2 nd INTERNATIONAL PRECIPITATION WORKING GROUP WORKSHOP. Monterey (CA) USA 25 – 28 October 2004

1. 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.

2. 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 (1/4) 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.

3. 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 (2/4) 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:

4. 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 (3/4) 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.

5. 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 (4/4) 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.

6. 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

7. 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 (1/4) 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:

8. 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 (2/4) 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:

9. 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 (3/4) • 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.

10. 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 (4/4) 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.