# Presentation: 080330

## Progress in estimating analysis error variances from GSI  Mozheng Wei, Manuel Pondeca and Zoltan Toth NOAA/NCEP/EMC, MD, USA NOAA Thorpex Ensemble Data Assimilation Teleconference,  April 1,  2008Acknowledgment:   John Derber

**Notes:**
Outline slide

## Method (1)

**It is based on Fisher and Courtier (1995) “Estimating the covariance matrices of analysis and forecast error in variational data assimilation”  ECMWF Tech. Memo, #220.**
**It is applied to NCEP GSI.**

## Method (2)

- Define

- In summary, (1)  pre-conditioned conjugate gradient methods in GSI deliver two sets of gradients vectors, one is gradient with respect to **x** (**g**) and the other is with respect to the transformed vector **z** (**h**) (subroutine pcgsoi.f90 was modified). (2) These two sets form a bi-orthogonal system                     (3) The method consists of using sub-sets of                    vectors to estimate the Hessian matrix.   It capitalizes on relationship between conjugate gradient and the Lanczos algorithms  for solving large scale eigenvalue problem (Pondeca 2008).

## Method (3) 

- The final analysis error covariance matrix can be expressed as:

## Experiment 

- The diagonal parts of **C** represent the reductions of error variances due to the
- observations in DA.  The following figures show the variance reduction
- distributions only (at 2007041000).
-  miter=1, niter(1) = 30
-  which means that there will be only 30 dominant eigenvectors produced.
-  &OBS_INPUT
-    dmesh(1)=180.0,dmesh(2)=145.0,dmesh(3)=240.0,dmesh(4)=160.0,dmesh(5)=180.0,
-    time_wind_ow_max=3.0,
-    dfile(01)='prepbufr',  dtype(01)='ps',        dplat(01)=' ',         dsis(01)='ps', \
-                   dval(01)=1.0,  dthin(01)=0,
-    dfile(02)='prepbufr'   dtype(02)='t',         dplat(02)=' ',         dsis(02)='t',  \
-                   dval(02)=1.0,  dthin(02)=0,
-    dfile(03)='prepbufr',  dtype(03)='q',         dplat(03)=' ',         dsis(03)='q',  \
-                  dval(03)=1.0,  dthin(03)=0,
-    dfile(04)='prepbufr',  dtype(04)='uv',        dplat(04)=' ',         dsis(04)='uv', \
-                  dval(04)=1.0,  dthin(04)=0,

## Slide 6

## Slide 7

##  Summary and challenges (1)

- (a).  With a small number of conventional obs data, the reductions of error  variances are identified with 30 dominant eigenvectors.
- (b).  If larger number of obs data especially satellite data are used, the number
-         of dominant eigenvectors needs to be very large, it is expected that this
-         method may have difficulty to identify all the impacts. More experiments
-         are needed to test this.
- (c).  The  variances for analysis variables stream function and velocity
-         potential are not very useful for ensemble initialization, we need more
-         physically meaningful variables like **u**, **v**, **t** to build kinetic or total
-         energy.

**Notes:**
CIRA operational and research experience

##  Summary and challenges (2)

-       The equations linking these analysis variables are expressed in
-       Helmholtz’s theorem as
-         The variances relationship between them is hard to establish:
-      One possible way is to get a statistical mapping from variances of stream
-      function and velocity potential to the variances of **u** and **v **by using the
-      variances data of 6-hour fcst from global ensemble system.
-      Any suggestions and advices will be appreciated!!

**Notes:**
CIRA operational and research experience

##  Summary and challenges (3)

- (d). Even if we estimate the variance reduction reasonably well, we still need
-       the good estimate of background error variance info to get a good estimate
-      of analysis error info:
-      Right now, background variances in operational global GSI are estimated
-      using the NMC method with recursive filter and they are dependent on
-      latitude and sigma level only. They do not show more realistic spatial
-      variation.
-      One solution is to use 6-hour ensemble fcst variance which is much better
-      defined  than ensemble constructed covariance (which has a problem of
-      long-distance spurious correlation).
- (e).  Another completely different method is to use the analysis data from different NWP centers like NCEP, ECMWF, UKM, CMC, JMA etc.

**Notes:**
CIRA operational and research experience