 7

8 Conditioning on burn suitability the sample mean operation is modified so that xn at time n is acceptable for a burn: x = NC 1 n=1nxnc(xn), where C(x) is 1 or 0 as x meets all the criteria, Ci(x) = (Tli T Tui) & ( li ui ) & (sli s sui), NC = n=1nc(xn) and i indexes all of potential burn location, day in burn window, and hours in window per day. 8

9 Typical initial conditions... Standardize all Ranges to (, ) By transforming (,s,d,g) to: dew-point temperature Td; horizontal wind vector (u,v) = (sin d,cos d)s; gust logarithm ℓ = log10 g. 9

10 Typical initial conditions... x = (T,Td,u,v,ℓ) is internally incommensurate typical defined as small Mahalanobis distance ǁδǁ = (δtδ)1/2, where δ = Σ 1(x x ), and (x x )(x x )t = ΣΣt is a Cholesky (or other) factorization of covariance. Null sample-mean vector δ = 0. Identity covariance-matrix δδt = I. x N( x,σσt) ǁδǁ 5, ǁδǁ 2.13, (ǁδǁ ǁδǁ )2 1/

11 Mahalanobis distance ǁδ = Σ 1(x x )ǁ 11

12 Sensitivity analysis Compute rn coupled fireatmosphere simulations. r Latin hypercube samples select L parameters (e.g., L = 7 rows). N equally probable parameter-value Intervals (e.g., N = 5). N parameter sets sampling each parameter without repetition. Drastic reduction in the number of simulations. 12

13 Sensitivity analysis... Global sensitivity analysis (McKay et al. 1979; McKay 1995; Saltelli et al. 2004; Sobol 2001) isolates the variance of simulated measurements due to each parameter. The total variance V of a model output (e.g., the vertical wind velocity at a time and location) decomposes into a sum of variances V Xi conditioned on each parameter Xi, i = 1,..., L. The sensitivity index (V Xi)/V estimates the relative effect of parameter Xi on the output. 13

14 Sensitivity analysis... 14

15 Optimal sensor placement Maps of variance and sensitivity index of a parameter show locations where suitable measurements would improve that parameter. 15

16 Conclusions Typical-day statistics identified statistically similar days w.r.t. plume rise and dispersion. Statistical analysis of runs with sampled parameter sets provided clear guidance for placing measurements in space and time. Some measurements were more affected by certain parameters what parameters are constrained by observations. Sampling right above the fire (optimal from the fire heat-flux measurement standpoint) may not be optimal for sampling most vigorous parts of the plume. For more information, see 16

17 Acknowledgements Work partially supported by Joint Fire Science Program grant HPC (Yellowstone, Cheyenne supercomputers) was provided by NCAR, sponsored by the National Science Foundation. 17