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QPF from Non-Meteorologists’ Point of View ---- S-PROG and Other Algorithms. Huaqing Cai NCAR/RAL. A Meteorologist’s Approach to QPF: Deterministic Dynamical Modeling. Navier-Stokes equations PDEs to ODEs Drastic scale truncations; studying one scale independently of another
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QPF from Non-Meteorologists’ Point of View ---- S-PROG and Other Algorithms Huaqing Cai NCAR/RAL
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
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
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
What is a Multifractal Process ? A time series : When Non-constant function of q ----- Multifractal
Rain Fields as Multifractal Processes D(t) versus t q = 2 K(q) versus q
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
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
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
S-PROG • Field advection • Spectral decomposition • Temporal evolution in Lagrangian space • Forecasting
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
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 ???
S-PROG ---- (c) Temporal Evolution in Lagrangian Space AR(p) model : Yule-Walker equation:
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
S-PROG Case Study Results ----Forecasts T = 0 Min T = 10 Min T = 60 Min
S-PROG Case Study Results Dynamic Scaling RMS Error
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
Another Perspective to Look at Rainfall Fields I I’ L (I,J) L (I,J) L L t t+t Time Time
Temporal Variation of Mean and Standard Deviation of DlnI Stationary
PDFs Remain Statistically Invariant Under Transformation t/Lz =constant
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.
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