1. Providing distributed forecasts of precipitation using a Bayesian nowcast scheme Neil I. Fox & Chris K. Wikle University of Missouri - Columbia

2. Contents §Reasoning §Method / model l Statistical method l Dynamics l More reasoning §Products §Case study example §Development

3. Reasoning §Need realistic representation of uncertainty in precipitation forecasts §Previous methods too deterministic (no measure of uncertainty) or too probabilistic (stochastic) §This methodology allows for the integration of some real physics with a realistic statistical formulation that can provide genuine information on forecast uncertainty

4. Hierarchical Model §5 stage model l Data l Process l Spatial distributions l Parameters

5. Spatio-Temporal Dynamic Models Hierarchical Space-Time Framework:

6. Used in §Ecology: e.g. Model species dispersion §Data sparse obs: e.g. Scatterometer winds §Long-term modeling: e.g. SST prediction

7. Model formulation §Stage 3: The integro-difference equation (IDE) k s (r;θ s ) is a redistribution kernel that describes how the process Y at time t is redistributed spatially at time t+1.

8. IDE Kernel Parameterization For 2-D space, consider the multivariate Gaussian kernel for location s: The kernel is centered at s + µ(θ s ) and thus the µ parameters control the translation and the covariance parameters control the dilation and orientation. * These parameters can be given spatial distributions at the next level of the hierarchy!! Alternative kernel parameterization: ellipse foci, e.g., Higdon et al. 1999

9. Spatial distribution §Model the θ s parameters with a spatial distribution at the next level of the hierarchy §Gaussian random field Diffusive wave fronts; shape and speed of diffusion depend on kernel width and tail behavior (dilation); (e.g., Kot et al. 1996) Non-diffusive propagation via relative displacement of kernel (translation); e.g., Wikle (2001; 2002)

10. Model implementation: MCMC §Markov Chain Monte-Carlo §Gibbs sampler

11. Things this can do §Full spatial variance field l Where do we have least confidence in the forecast l Quantitative uncertainty for defined points and areas (i.e. catchment QPF uncertainty)

12. More things we can do §Incorporation of physics l γ can become a spatially varying growth parameter l Kernel can incorporate windfield information

13. Products - domain §Nowcast fields l Mean nowcast l to T+60 (10 minute intervals at present) §Variance fields l Uncertainty

14. Mean nowcast fields

15. Indication of uncertainty in space

16. Products - point / catchment §Nowcast reflectivity l 10 minute intervals to T+60 l With variance §Nowcast Rainfall l Point or group of points l Mean or median nowcast rainfall or accumulation out to T+60 l Cumulative frequency / probability distributions

18. Rainrate distribution

19. Cumulative frequency of nowcast rainrate Pixel 1Pixel 2Pixel 3 3 pixel aggreg

20. Cumulative frequency of nowcast rain accumulations Pixel 1Pixel 2Pixel 3 3 pixel aggreg

21. In the future §Verification and adjustment §Incorporation of physics §Computational efficiency §Hydrology l lumped model probabilities l distributed probabilistic input

22. References §Wikle, C.K., Berliner, L.M., and Cressie, N. (1998). Hierarchical Bayesian space-time models. Environmental and Ecological Statistics, 5, 117-154. §Wikle, C.K., Milliff, R.F., Nychka, D., and L.M. Berliner, 2001: Spatiotemporal hierarchical Bayesian modeling: Tropical ocean surface winds, Journal of the American Statistical Association, 96, 382-397. §Berliner, L.M., Wikle, C.K., and Cressie, N., 2000: Long-lead prediction of Pacific SSTs via Bayesian dynamic modeling. Journal of Climate, 13, 3953- 3968. §Xu, B., Wikle, C.K., and N.I. Fox, 2003: A kernel-based spatio-temporal dynamical model for nowcasting radar precipitation. Journal of the American Statistical Association. In review. Available at: http://solberg.snr.missouri.edu/People/fox/research/xuetal2003.pdf