A Framework for Daily Spatio-Temporal Stochastic Weather Simulation

Transcription

1 A Framework for Daily Spatio-Temporal Stochastic Weather Simulation, Rick Katz, Balaji Rajagopalan Geophysical Statistics Project Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder, CO Seattle, WA, August 6, 2012

2 Introduction Precipitation Temperature Conclusion History Framework Agricultural, ecological, hydrological models often require daily weather (e.g. precipitation, minimum/maximum temperature, solar radiation) On grid In the past or future (SWGs) are statistical models whose simulated values look like observed weather (approaches: empirical, model-based) SWGs are not forecast models SWGs can be used to produce infinitely long series of synthetic weather, for observation network infilling, or climate model downscaling

3 Introduction Precipitation Temperature Conclusion History Framework Short History Gabriel and Neumann (1962): First SWG for rainfall occurrence (persistence via Markov chain) p 01 = P(rain today no rain yesterday) p 11 = P(rain today rain yesterday) Todorovic and Woolhiser (1975): Rainfall amounts (skewed distribution via exponential pdf) f (y rain) = a exp( ay) for y > 0

4 Introduction Precipitation Temperature Conclusion History Framework Short History Richardson (1981): Full weather generator (precipitation, minimum temperature, maximum temperature, solar radiation) N(t) X(t) MVNormal(µ(t), Σ(t)) R(t) Current research aimed at spatial and temporal simulations that honor observed spatial correlations

5 Introduction Precipitation Temperature Conclusion History Framework General Framework For a spatially consistent stochastic weather generator, think of generating Local Climate + Weather Local Climate: Generalized linear model (include covariates, model parameters spatially varying, portable between domains) Weather: Spatial Gaussian process (flexible spatial model)

6 Introduction Precipitation Temperature Conclusion History Framework Gaussian Processes We say W(s), indexed by location s R d, is a isotropic Gaussian process if i) For any s 1,..., s n R d, (W(s 1 ),..., W(s n )) is multivariate normal ii) E W(s) = µ for all s R d iii) Cov(W(s + h), W(s)) = C( h ) for all s, h R d Model is complete with µ and C( h ).

7 Introduction Precipitation Temperature Conclusion History Framework Gaussian Process Simulations Exponential correlation: ( C( h ) = exp h ) λ λ = 100

8 Introduction Precipitation Temperature Conclusion History Framework Gaussian Process Simulations Exponential correlation: ( C( h ) = exp h ) λ λ = 10

9 Introduction Precipitation Temperature Conclusion History Framework Gaussian Process Simulations Matérn correlation: ( ) C( h ) = 21 ν h ν ( ) h K ν Γ(ν) λ λ λ = 10 and ν = 2

10 Introduction Precipitation Temperature Conclusion History Framework Gaussian Process Simulations Nonstationary: Cov(W(s 1 ), W(s 2 )) = C(s 1, s 2 ) C( s 1 s 2 )

11 Introduction Precipitation Temperature Conclusion History Framework Kriging Suppose we observe but want W(s 0 ). W = (W(s 1 ),..., W(s n )) The kriging estimator or interpolator for W(s 0 ) is with interpolation variance Ŵ(s 0 ) = c Σ 1 (W µ) C(s 0, s 0 ) c Σ 1 c (these coincide with conditional expectation and conditional variance of a multivariate normal).

12 Introduction Precipitation Temperature Conclusion History Framework Kriging

13 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Daily Precipitation

14 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Precipitation at NCAR

15 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Study Regions Data: Total daily rain at 22 stations from the United States Historical Climatology Network over the years (a) (b) Latitude Latitude Longitude Longitude

16 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Frontal Precipitation

17 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Convective Precipitation

18 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Convective Precipitation

19 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Occurrence Location s, day t, occurrence O: O(s, t) = 0 if W(s, t) < 0 O(s, t) = 1 if W(s, t) 0 where W (weather) is a Gaussian process with mean µ(s, t) = β µx µ (s, t) ( = local climate) and covariance C(h, t) = exp( h /λ(t)).

