NASA Technical Memorandum 104606, Vol. 4

Technical Report Series on

Global Modeling and Data Assimilation

Max J. Suarez, Editor

Volume 4

(NA_A-TM-IO4606-VQ!_4) TECHNICAL

REPORT SERIES ON GLOBAL MODELING

AND DATA ASSIMILATION, VOLUME 4:

DOCUMENTATION OF THE G0_DARD EARTH

DBSERVING SYSTEM (G_O_) 0ATA

ASSIMILATION SYSTEM, VERSION I

_NASA. Goddard Space Flight Center)

_- N?fi-2)529

Unclas

G3/46 004089S

Documentation of the

Goddard Earth Observing System (GEOS)

Data Assimilation System - Version 1

James Pfaendtner, Stephen Bloom, David Lamich, Michael Seablom,

Meta Sienkiewicz, James Stobie, and Arlindo da Silva

January 1995

DATA ASSIMILATION OFFICE

NASA Technical Memorandum 104606, Vol. 4

Technical Report Series on

Global Modeling and Data Assimilation

Max J. Suarez, Editor

Goddard Space Flight Center

Greenbelt, Maryland

Volume 4

Documentation of the

Goddard Earth Observing System (GEOS)

Data Assimilation System - Version I

James Pfaendtner

Arlindo da Silva

Goddard Space Flight Center

Greenbelt, Maryland

Stephen Bloom

David Lamich

Michael Seablom

Meta Sienkiewicz

James Stobie

General Sciences Corporation

Laurel, Maryland

National Aeronautics and

Space Administration

Goddard Space Flight Center

Greenbelt, Maryland

1995

Abstract

This report describes the analysis component of the Goddard Earth Observing System, Data Assimilation System, Version 1 (GEOS-1 DAS). The general features of the

data assimilation system are outlined, followed by a thorough description of the statistical interpolation algorithm, including specification of error covariances and quality

control of observations. We conclude with a discussion of the current status of development of the GEOS data assimilation system.

The main components of GEOS-1 DAS are an atmospheric general circulation model

and an Optimal Interpolation algorithm. The system is cycled using the Incremental

Analysis Update (IAU) technique in which analysis increments are introduced as time

independent forcing terms in a forecast model integration. The system is capable of producing dynamicMly balanced states without the explicit use of initialization, as well as a

time-continuous representation of non-observables such as precipitation and radiational

fluxes. This version of the data assimilation system was used in the five-year reanalysis

project completed in April 1994 by Goddard's Data Assimilation Office (Schubert et

al. 1993). Data from this reanalysis are available from the Goddard Distributed Active Archive Center (DAAC) which is part of NASA's Earth Observing System Data

and Information System (EOSDIS). For information on how to obtain these data sets

contact the Goddard DAAC at (301) 286-3209, EMAIL daac@gsfc.nasa.gov, or consult

DAO's Mosaic Home Page URL http ://hera. gsfc. nasa. gov/dao, home_page, html.

This document is available electronically via anonymous ftp from dao. gsf c. nasa. gov,

directory pub/tech_memos, file volume_4.ps. Z.

,,.

III

"" INTENTIONALLY B'.A.NK

Contents

List of Figures vii

List of Tables viii

1 Introduction 1

2 General Features of GEOS-1 DAS

3 The GEOS-1 Model 4

4 The Analysis Scheme 6

Quality Control of Observations 8

5.1 Pre-analysis Data Checks ............................ 8

5.2 Analysis Data Checks .............................. 10

6 Localization of the Analysis Problem 10

? Specification of Observation Error Covariances 11

8 Specification of Background Error Covariances 13

8.1 Background Error Variances ........................... 14

8.2 Background Error Correlations ......................... 17

9 Current Development Status of GEOS DAS 19

Ir ".,_ -_J_ •. r

v

|NTENIIONXLLYBLANK
Acknowledgments 21

Appendix A: Chord-length/Great Circle Distance Approximation 23

Appendix B: Damped Cosine Correlation Model 25

Appendix C: Height-Wind and Wind-Wind Background Error Covariances with Damped Cosine Model Function

C.1 Correlation modeling assumptions for background errors ...........

