A Cloud Motion Winds Diffusion Scheme for Quantitative Rainfall Estimation

1. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 A Cloud Motion Winds Diffusion Schemefor Quantitative Rainfall Estimation Francisco J. Tapiador Department of Geography University of Lleida, Spain f.tapiador@geosoc.udl.es with contributions from M. Castro, M.A. Gaertner, C. Gallardo, A. Roselló (University of Castilla-La Mancha, UCLM, Toledo, Spain); M.A. Martínez (Instituto Nacional de Meteorología, INM, Spain) and C. Gonzalo (Polytechnic University of Madrid, UPM, Spain)

2. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work • Improving Precipitation Estimates Resolutions • The GPM horizon is a 3-hour coverage, but this period can be improved using combined approaches. • More frequent and better resolution estimates are required for applications such as nowcasting and hydrological flood forecasting models. • General Circulation Models (GCM) may benefit of assimilating timely rain rates (e.g. EuroTRMM project).

3. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work • Combined Precipitation Approaches • IR + ( PMW | MW ) • Goals: • Increase the PMW spatial and temporal resolutions • Maintain the PMW ability to sense rainfall • Approaches • Neural networks (Hsu et al. 1997) • Histogram matching techniques (Turk et al. 2000) • Motion vectors approaches (Joyce et al. 2004)

4. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work • Motion Vector Approaches • Basic idea: • If we can accurately estimate rainfall at a given time, this good estimate can be “propagated” forward (and backwards). • Underlying assumption is that error in propagating precipitation is lower than the error in using IR to directly estimate precipitation. • PMW gives the good estimate while the IR provides a means to calculate the movement (and maybe the rain evolution)

5. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work Motion Vector Approaches [RAIN ESTIMATE] + [MOVEMENT] + [RAIN EVOLUTION] • MW • PMW • Blended PMW+IR • Radar • Raingauge • Cloud Motion Winds • Measured Winds (eg TOVS) • Modeled Winds • IR-based • Model-based • Forward-Backward approach

6. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work • A CMW Model • CMW approach is a sensible choice for improving estimates. It uses a IR+PMW combination of strengths. • Motion-Wind schemes can be improved if the motion vectors could be calculated with higher precision. • Current approaches use correlation-based methods (statistical approaches). More physically-based methods should also be investigated. • Of course there is a difference between top-cloud CMW, geostrophic winds and rainfall motion. The method have to deal with this fact.

7. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work • A CMW Diffusion Scheme • Description and assumptions • We would like to have a physically-base model instead of a image-processing procedure. • The proposed diffusion scheme uses basically the same equations that GCM does, but which different assumptions. • We model as if the IR brightness temperature field could be considered as a fluid → We need first to demonstrate this. • Quite different (in theory and in practice) to correlation-based approaches.

8. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work CMW Diffusion Scheme (1) One side: Navier-Stokes modeling of the actual cloud movement seen as a fluid Where (u,v,w) are the components in x,y,z of the velocity, p is the pressure, r is the fluid density, n is the viscosity and gx,y,z are the gravity vector components.

9. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work CMW Diffusion Scheme (2) The other side: cloud movement as an IR image The variations of the IR brightness temperature (P) in the x and y dimensions from time t0 to t1 (t1 very close to t0) are equivalent to an affine transformation –a transformation that preserves lines and parallelism-. So Where P is the “IR matrix”, and A and B are affine transformation matrices. The velocity for the unit of time is: Where I is the singular matrix. By taking derivatives and using the properties of affine matrices: is obtained from the right side of the equation. From the left side and after some algebra we get that:

10. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work CMW Diffusion Scheme (3) Cloud movement – Brightness Temperature movement equivalence So we have that Substituting (gravity is negligible; density~Tb) Image Physics Meaning that the divergence of the pressure in a cloudy area is a linear combination of the area velocity components.

11. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work CMW Diffusion Scheme (4) Cloud movement – Brightness Temperature equivalence • Thus, working with the IR image provided by the satellite is equivalent (with the mentioned simplifications) to the motion of the cloud movement from the point of view of fluid dynamics. • This is important since we can now model the problem of the cloud movement as equivalent to the flow of the brightness temperature as seen by the satellite, and we can use image processing techniques.

12. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work CMW Algorithm The actual algorithm • Multi-scale approach to avoid local minimums in the constrained minimization algorithm used. • Image segmentation • Iterative algorithm • Valid for atmospheric motion and cloud-only motion

13. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work CMW Algorithm (1) First, we consider that the brightness temperature of an area remains constant after a short period of time, 30 minutes for example: Expanding the rhs and gathering the terms of the d increments above the second in e: Applying the chain rule:

14. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work CMW Algorithm (2) Since the elapsed time is negligible: Simplifying the notation by naming the components of the velocity as u and v and the partial derivatives of the brightness temperature in x and y by Tx and Ty we have this conservation law to be satisfied:

15. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work CMW Algorithm (3) • We need additional constraints to solve the problem • Horn and Schunck (1980) proposed as a functional to be minimized the sum of the squares of the Laplacians of the x and y components of the movement. • Including the conservation law to ensure that the conservation of irradiance is satisfied, we obtain this functional: Where a is a proportionality factor that gives the relative weight of the two constraints, and is related with the noise of the image sequence.

16. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work CMW Algorithm (4 at last!) Using the method of Lagrange multipliers to minimise the functional, we obtain that: • This is solved at multi-scale using an iterative procedure • This modeling produces a smooth field. If only the cloud models are desired, additional constraints need to be used.

17. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work Example 31/OCT/2003 Iberian Peninsula METEOSAT-7 IR 10.5-12.5 µm  EUMETSAT 2003

18. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work

19. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004

20. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work • GPM scenario: • We have high-quality 3-hours rainfall estimates • The goal here is to test the CMW scheme, not the performance of the rainfall estimate to be propagated. • We will now simulate realistic hourly rainfall estimates using a regional area GCM. • We will use our CMW to propagate this rainfall. • We will test the CMW scheme performances with independent rainfall estimates from the model: this will solve many of the validation problems.

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23. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work • PROMES model • Limited-Area Model of primitive equations, hydrostatic and totally compressible. • Boundary conditions from NOAA’s GFS AVN • Horizontal resolution: 15km x 15km. • Seven vertical layers (3 in the first 100 m). • References: • Gaertner, Miguel A., Fernández, Casimiro, Castro, Manuel. 1993: A Two-Dimensional Simulation of the Iberian Summer Thermal Low. Monthly Weather Review: Vol. 121, No. 10, pp. 2740–2756. • Arribas, A., C. Gallardo, M. A. Gaertner, and M. Castro, 2002: Sensitivity of the Iberian Peninsula climate to a land degradation. Climate Dynamics, 20 • (…)

24. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work PROMES model • Physical parameterizations: • Soil processes: SECHIBA Model (Decoudré et al., 1990) • Surface processes: Force-restore (Blackadar, 1976) • Vertical exchanges: (Zhang & Anthes, 1982) • PBL: Blackadar model (non-local fluxes) • Above PBL: K theory (local fluxes) • Horizontal diffusion: Flux deformation (Smagorinsky,1965) • Radiation (Dudhia, 1989) • Shortwave: Absorption and dispersion (total spectra) • Longwave: Radiative fluxe divergence • Hydrological processes: Explicit cloud and precipitation model at the resoluble scale (Hsie et al., 1984). Implicit at sub-scale (Kain &Fritsch,1998) • Numerical scheme (finite differences) • Arakawa C grid • Cubic-spline upstream advection • Fourth order explicit horizontal diffusion • Implicit vertical diffusion scheme • Boundary conditions: • Variable relaxation (Davies, 1976) • Running in a BULL NovaScale 6320(32 parallel processors).

25. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work

26. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 02:30 03:00 03:30 04:30 05:00 05:30 ACTUAL RAIN MEASUREMENT RAIN ESTIMATE CMW Diffusion ACTUAL RAIN MEASUREMENT CMW Diffusion RAIN ESTIMATE IndependentValidation

27. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work 04:30 TUC

28. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work Comparison between CMW estimate and (independent) reference rainfall for 04:30 TUC (forward propagation)

29. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work 16:30 TUC

30. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work Comparison between CMW estimate and (independent) reference rainfall for 16:30 TUC (forward propagation)

31. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work

32. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work What if we use the 02:30 measure instead of the 04:30 CMW-scheme estimate when comparing @ 04:30? So, the CMW scheme is actually transporting rainfall

33. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work Time degradation: Average for 31/OCT/2003 Using the CMW, we can maintain correlations > 0.80 for up to 2.5 hours The performances of the method when compared with ground rainfall at instantaneous scale will be linked with the performances of the rainfall to be transported

34. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Current work • In house operative procedure using SSM/I and AMSU-B for Europe (7 months worth of data) • Processing of the NCEP Global-IR database to generate CMWs at pixel (4km) resolution (7 months worth of data) • In house operative procedure using TRMM (3 months worth of data) • Problems found: • Storage • Available global validation data at 30 minutes interval • (Processing time is not a problem) Introduction CMW model GCM descrip GPM experim Results Future work

35. Department of Geography University of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004 Introduction CMW model GCM descrip GPM experim Results Future work • Future work • Near-real-time radar validation for instantaneous estimates • Validation (daily and monthly) with 12 months worth of data • Extensive ground-truth validation and several comparisons