20 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Occurrence Location s, day t, occurrence O: O(s, t) = 0 if W(s, t) < 0 O(s, t) = 1 if W(s, t) 0 where W (weather) is a Gaussian process with mean µ(s, t) = β µ X µ (s, t) ( = local climate) and covariance C(h, t) = exp( h /λ(t)).

21 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Occurrence Location s, day t, occurrence O: O(s, t) = 0 if W(s, t) < 0 O(s, t) = 1 if W(s, t) 0 where W (weather) is a Gaussian process with mean µ(s, t) = β µ (s) X µ (s, t) ( = local climate) and covariance C(h, t) = exp( h /λ(t)).

22 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Coefficients We suppose β Gaussian process with constant mean and Matérn covariance function.

23 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Coefficients We suppose β Gaussian process with constant mean and Matérn covariance function. Locally, estimate ˆβ(s) by maximum likelihood, and conditional on these, estimate {mean, variance, smoothness, range} by maximum likelihood. ˆβ(s) available on the simulation grid by kriging local estimates.

24 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Occurrence Mean Residual Mean + Residual Occurrence

25 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Amounts Model amount Y(s, t) with gamma cdf G s,t with scale α and shape γ: log α(s, t) = β α (s) X α (s, t) log γ(s, t) = β γ (s) X γ (s, t).

26 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Amounts Model amount Y(s, t) with gamma cdf G s,t with scale α and shape γ: log α(s, t) = β α (s) X α (s, t) log γ(s, t) = β γ (s) X γ (s, t). For spatially correlated fields of precipitation, use Gaussian process W(s, t) Y(s, t) = G 1 s,t (Φ(W(s, t))) where Φ is the standard normal cdf.

27 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Amounts Shape Parameter Gaussian Realization Probability Integral Transform Rain

28 Introduction Precipitation Temperature Conclusion Daily Precipitation Occurrence Amounts Daily Simulation Day 1 Day Day 3 Day

29 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Colorado Data: 145 stations from the Global Historical Climatology Network. Daily minimum and maximum temperature between Longitude Latitude

30 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Stevenson Screen and Rain Gauge

31 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Statistical Model Model min (N) and max (X) temperature as N(s, t) = β N (s) X N (s, t) + W N (s, t) X(s, t) = β X (s) X X (s, t) + W X (s, t) ( = Local Climate + Weather). As with precipitation, β N, β X Gaussian processes.

32 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Min/Max Temperature Complicated relationship between min and max temperature: Elevation Cross Correlation Latitude Latitude Longitude Longitude Evidence of nonstationary cross-correlation.

33 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Bivariate Gaussian Process The matrix-valued covariance function ( CNN (x, y) C C(x, y) = NX (x, y) C XN (x, y) C XX (x, y) with diagonals direct-covariance functions and off diagonal cross-covariance functions, e.g. C NN (x, y) = Cov(N(x), N(y)) C NX (x, y) = Cov(N(x), X(y)). ),

34 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Covariance for Temperature The covariance at sites x, y and day t is Cov(W N (x, t), W N (y, t + 1)) = 0

35 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Covariance for Temperature The covariance at sites x, y and day t is Cov(W N (x, t), W N (y, t + 1)) = 0 Cov(W N (x, t), W N (y, t)) = C NN (x, y, d(t)) + τ N (x, y) 2 I [x=y]

36 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Covariance for Temperature The covariance at sites x, y and day t is Cov(W N (x, t), W N (y, t + 1)) = 0 Cov(W N (x, t), W N (y, t)) = C NN (x, y, d(t)) + τ N (x, y) 2 I [x=y] Cov(W N (x, t), W X (y, t)) = C NX (x, y, d(t))

37 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Nonparametric Estimator For i, j = N, X, estimator of C ij (x, y, d(t 0 )) is P T t=1 P n P n k=1 l=1 K λ t ( d(t 0), d(t) d ) K λ ( x s k ) K λ ( y s l ) W i(s k, t)w j(s l, t) P T P n P n t=1 k=1 l=1 K λ t ( d(t 0), d(t) d )K λ ( x s k )K λ ( y s l ) for kernel functions K and bandwidths λ and λ t.

