A Cloud Motion Winds Diffusion Scheme for Quantitative Rainfall Estimation

A Cloud Motion Winds Diffusion Scheme for Quantitative Rainfall

Estimation

Department of GeographyUniversity of Lleida, Spain

Francisco J. Tapiador

Department of GeographyUniversity of Lleida, Spain

[email protected]

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)

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

Introduction GCM descrip GPM experim Results Future workCMW model

Department of GeographyUniversity of Lleida, Spain 2nd INTERNATIONAL PRECIPITATION WORKING GROUP

WORKSHOP. Monterey (CA) USA 25 – 28 October 2004

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).



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)




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)




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




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.




A CMW Diffusion SchemeDescription 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.




CMW Diffusion Scheme (1)One side: Navier-Stokes modeling of the actual cloud movement seen as a fluid

z

y

x

gz

w

y

w

x

w

z

p

dt

dw

gz

v

y

v

x

v

y

p

dt

dv

gz

u

y

u

x

u

x

p

dt

du

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Where (u,v,w) are the components in x,y,z of the velocity, p is the pressure, is the fluid density, is the viscosity and gx,y,z are the gravity vector components.

gvpdt

vd

2




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

)(01

BAPP tt

Where P is the “IR matrix”, and A and B are affine transformation matrices. The velocity for the unit of time is:

BPIAPBAPv

Where I is the singular matrix. By taking derivatives and using the properties of affine matrices:

vIAdt

vd

is obtained from the right side of the equation. From the left side and after some algebra we get that:

022 BPIAv




CMW Diffusion Scheme (3)Cloud movement – Brightness Temperature movement equivalence

So we have that

Substituting (gravity is negligible; density~Tb)

vIAp

Meaning that the divergence of the pressure in a cloudy area is a linear combination of the area velocity components.

vIAdt

vd

022 BPIAv

gvpdt

vd

2

Ima

ge

Ph

ysic

s




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.




CMW AlgorithmThe 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




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:

ttyyxxTtyxT ,,,,

Expanding the rhs and gathering the terms of the increments above the second in :

t

Tt

y

Ty

x

TxtyxTtyxT ,,,,

Applying the chain rule:

0

tOt

T

y

T

t

y

x

T

t

x




Since the elapsed time is negligible:

0

t

T

dt

dy

y

T

dt

dx

x

T

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:

0 tyx TvTuT

CMW Algorithm (2)




• 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:

dxdyy

v

x

v

y

u

x

uTvTuTE tyx

2

2

2

2

2

2

2

2

Where is a proportionality factor that gives the relative weight of the two constraints, and is related with the noise of the image sequence.

CMW Algorithm (3)




tyxy

tyxx

TvTuTTvt

v

TvTuTTut

u

2

2

• 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.

CMW Algorithm (4 at last!)


Using the method of Lagrange multipliers to minimise the functional, we obtain that:



Example

31/OCT/2003

Iberian Peninsula

METEOSAT-7 IR

10.5-12.5 µm

EUMETSAT 2003









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.


06:00

10:30

08:00

09:00

17:00

20:30

19:00



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

• (…)




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).

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)

PROMES model







02:30 03:00 03:30 04:30 05:00 05:30

ACTUAL RAIN

MEASUREMENTRAIN ESTIMATE

ACTUAL RAIN

MEASUREMENTRAIN ESTIMATE

Independent Validation

CMW Diffusion

CMW Diffusion



04:30 TUC




Comparison between CMW estimate and

(independent) reference rainfall for

04:30 TUC

(forward propagation)




16:30 TUC




Comparison between CMW estimate and

(independent) reference rainfall for

16:30 TUC

(forward propagation)







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




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




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)




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


Documents

A Cloud Motion Winds Diffusion Scheme for Quantitative Rainfall Estimation