Implementation of WRF-3DVAR Data
Assimilation over East Africa.
A case study of Tanzania.
Mr. Chuki .A. Sangalugembe (MSc. student)
Tanzania Meteorological Agency
03.09.2012
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
Outline
1
Introduction
2
Short history of data assimilation
3
3D-Var Data Assimilation Method
4
WRF model
What is WRF model?
Domain and Input data
Initialization and boundary condition
Governing equations of the model
5
3D-Var in WRFDA
3D-Var in WRFDA
Results and discussion
6
Summary
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
Introduction
What is data assimilation?
Data assimilation is a technique that uses all available
information (data) to determine as accurately as we can the
state of an evolution of model.
Also data assimilation can be defined as the technique by which
observations are combined with an NWP (Numerical Weather
Prediction) product (the first-guess or background forecast)
and their respective error statistics to provide an improved
estimated (analysis) of the atmospheric (or oceanic) state.
Data here implies both observations and the background
information about the current state.
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
Short history
Subjective analysis (19th century):
Early work by Lewis Richardson et al.,(1922) used data
assimilation based on hand interpolations. They combined
present and past observations from the model and hence the
forecast is adjusted by their expertise. Since this was a rather
tedious procedure, effort to obtain automatic objective analysis
have been developed
Cressman’s objective analysis (1950’s):
The correction at the grid point j with an observation at i, is
given by
Pn
w (i, j)(yi − xib )
b
xj = xj + i=1Pn
i=1 w (i, j)
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
short history
where yi is the observation at the grid point i and w (i, j) is the
the weight of yi at the point j.
To prescribe the weights, Cressman process
w (i, j) =
2
R 2 −ri,j
2
R 2 +ri,j
if ri,j ≤ R and w (i, j) = 0 if ri,j > R
ri,j is the distance between the points i and j. R is an influence
radius to be prescribed.
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
short history
Nudging (1970’s).
The idea is to force the numerical model toward observations
with extra term for elastic relaxation. If the model is defined by
dx
= M(x)
dy
then the nudging equation is given by
dx
= M(x) + α(y − x)
dy
where y is a direct observation of x.
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
short history
After (1970’s).
Recent sophisticated methods include 3D-Var and Optimal
interpolation (1980’s), then 4D-Var and the Kalman filter
(1990’s).
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var data assimilation
This technique finds an optimal estimate of the dynamic model
by assuming that errors are unbiased Gaussian distributed, with
the following pdfs, Pb for background and Po for observations.
−1
1
b T −1
b
Pb = p
e 2 (x−x ) B (x−x )
(2π|B|)
−1
1
T −1
Po = p
e 2 (y −H(x)) R (x−H(x)
(2π|R|)
where
x b is the background vector,
x is the analysis vector,
y is the observation vector,
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var data assimilation
B is the background error covariance,
R is the observation error covariance,
H is the observation error operator,
When we assume a perfect model and independent
perturbation, the total pdf is the product of Pb and Po
P = αe
−1
(x−x b )T B −1 (x−x b )− 21 (y −H(x))T R −1 (y −H(x))
2
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var data assimilation
where α is a product of normalization constant for Pb and Po
Therefore the cost function J(x) become
1
1
J(x) = (x − x b )T B −1 (x − x b ) + (y − H(x))T R −1 (y − H(x))
2
2
The cost function can be written as
J(x) = Jb + Jo
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var data assimilation
where Jb is a measure of the distance of the initial state from
the background estimate
Jo is a measure of distance between the model trajectory and
observation over the assimilation window
The 3D-Var solve J(x) iteratively using conjugate gradient or
by Quasi-Newton algorithm.
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
Incremental formulation of 3D-Var
Let us defined the analysis increment as δx = x − x b
Making use of the tangent-linear hypothesis
H(x b + δx) − Hx b ≈ Hδx
the function becomes
1
1
J(δx) = δx T B −1 δx + (Hδx − d)T R −1 (Hδx − d)
2
2
where d = y − Hx b
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
Incremental formulation of 3D-Var
The analytical solution which minimize J by setting
∇x J(δx) = 0 gives
x a = x b + BH T (HBH T + R)−1 (y − Hx b )
b
b
= x + K (y − Hx )
(1)
(2)
where y − Hx b is known as the observation innovation vector or
departure vector.
K = BH T (HBH T + R)−1 is the gain matrix.
Both Optimal Interpolation (OI) and Kalman Filter (KF)
utilizes eqn(2).
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
What is WRF model?
Domain and Input data
Initialization and boundary condition
Governing equations of the model
What is WRF model?
WRF model is a Weather Research and Forecasting model.
Is a meso-scale developed by several collaborating institutions
It is supported as community model for research but also run
operationally at many private and public institutions
The model fully portable and comes with initialization routine,
making it suitable for both real and idealized experiment.
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
What is WRF model?
Domain and Input data
Initialization and boundary condition
Governing equations of the model
WRF modelling system flow chart
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
What is WRF model?
Domain and Input data
Initialization and boundary condition
Governing equations of the model
Domain and Input data
DomainWizard is used to create the domain of interest.
The WRF Preprocessing system (WPS) is made up of the
routines geogrid, ungrib and metgrid
geogrid defines model domains and interpolates static
geographical data to the grids
ungrib extracts meteorological fields from GRIB formatted files
metgrid horizontally interpolates the meteorological fields
extracted by ungrib to the model grids defined by geogrid
the output file from metgrid is netcdf.
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
What is WRF model?
Domain and Input data
Initialization and boundary condition
Governing equations of the model
Initialization and boundary condition
The program REAL creates initial and boundary condition files
for WRF
REAL interpolates atmospheric inputs vertically onto
hydrostatic pressure coordinate η
The prognostic variables are thus in exact hydrostatic balances
for the model equations
The initialization file outputted by REAL is a netcdf file.