C.2 Derivation of wind component error variances .................

C.3 Formulas for computing background error correlations for the upper-air analysis .........................................

C.4 Decoupling of the geostrophic assumption in the tropics ...........

26

27

28

29

31

Appendix D: List of Acronyms 39

References 41

vi
List of Figures

2

3

4

5

Vertical extent and distribution of layers for both the analysis and model

components of GEOS-1 DAS. Solid lines are analysis levels and hatched

lines are model sigma levels ............................

Schematic diagram of Incremental Analysis Update (IAU) procedure .....

Number of NESDIS TOVS retrievals for August 1985 .............

Observed and modelled forecast error standard deviations ...........

Fit of scaled damped cosine correlation function to radiosonde minus background 500 hPa height difference correlations using North American radiosondes for January and February 1979 ........................

The functions #, -tt'/(s¢t2), and -it"l:12 calculated using the parameter

values given in the text ..............................

7 Graphs of the functions #ijzu, laij,_z,t'tzvij,and #ijw for _i = 10° and hi = 0°.

8 Graphs of the functions #_j_, !'tuvij, #ijwt, and Pij'" for _i = 10 ° and hi = 0 °.

9 Graphs of the functions Pijzu, It_ij, _'tzvij,and #ij'z for _i = 40 ° and hi = 0 °.

v,, and vv 0 o. l0 Graphs of the functions p_, #_', #ij , Pij for _i = 40 ° and hi =

11 Graphs of the functions P_, tt_ij, #ijt°, and #ijvz for _i = 900 and hi = 0 °.

12 Graphs of the functions I_'_, #_f, Pi_", and lqujv for _Pi= 90 ° and hi = 0 °.

2

3

9

15

19

27

33

34

35

36

37

38

vii
List of Tables

1

2

3

4

5

6

Observation error standard deviations ......................

Vertical correlation of rawinsonde errors .....................

Vertical correlation of TOVS errors .......................

Growth model parameters .............................

Tropical upper air wind forecast error standard deviations .............

Vertical correlation of background errors for geopotential height and water

vapor mixing ratio .................................

12

13

14

16

17

18

°°.

Vlll
1 Introduction

A recent NRC panel report (National Research Council 1991) provides an overview of the

issues involved in using data assimilation in climate research. In its report the panel recommends the routine generation of research-quality, model-assimilated and tested geophysical

data sets to serve a broad range of national endeavors, including climate and global change

research and prediction. In early 1991 the Data Assimilation Otfice (DAO) was formed

within the NASA/Goddard Laboratory for Atmospheres (GLA) and charged with producing research-quality analyses for general Earth Science applications.

In April 1994 the DAO completed a five-year reanalysis for the years 1985 through 1989

(Schubert et al. 1993). While providing an important tool for climate research, this analysis

will also serve as a baseline for further assimilation system development. The DAO's longer

term goal is to develop an assimilation system capable of making optimal use of the large

volume of high-quality observations expected near the turn of the century from NASA's

Earth Observing System (EOS) satellites.

The main objective of this Technical Memorandum is to describe the analysis component

of the system used for the five-year reanalysis mentioned above. This system has been

named the Goddard EOS Version 1, Data Assimilation System, or GEOS-1 DAS. Its immediate predecessor is the optimum interpolation (OI) analysis scheme described in Baker

et al. (1987), hereafter denoted B87. The atmospheric general circulation model (AGCM)

component of the system is referred to as GEOS-1 GCM, and is documented in Volume 1

of this Technical Memorandum series (Takacs et al. 1994).

This report describes the analysis component of GEOS-1 DAS with particular emphasis on

those features which differ significantly from B87. Sections 2-3 present a general outline

of the data assimilation system and forecast model, followed by a description of the statistical interpolation algorithm (section 4), quality control of observations (section 5) and

localization of the analysis problem (section 6). The specification of forecast and observational error statistics is covered in sections 7 and 8. We conclude with a brief discussion

of the improvements made to GEOS DAS since its version 1. The appendix gives detailed

derivations of the statistics used in the GEOS-1 DAS and provides a list of acronyms.

