QPF from Non-Meteorologists’ Point of View ---- S-PROG and Other Algorithms

1. QPF from Non-Meteorologists’ Point of View ---- S-PROG and Other Algorithms Huaqing Cai NCAR/RAL

2. A Meteorologist’s Approach to QPF:Deterministic Dynamical Modeling • Navier-Stokes equations • PDEs to ODEs • Drastic scale truncations; studying one scale independently of another • Performing ad hoc parameterizations • Arbitrarily hypothesizing the homogeneity of subgrid-scale fields • Computationally expensive

3. A Hydrologist’s Approach to QPF:Phenomenological Stochastic Modeling • The stochastic approaches were designed to mimic the rain phenomenology • Largely ad hoc • Lots of empirical parameters tuned for the narrow range of time scales and space scales

4. Multiplicative Cascade Models • Physically based models involving huge ratios of scale and intensity of rain fields • Cascades preserve the fundamental dynamical scaling symmetries • Overcome the limitations of both conventional approaches and bridge the gap between them

5. What is a Multifractal Process ? A time series : When Non-constant function of q ----- Multifractal

6. Rain Fields as Multifractal Processes D(t) versus t q = 2 K(q) versus q

7. Process to Construct a Discrete Multiplicative Cascade R0 , T0 Step 0 R1 , T0/2 R2 , T0/2 Step 1 R1 = R0W(1) R2 = R0W(2) If <W> =1, canonical cascade, multifractal If ∑ W = 1, microcanonical cascade, monofractal

8. Turbulence and Dynamical Scaling of Rain fields • Rain fields scaling on spatial and temporal scales have been studied extensively • The spatiotemporal organization of rain fields has also been explored • Dynamical scaling, i.e., the rate of evolution of rainfall remains invariant under space-time transformation of the form t ~ Lz

9. S-PROG (Spectral Prognosis)An Advection-Based Nowcasting System • Rain fields commonly exhibit both spatial and dynamic scaling properties, i.e., the life time of a feature is dependent on the scale of the feature; large features evolve more slowly than small features • Assuming rain fields are NOT organized as a collection of cells or objects, each with a characteristic scale, but are a continuum or hierarchy of structures over all scales

10. S-PROG • Field advection • Spectral decomposition • Temporal evolution in Lagrangian space • Forecasting

11. S-PROG ---- (a) Field Advection • A single advection vector is obtained for the whole rain field • Only valid for small domain covered by a single radar, more work need to be done if a large domain is involved • The advection vector is crucial for the accurate estimates of autocorrelation coefficients used in the forecasting

12. S-PROG ---- (b) Spectral Decomposition

13. S-PROG ---- (b) Spectral Decomposition • Big assumption: mean and standard deviation of the kth level are assumed to be constant during the forecasting period • Similar to the stationary assumption in other studies ???

14. S-PROG ---- (c) Temporal Evolution in Lagrangian Space AR(p) model : Yule-Walker equation:

15. S-PROG ---- (c) Temporal Evolution in Lagrangian Space: Why AR(2) ? • It is causal • It is parsimonious with two parameters per level that can be estimated in real time • It is able to approximate the long memory and dynamic scaling observed in rainfall • Most importantly, you should not go beyond AR(2) because the autocorrelation calculations become less accurate for longer time lags

16. S-PPROG ---- (d) Forecasting

17. S-PROG Case Study Results ----Spectral Decomposition

18. S-PROG Case Study Results ----Forecasts T = 0 Min T = 10 Min T = 60 Min

19. S-PROG Case Study Results Dynamic Scaling RMS Error

20. Summary for S-PROG 1. Conceptually advanced, slightly better than extrapolation in terms of RMS error 2. Very parsimonious, only take a few seconds to run 3. The forecasts tend to become smooth and approach the mean as forecasting lead time increases

21. Temporal Variations of Rainfall

22. Another Perspective to Look at Rainfall Fields I I’ L (I,J) L (I,J) L L t t+t Time Time

23. Temporal Variation of Mean and Standard Deviation of DlnI Stationary

24. Selected PDFs of DlnI

25. Evidence of Dynamical Scaling

26. PDFs Remain Statistically Invariant Under Transformation t/Lz =constant

27. Implications of Dynamical Scaling • Rainfall variations in space and time are not independent of each other but depend in a way particular to the process at hand. The dependence of the statistical structure of rainfall on space (L) and time (t) can be reduced to a single parameter t/Lz . How to link z to large scale atmospheric conditions is yet to be investigated • Simple relationships which might connect the rate of rainfall pattern evolution at small space and time scales to that at larger scales may exist.

28. Applications of Dynamic Scaling: A Space-Time Downscaling Model t min t+10 min t + 20 min L=32 km Given Predicted L= 2 km t t+5 t+10 t+15 t+20 min Observed