38 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Nonparametric Estimator For i, j = N, X, estimator of C ij (x, y, d(t 0 )) is P T t=1 P n P n k=1 l=1 K λ t ( d(t 0), d(t) d ) K λ ( x s k ) K λ ( y s l ) W i(s k, t)w j(s l, t) P T P n P n t=1 k=1 l=1 K λ t ( d(t 0), d(t) d )K λ ( x s k )K λ ( y s l ) for kernel functions K and bandwidths λ and λ t.

39 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Nonparametric Estimator For i, j = N, X, estimator of C ij (x, y, d(t 0 )) is P T t=1 P n P n k=1 l=1 K λ t ( d(t 0), d(t) d ) K λ ( x s k ) K λ ( y s l ) W i(s k, t)w j(s l, t) P T P n P n t=1 k=1 l=1 K λ t ( d(t 0), d(t) d )K λ ( x s k )K λ ( y s l ) for kernel functions K and bandwidths λ and λ t.

40 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Nonparametric Estimator For i, j = N, X, estimator of C ij (x, y, d(t 0 )) is P T t=1 P n P n k=1 l=1 K λ t ( d(t 0), d(t) d ) K λ ( x s k ) K λ ( y s l ) W i(s k, t)w j(s l, t) P T P n P n t=1 k=1 l=1 K λ t ( d(t 0), d(t) d )K λ ( x s k )K λ ( y s l ) for kernel functions K and bandwidths λ and λ t.

41 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Nonparametric Estimator What is Ĉ ii (s, s) estimating, C ii (s, s) or C ii (s, s) + τ i (s) 2?

42 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Nonparametric Estimator What is Ĉ ii (s, s) estimating, or C ii (s, s) C ii (s, s) + τ i (s) 2? The estimator Ĉ ii (s, s) is a smoothed version of the empirical matrix W i (s 1 )W i (s 1 ) W i (s 1 )W i (s 2 ) W i (s 2 )W i (s 1 ) W i (s 2 )W i (s 2 )...

43 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Nonparametric Estimator What is Ĉ ii (s, s) estimating, or C ii (s, s) C ii (s, s) + τ i (s) 2? The estimator Ĉ ii (s, s) is a smoothed version of the empirical matrix, which has expectation C ii (s 1, s 1 ) + τ i (s 1 ) 2 C ii (s 1, s 2 ) C ii (s 2, s 1 ) C ii (s 2, s 2 ) + τ i (s 2 ) 2...

44 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Last Theory It can be shown that, with an increasing number of replications of W i (s, t), Ĉ ii (s, s) C ii (s, s). Hence, we can estimate τ i (s) 2 from the following facts: Local Empirical Variance C ii (s, s) + τ i (s) 2 Ĉ ii (s, s) C ii (s, s).

45 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Spatial Correlation Min Temperature Correlation Max Temperature Correlation Cross Correlation Latitude Latitude Latitude Longitude Longitude Longitude Min Temperature Correlation Max Temperature Correlation Cross Correlation Latitude Latitude Latitude Longitude Longitude Longitude

46 Introduction Precipitation Temperature Conclusion Colorado Model Estimation Example Nonstationary vs. Stationary

47 Introduction Precipitation Temperature Conclusion Conclusions Spatial SWG framework: Local Climate + Weather Generalized linear models make SWG portable Gaussian processes are flexible spatial models for climate and weather need nonstationary multivariate models! Kleiber, W., R. W. Katz and B. Rajagopalan, 2012: Daily spatiotemporal precipitation simulation using latent and transformed Gaussian processes. Water Resources Research, 48, doi: /2011wr Kleiber, W., R. W. Katz and B. Rajagopalan, 2012: Daily minimum and maximum temperature simulation over complex terrain. Under review.