This document is available electronically via anonymous ftp from dao.gsfc.nasa.gov,

directory pub/tech_memos, file volumeA.ps.Z or from DAO's Mosaic Home Page URL

http ://hera. gsfc. nasa. gov/dao, home_page,html.

2 General Features of GEOS-1 DAS

The GEOS-1 DAS analyzes global sea level pressure and near surface winds over the oceans,

as well as geopotential height, vector wind, and water vapor mixing ratio on constant

pressure surfaces. The upper air height/wind analyses and the sea level pressure/surface

wind analyses are done using multivariate statistical interpolation algorithms in which mass

(height/pressure) and wind data affect both the mass and wind analyses. The moisture
analysisisdonewith a univariatestatisticalalgorithm,andonly at levelsfrom 1000hPato

300hPa.The basicGEOS-1DASconfigurationconsistsof a 2° latitude by 2.5° longitude,

14-levelanalysis(20, 30, 50, 70, 100,150,200,250,300,400,500,700,850,1000hPa)

coupledto a 20-level,2° by 2.5° modelforthe troposphereandlowerstratosphere.Another

versionof thesystem(not usedin the 5-yearreanalysis)consistsof a 2° by 2.5°, 18-level

(surfaceto 0'4hPa)analysiscoupledto a 46-level,2° by 2.5 ° model. This later system has

been named STRATAN for stratospheric analysis (Coy et al. 1994 and references therein).

Figure 1 shows the vertical extent and distribution of layers for both the analysis and model

components of GEOS-1 DAS.

..., ............ .oo. .........................................

50_ ...................... "" o/ "_'-" -_,

150 200 ..................... "

a. 250 .......................

300 -"'/ ....

400 ..................... * " """'-" " ""

500 ..................... "" """

.................... _°1"j.

700 .................... " """

,Nt

Figure 1: Vertical extent and distribution of layers for both the analysis and model components of GEOS-1 DAS. Solid lines are analysis levels and hatched lines are model sigma

levels.

The analysis scheme

The GEOS-1 DAS statistical analysis scheme uses up to 75 observations to analyze all

grid-points within a small three-dimensional cluster. The data selection algorithm, which

chooses the observations to be used from those that have passed the quality control procedures, is an empirically tuned decision tree which uses a priori observation error estimates

in making its choices. All observations used in the analysis have passed a two stage quality

check (Seablom et al. 1991). The first stage gross check makes use of the assimilation's

forecast error variance fields to estimate expected innovation vector variances. In a second

stage a buddy check compares suspect observations with neighboring data by means of a

2
successivecorrection analysis to the location of the suspect datum. Forecast error correlations are modeled with the damped cosine function as in B87, but the fit parameters have

been recalculated using GEOS-1 DAS. Details are given in section 8 and in the Appendix.

Schematic of Incremental Analysis Update (IAU)

I I I I

_analysi nalysi nalys nalysis

1

03Z, "=- 06Z -- 09Z]'-- 12Z -- 15Z2-- 18Z _']

I

Figure 2: Schematic diagram of Incremental Analysis Update (IAU) procedure.

Incremental Analysis Updates

The assimilation scheme does not include an explicit initialization step but rather relies

on the damping properties of a Matsuno time differencing scheme to control initial imbalances generated by the insertion of analysis increments. However, the initial imbalances

and spin-up have been greatly reduced over earlier versions by the introduction of an incremental analysis update (IAU) procedure (Bloom et al. 1991). As shown in Fig. 2, in the

