Providing distributed forecasts of precipitation using a Bayesian nowcast scheme

Neil I. Fox & Chris K. Wikle

University of Missouri - Columbia

Contents

ReasoningMethod / model

Statistical method Dynamics More reasoning

ProductsCase study exampleDevelopment

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

Hierarchical Model

5 stage model Data Process Spatial distributions Parameters

Spatio-Temporal Dynamic Models

)]()][([:s][Parameter

))(),(,),(()(:]Parameters|[Process

))(),(()(:]ParametersProcess, | [Data

11

tt

tYYfY

tYfZ

pd

ptpt

dtdt

Hierarchical Space-Time Framework:

Theorem Bayes' via

Data]|Parameters [Process,

:Obtain

Used in

Ecology: e.g. Model species dispersionData sparse obs: e.g. Scatterometer winds Long-term modeling: e.g. SST prediction

Model formulation

Stage 3: The integro-difference equation (IDE)

);()1;();()( tsdrtryrksy sst

ks(r;θs) is a redistribution kernel that describes how the process Y at time t is redistributed spatially at time t+1.

)()()(

2

1exp

)(2

1),( 1'

2/1 sss

s

ss srsrrk

IDE Kernel ParameterizationFor 2-D space, consider the multivariate Gaussian kernel for location s:

The kernel is centered at s + µ(θs) and thus the µ parameterscontrol the translation and the covariance parameters control thedilation and orientation.

* These parameters can be given spatial distributions at thenext level of the hierarchy!!

Alternative kernel parameterization: ellipse foci, e.g., Higdon et al. 1999

Spatial distributionModel the θs

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)

Model implementation: MCMC

Markov Chain Monte-CarloGibbs sampler

Things this can do

Full spatial variance field Where do we have least confidence in the

forecast Quantitative uncertainty for defined points and

areas (i.e. catchment QPF uncertainty)

More things we can do

Incorporation of physics γ can become a spatially varying growth

parameter Kernel can incorporate windfield information

Products - domain

Nowcast fields Mean nowcast to T+60 (10 minute intervals at present)

Variance fields Uncertainty

Mean nowcast fields

Indication of uncertainty in space

Products - point / catchment

Nowcast reflectivity 10 minute intervals to T+60 With variance

Nowcast Rainfall Point or group of points Mean or median nowcast rainfall or

accumulation out to T+60 Cumulative frequency / probability distributions

Rainrate distribution

Cumulative frequency of nowcast rainrate

0 500

50

100

Fre

quen

cy (

%) T+10

0 500

50

100

Fre

quen

cy (

%) T+20

0 500

50

100

Fre

quen

cy (

%) T+30

0 500

50

100

Fre

quen

cy (

%) T+40

0 500

50

100

Fre

quen

cy (

%) T+50

0 500

50

100

Fre

quen

cy (

%)

Rainrate (mm/hr)

T+60

0 500

50

100T+10

0 500

50

100T+20

0 500

50

100T+30

0 500

50

100T+40

0 500

50

100T+50

0 500

50

100

Rainrate (mm/hr)

T+60

0 500

50

100T+10

0 500

50

100T+20

0 500

50

100T+30

0 500

50

100T+40

0 500

50

100T+50

0 500

50

100

Rainrate (mm/hr)

T+60

0 500

50

100T+10

0 500

50

100T+20

0 500

50

100T+30

0 500

50

100T+40

0 500

50

100T+50

0 500

50

100

Rainrate (mm/hr)

T+60

Pixel 1 Pixel 2 Pixel 3 3 pixel aggreg

Cumulative frequency of nowcast rain accumulations

0 10 20 300

50

100

Fre

quen

cy (

%) T+10

0 10 20 300

50

100

Fre

quen

cy (

%) T+20

0 10 20 300

50

100

Fre

quen

cy (

%) T+30

0 10 20 300

50

100

Fre

quen

cy (

%) T+40

0 10 20 300

50

100

Fre

quen

cy (

%) T+50

0 10 20 300

50

100

Fre

quen

cy (

%)

Rain (mm)

T+60

0 10 20 300

50

100T+10

0 10 20 300

50

100T+20

0 10 20 300

50

100T+30

0 10 20 300

50

100T+40

0 10 20 300

50

100T+50

0 10 20 300

50

100

Rain (mm)

T+60

0 10 20 300

50

100T+10

0 10 20 300

50

100T+20

0 10 20 300

50

100T+30

0 10 20 300

50

100T+40

0 10 20 300

50

100T+50

0 10 20 300

50

100

Rain (mm)

T+60

0 10 20 300

50

100T+10

0 10 20 300

50

100T+20

0 10 20 300

50

100T+30

0 10 20 300

50

100T+40

0 10 20 300

50

100T+50

0 10 20 300

50

100

Rain (mm)

T+60

Pixel 1 Pixel 2 Pixel 3 3 pixel aggreg

In the future

Verification and adjustmentIncorporation of physicsComputational efficiencyHydrology

lumped model probabilities distributed probabilistic input

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