BE, gen_be andSingle ob experiments

BE, gen_be andSingle ob experiments

Syed RH RizviNational Center For Atmospheric Research

NCAR/MMM, Boulder, CO-80307, USA

Email: rizvi@ucar.edu

Syed RH RizviNational Center For Atmospheric Research

NCAR/MMM, Boulder, CO-80307, USA

Email: rizvi@ucar.edu

A Description of the Advanced Research WRF Version 3 (chapter 9) NCAR/TN-475+STR NCAR TECHNICAL NOTE

gen_be technote by Dale Barker Wu, W. -S., R. J. Purser, and D. F. Parrish, 2002: Three-Dimensional

Variational Analysis with Spatially Inhomogeneous Covariances. Mon. Wea. Rev., 130, 2905-2916.

Talk overview:Talk overview:

• What is Background Error (BE) ? • Role of BE in WRF-Var • Importance of BE• How is it computed (“gen_be” utility)?• Impact of Background Error on minimization• Single Observation Tests• Tuning of background error

Where BE fits in WRF-modelling SystemWhere BE fits in WRF-modelling System

What is BE?What is BE?• It is the covariance of (forecast-truth) for analysis control

variables

BE = <(x-xt), (x-xt)>

• Since “truth” is not known, BE needs to be estimated

• Common methods to estimate BE

a) Innovation Based approach

b) NMC Method: (x-xt) ≈ (xt1 - xt2)

(Forecast differences valid for same time)

c) Ensemble Method: (x-xt) ≈ (xens - <xens>)

(Ensemble - Ensemble mean)

d) Flow dependent (adaptive approach)

Role of BE in WRF-Var cost Function:Role of BE in WRF-Var cost Function:

• Basic WRF-Var cost function (J):

J = 1/2[ (x-xb)TB-1(x-xb) +(yo - H(x))TR-1(yo-H(x))) ]

x - Analysis control variable

xb - Background (FG)

B - Background Error covariance

H - Full (Non-linear) Forward Observation Operator

yo - Observations

R - Observation error covariance

Role of BE:Role of BE:• BE is used for preconditioning the analysis equation by

representing it as followsB = U UT with a suitable choice of U

U = UpUvUh

Uh Horizontal Transform

Uv Vertical Transform

Up Physical Transform

• Horizontal transformation (Uh) is via Regional ----- Recursive filters Global ----- Power spectrum• Vertical transformation (Uv) is via EOF’s • Physical transformation (Up) depends upon the choice of the

analysis control variable

for which the variable Ux’ has covariance matrix equal to the identity matrix.A natural way to build U is as a sequence of steps, each of which removes some correlation from the background error.

The background error covariance matrix is defined implicitly by U

How BE is represented?How BE is represented?• In true sense the size of B is typically of the order of 107x107

• Thus it is not possible to handle such huge matrix• Size of B is reduced by designing the analysis control variables

in such a way that cross covariance between these variables are minimum

• Currently the analysis control variables for WRF-Var are the amplitudes of EOF’s of

stream function ()

Unbalanced part of velocity potential (u)

Unbalanced part of temperature (Tu) Relative Humidity (q)

Unbalanced part of surface pressure (psfc_u)

• With this choice of analysis control variables off-diagonal elements of BE is very small and thus its size typically reduces to the order of 107

How BE is represented Contd.How BE is represented Contd.

Up

B

Uv

Uh. . . .

B = UpUvUh UhTUv

TUpT

Basic statistical parameters of BEBasic statistical parameters of BECorresponding to each control variables, following parameters constitutes the basic statistics of BE

a) Regression Coefficient for estimating balanced (statistical) part

of Velocity potential, Temperature and Surface pressure

b) Eigen vectors and Eigen values

c) Scalelength for regional and power spectrum for global option

In WRF-Var, background error covariances are specified in a control variable space related to the model-space x’ via a control variable transform U defined by

x’ Uv UpUvUhv

In gen_be, the inverse transform v U-1 x’ Uh

-1 Uv-1 Up

-1 x’ is performed in order to accumulate statistics for each component of the control vector v.

WRF-Var “gen_be” utility:WRF-Var “gen_be” utility:• Computes various components of BE statistics needed for the

WRF-Var

• Resides in WRF-Var top directory

• Designed both for NMC and Ensemble methods

• Consists of five stages (stage 0 - stage 4)

WRFDA/var/gen_be> lsMakefile gen_be_stage0_wrf.f90gen_be_cov2d.f90 gen_be_stage1.f90gen_be_cov3d.f90 gen_be_stage1_1dvar.f90gen_be_diags.f90 gen_be_stage2.f90gen_be_diags_read.f90 gen_be_stage2_1dvar.f90gen_be_ensmean.f90 gen_be_stage2a.f90gen_be_ensrf.f90 gen_be_stage3.f90gen_be_ep1.f90 gen_be_stage4_global.f90gen_be_ep2.f90 gen_be_stage4_regional.f90gen_be_etkf.f90 gen_spectra.f90gen_be_read_regcoeffs.f90

be_method=‘NMC’be_method=‘ENS’

WRFDA/var/gen_be/*.f90WRFDA/var/da/da_gen_be

“gen_be” - Stage0:“gen_be” - Stage0:

• Computes ( , ) from (u,v)• Forms desired differences for the following fields

- Stream Function

- Velocity potential

T - Temperature

q - Relative Humidity

ps - Surface Pressure

“gen_be” - Stage1:“gen_be” - Stage1:• Reads “gen_be_stage1” namelist• Fixes “bins” for computing BE statistics• Computes “mean” of the differences formed in stage0• Removes respective “mean” and forms perturbations for

Stream Function (´)

Velocity potential (´)

Temperature (T´)

Relative Humidity (q´)

Surface Pressure (ps´)

“gen_be” bins structure“gen_be” bins structure• Currently “gen_be” utility has provisions of following seven (0-

6) “bin_types”

0: No binning (each grid point is a bin)1: mean in X-direction (Each latitude is a bin)2: bins with binwidth_lat/binwidth_hgt3: bins with binwidth_lat/nk4: bins with binwidth_lat/nk (binwidth_lat (integer) is defined in terms of latitudinal grid points) 5: bins with all horizontal points (nk bins)6: Average over all points (only 1 bin)

nk - Number of vertical levels

Default option is “bin_type=5”

• Only bin_type=1 and bin_type=5 have been tested.