IAU procedure the analysis increments are computed in a conventional way at the analysis

times (00, 06, 12, 18 UTC). These increments are then inserted gradually into the model

by restarting the short-term forecast that provided the analysis background and adding a

fraction of the analysis increment at each model step. Over the six hour period centered on

the analysis time, the full effect of the analysis increment is realized. The final assimilation

product thus effectively consists of a model forecast (heavy solid line in Fig. 2) produced

using additional heat, momentum, moisture and mass tendency terms which are updated

every six hours from observations. This update scheme for the assimilation is similar to

the way the forcing tendencies from the model's physical parameterizations and filters are

recomputed intermittently (usually every 30 minutes) and gradually introduced into the

ongoing integration at every time-step. An important difference between the IAU scheme

and the usual Newtonian nudging procedure (Anthes 1974; Stauffer and Seaman 1990) is

that the IAU forcing terms are held constant over the insertion period, while in Newtonian nudging they are proportional to the changing difference between a target analysis

and the instantaneous current model state. Significant improvements in terms of assimilation accuracy, noise control, and the hydrologic cycle spin-up are obtained using the IAU

technique.
The actualimplementationof IAU in GEOS-1 DAS involves the following steps:

1. The analysis fields must be interpolated from mandatory levels to the sigma-levels

required by the GEOS GCM. To minimize the adverse effects of the vertical interpolation, only the analysis increments are vertically interpolated, and then added to

the original model first guess in sigma-coordinates. For details of the interpolation

algorithm refer to Takacs et al. (1994).

2. The analysis variables must be converted to the GCM prognostic variables. For example, the potential temperature field is first computed from analyzed heights and mixing

ratio. Because the model thermodynamic equation is formulated in flux form (Takacs

et al. 1994, eq. 9), the after analysis potential temperature must be mass-weighted,

i.e., the potential temperature field must be multiplied by the factor II = Ps - PT,

where Ps is the surface pressure and PT = 10 hPa is the pressure at the top of the

model.

3. The IAU forcing terms are formed by subtracting the after analysis prognostic fields

from the corresponding first guess fields, and then dividing this difference by the

number of seconds in 6 hours. IAU forcing fields are produced for surface pressure,

wind components, mass-weighted temperature and mass-weighted moisture.

4. The model integration is restarted 3 hours before the synoptic time and continues for

6 hours with the IAU forcing held constant in time. After these 6 hours, the forcing

terms are set to zero and the model integration continues for another 3 hours up to

next synoptic time when the model state is used as first guess for the next analysis.

3 The GEOS-1 Model

This section presents a summary of the main features of the GEOS-1 GCM. This model is

fully documented by Takacs et al. (1994) which is available electronically via anonymous ftp

from dao.gsfc.nasa.gov, directory pub/tech_memos, file volume_l .ps .Z or from DAO's

Mosaic Home Page URL http ://hera.gsfc .nasa. gov/dao .home_page .html.

The GEOS-1 GCM uses the second-order potential enstrophy and energy conserving horizontal differencing scheme on a C-grid developed by Sadourney (1975), and further described by Burridge and Haseler (1977). An 8th-order Shapiro filter with a reduced coefficient is applied to the wind, potential temperature and specific humidity to avoid non-linear

computational instability. The reduced_:oefficient filter is applied at every step in such a

way that the amplitude of the two-grid interval wave is essentially removed in six hours.

Applying the filter weakly at each step way eliminates the shock that occurred in earlier

assimilations using an intermittent application of the filter. The model also uses a polar

Fourier filter to avoid linear instability due to violation of the CFL condition for the Lamb

wave and interval gravity waves. This polar filter is applied only to the tendencies of the

winds, potential temperature, specific humidity and surface pressure. The model's vertical

finite differencing scheme is that of Arakawa and Suarez (1983). The dynamics routines are

organized into a plug-compatible module called the "ARIES/GEOS dynamical core" which

is described in Suarez and Takacs (1995).
The infrared and solar radiation parameterizations follow closely those described by Harsh°

vardhan et al. (1987). In the longwave, water vapor absorption is parameterized as in Chou

(1984), the 15 micron band of CO2 as in Chou and Peng (1983), and ozone absorption as

in Rogers (1968) with modifications suggested by Rosenfield et al. (1987). The shortwave

follows Davies (1982), as described in Harshvardhan et al. (1987). Shortwave absorption by

water vapor uses a k-distribution approach as in Lacis and Hansen (1974). Cloud albedo

and transmissivity for the model layers are obtained from specified single-scattering albedo

and cloud optical thickness using the delta-Eddington approximation (Joseph et al. 1976;

King and Harshvardhan 1986).