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
What is WRF model?
Domain and Input data
Initialization and boundary condition
Governing equations of the model
Governing equations of the model
The equations are formulated using terrain following
hydrostatic pressure vertical coordinate defined as
η=
(Ph − Pht )
(Phs − Pht )
(3)
where
Ph is the hydrostatic component of pressure
Phs and Pht are surface and top boundary values.
V = µv = (U, V , W ), ω = µη, θ = µθ
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
(4)
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
What is WRF model?
Domain and Input data
Initialization and boundary condition
Governing equations of the model
Governing equations of the model
The prognostic Euler equations
∂U
∂(Pφη ) ∂(Pφx )
+ (∇.Vu) −
+
= Fu
∂t
∂x
∂η
(5)
∂V
∂(Pφη ) ∂(Pφy )
+ (∇.Vv ) −
+
= Fv
∂t
∂x
∂η
(6)
∂W
∂P
+ (∇.Vw ) − g (
− µ) = Fw
∂t
∂η
(7)
∂θ
+ (∇.V θ) = Fθ
∂t
(8)
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
What is WRF model?
Domain and Input data
Initialization and boundary condition
Governing equations of the model
Governing equations of the model
The prognostic Euler equations (cont...)
∂µ
+ (∇.V ) = 0
∂t
(9)
∂φ
+ µ−1 [(∇.V φ) − gW ] = 0
∂t
(10)
where
g is the gravitational constant
φ = gz is the geopotential
α = ρ1 is the inverse density
Fu , Fv and Fθ representing forcing terms.
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
What is WRF model?
Domain and Input data
Initialization and boundary condition
Governing equations of the model
Governing equations of the model
The diagnostic relation for the inverse density and equation of
state resp. are
∂φ
= −αµ
∂µ
P = Po (
Rd θ γ
)
Po α
(11)
(12)
where γ = CCvp = 1.4 for dry air, Rd is the gas constant for dry
air and Po is the reference pressure
The equations (5)-(12) are solved using a time-split integration
scheme using 3rd order Runge-kutta (RK3)
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
3D-Var in WRFDA
In this research we will use 3D-Var because it is most widely
method for data assimilation in NWP and is packaged with
WRF model.
3D-Var in WRFDA seek to solve the cost function
1
1
J(x) = (x − x b )T B −1 (x − x b ) + (y − H(x))T R −1 (y − H(x))
2
2
(13)
The control variables are defined for the background state to
efficiently approximate B using a preconditioner U
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
3D-Var in WRFDA
0
The analysis increment X is given by
0
X = Uv
(14)
where
U represents the various stages of covariance modeling
v is the control variable and with suitable defined U, UU T is an
approx. to B that is easy to compute and converges faster
The increment cost function is thus written as
0
0
J(v) = vt v + (Yo − HUv)T R −1 (Yo − HUv)
(15)
This is the form of quadratic objective function in WRFDA.
Control variables are stream-function ψ, velocity potential φ,
surface pressurePu , temp. T and relative humidity, r.
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
WRFDA in the WRF modeling system
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
WRFDA in the WRF modeling system
x b is the first gues, either from a previous WRF forecast or
from WPS/REAL output
x lbc is the lateral boundary from WPS/REAL output
x a is the analysis from the WRFDA data assimilation system
x f is the WRF forecast output
y o is the observation processed by OBSPROC
Bo is the background error covariance
R is observation error covariance
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Observation processing in WRFDA
In order to be ingested by WRFDA, observations must be either
in little r or prepbufr format
prepbufr format observation do not go through OBSPROC
(Observation preprocessing program)
Observations accepted include U and V components, pressure,
temperature and relative humidity.
The little r file is processed by the program OBSPROC.
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
First guess and Background error statistics
WRFDA uses a wrf input format file for the first guess
In cold start, a file in wrf input is created by REAL program
For warm start, a forecast from WRF model become a first
guess in WRFDA.
Climatology background error covariance are included as the
default within WRFDA.
But background errors for the correct season and resolution can
be estimated by using utility GEN BE
Using NMC method (whereby at least a month of 24 hours and
12 hours interval are recommended), B can be estimated by
B ≈ [x f (T + 24) − x f (T + 12)][x f (T + 24) − x f (T + 12)T ]
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
The surface observations used in 3D-var analysis
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Output results from cost and gradient functions
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
Impact of U-component wind parameter in Assimilation
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
The analysis increment for U-component wind
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
Impact of V-component wind parameter in Assimilation
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
The analysis increment for V-component wind
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
Single Observation test
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
Single Observation test
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
850hpa wind forecasts before and after 3D-Var data assimilation
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
200hPa wind forecasts before and after 3D-Var data
assimilation
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
Precipitation forecasts before and after 3D-Var data
assimilation
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
Precipitable water forecasts before and after 3D-Var data
assimilation
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
Comparison of Mean Sea Level Pressure
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
Comparison of Mean Sea Level Pressure
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
Comparison of Mean Sea Level Pressure
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
3D-Var in WRFDA
Results and discussion
Results
Comparison of Mean Sea Level Pressure
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
Summary and conclusion
From analysis increments and single observation test by
different variables have shown some impacts on first guess from
the model.
The 3D-Var data assimilation has shown some promising results
even if very few observations have been used.
We hope if more observations will be used (such as upper air
observations) in data assimilation will there improve weather
forecast over Tanzania and East Africa as a whole
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation
Introduction
Short history of data assimilation
3D-Var Data Assimilation Method
WRF model
3D-Var in WRFDA
Summary
ASANTENI!
THANKS!
Sangalugembe
Implementation of WRF-3DVAR Data Assimilation