• bin_type=1 results seem questionable.

“gen_be” - Stage2 & 2a:“gen_be” - Stage2 & 2a:• Reads “gen_be_stage2” namelist• Reads field written in stage1 and computes covariance of the

respective fields

• Computes regression coefficient & balanced part of , T & ps

b = C´

Tb(k)= ∑lG(k,l) ´(l)

ps_b = ∑k W(k) ´(k)• Computes unbalanced part (stage 2a)

u´ = ´ - b

Tu´ = T´ - Tb

ps_u´ = ps´ - ps_b

Wu et al., 2002

Regression coefficients G(k,l) and W(k) are computed individually for each bin in order to allow representation of differences between e.g. polar, mid-latitude, and tropical dynamical and physical processes.The scalar coefficient C used to estimate velocity potential errors from those of streamfunction is calculated as a function of height to represent e.g. the positive correlation between divergence and vorticity in the PBL.

The summation over the vertical index relates to the integral (hydrostatic) relationship between mass fields and the wind fields.

WRF-Var Balance constraintsWRF-Var Balance constraints• WRF-Var imposes statistical balance constraints between

Stream Function & Velocity potential

Stream Function & Temperature

Stream Function & Surface Pressure• How good are these balanced constraints? Based on KMA

global model

€

b • χ / χ • χ

€

Tb • T / T • T

€

psb • ps / ps • ps

“gen_be” - Stage3:“gen_be” - Stage3:

• Reads “gen_be_stage3” namelist

• Removes mean for u´, Tu´ & ps_u´

• Computes eigenvector and eigen values for vertical error covariance matrix of ´ , u´, Tu´ & q

• Computes variance of ps_u´

Processing: For each variable: compute vertical component of B, perform eigenvector decomposition B EET, and compute projections of fields onto eigenvectors.

Output: eigenvectors E, eigenvalues , and projected 3D (i,j,m) control variable fields for calculation of horizontal correlations.

“gen_be” - Stage4:“gen_be” - Stage4:• Reads “gen_be_stage4” namelist• For each variable & each eigen mode for regional option,

computes “lengthscale (s)”

• For global option, computes “power spectrum (Dn)” €

B(r) = B(0)exp{−r2 /8s2}

€

y(r) = 2 2[ln(B(0) /B(r)]12 = r /s

€

Dn = Fnm( )2=

m=−n

n

∑ Fn0( )2+ 2 Re(Fn

m )( )2+ Im(Fn

m )( )2

[ ]m=1

n

∑

In WRF-Var (regional application), a recursive filter is used to provide the horizontal correlations. The error covariance input to the recursive filter is a correlation lengthscale and the number of passes through the recursive filter.

Processing: perform linear regression of horizontal correlations to calculate recursive filter lengthscales

var/graphics/ncl/gen_be

gen_be_corr_ps.nclgen_be_corr_yz.nclgen_be_corr_z.nclgen_be_global_evals.nclgen_be_global_evecs.nclgen_be_lengthscales.ncl

Reads in ASCII fort.1* output from gen_be/gen_be_diags_read.f90

Reads in ASCII ps_u.ps.dat, chi_u.chi.dat, t_u.t.dat,ASCII output from gen_be/gen_be_cov2d.f90and gen_be/gen_be_cov3d.f90

Lat_stats_option: .False. (default). Only set it true when be.dat is computed with i-dependence (approximately latitude-dependence) .

Namelist - WRFVAR11

lat_stats_option .true.Use latitude-dependent regression coefficients

lat_stats_option .false.Use domain-averaged regression coefficients

lat_stats_option=.false. lat_stats_option=.true.

700 hPa temperature increment

lat_stats_option=.false. lat_stats_option=.true.

20 hPa temperature increment

Single observation testSingle observation test• Through single observation, one can understand

a) structure of BE b) It identifies the “shortfalls” of BEc) It gives a broad guidelines for tuning BE

• “single obs utility” or “psot” may be activated by setting the following namelist parameters

num_pseudo = 1 pseudo_var =“ Variable name” like ”U”, “T”, “P”, etc. pseudo_x = “X-coordinate of the observation”pseudo_y = “Y-coordinate of the observation”pseudo_z = “Z-coordinate of the observation”

pseudo_val = “Observation value”, departure from FG”pseudo_err = “Observation error”

Single Observation (U) testSingle Observation (U) test

Single Observation (U) test - different BESingle Observation (U) test - different BE

BE tuningBE tuning• Horizontal component of BE can be tuned with following

namelist parameters

LEN_SCALING1 - 5 (Length scaling parameters)

VAR_SCALING1 - 5 (Variance scaling parameters)

• Vertical component of BE can be tuned with following namelist

parameter

MAX_VERT_VAR1 - 5 (Vertical variance parameters)

Tuning BE Tuning BE Len_scaling1=0.25No tuning

Tuning BE Tuning BE Len_scaling1 & 2 =0.25No tuning