The penetrative convection originating in the boundary layer is parameterized using the

Relaxed Arakawa-Schubert (RAS) scheme (Moorthi and Suarez 1992) which is a simplified Arakawa-Schubert (1974) type scheme. As an approximation to the full interaction

between the different allowable cloud types in the original AS scheme, many clouds are simulated frequently with each modifying the large scale environment some fraction of the total

adjustment. The parameterization thereby "relaxes" the large scale environment towards

neutrality. In addition to the RAS cumulus convection scheme, the GEOS-1 GCM employs

a Kessler-type scheme for the re-evaporation of falling rain (Sud and Molod, 1988). The

scheme accounts for the rainfall intensity, the drop size distribution, and the temperature,

pressure and relative humidity of the surrounding air.

Super-saturation or large-scale convection is defined in the GEOS-1 GCM whenever the

specific humidity in any grid-box exceeds its super-saturation value. The large-scale precipitation scheme rains at super-saturation, and re-evaporates during descent to partially

saturate lower layers in a process that accounts for some simple micro-physics.

The GEOS-1 GCM turbulence parameterization consists of elements which handle vertical

diffusion (Helfand and Labraga, 1988) and surface fluxes of heat, moisture and momentum

(Helfand, et al, 1991, and Helfand and Schubert, 1994). The vertical regime is divided

into a free atmosphere, a surface layer, and a viscous sub-layer above the surface roughness

elements. The turbulent eddy fluxes are calculated using a variety of methods depending

on the vertical location in the atmosphere.

Turbulent eddy fluxes of momentum, heat and moisture in the surface layer are calculated

using stability-dependent bulk formulae based on Monin-Obukhov similarity functions. For

an unstable surface layer, the chosen stability functions are the KEYPS function (Panofsky,

1973) for momentum, and its generalization for heat and moisture. The function for heat

and moisture assures non-vanishing heat and moisture fluxes as the wind speed approaches

zero. For a stable surface layer, the stability functions are those of Clarke (1970), slightly

modified for the momentum flux. The moisture flux also depends on a specified evapotranspiration coefficient.

Above the surface layer, turbulent fluxes of momentum, heat and moisture are calculated

by the Level 2.5 Mellor-Yamada type closure scheme of Helfand and Labraga (1988), which

predicts turbulent kinetic energy and determines the eddy transfer coefficients used for

vertical diffusion.
4 The Analysis Scheme

As mentioned above, the forcing terms for the IAU assimilation are obtained using a standard statistical analysis methodology. A short term 6-hour forecast to the analysis time,

tk (= 00, 06, 12, or 18 UTC) is used as the background, w[ (refer to Fig. 2). Using all

observations, w_, falling within a 6-hour data window centered on the analysis time, the

analysis, w_, is defined as the background forecast plus a correction expressed as a quasilinear combination of the differences between the observations and a priori estimates of the

observations computed from the background forecast,

= w[ + (1)

In GEOS-1 DAS the "forward problem" transformation 7/k is factored as 7/k = HkS where

S transforms from the model's variables (temperature, wind, specific humidity, surface

pressure) and sigma coordinates to the analysis wriables (geopotential height, wind, water

vapor mixing ratio, sea level pressure) and pressure coordinates. Hk is the multi-linear

interpolation of the pressure-level background vector _,k/ = S(wk/) to our best a priori

estimate of w_. Consistent with the factorization of 7t, the weights /Ck also factor, /Ck =

S+Kk, where Kk is the usual gain matrix (see eq. 5) used to define the increments for the

analysis variables on the constant pressure analysis levels, and S + is the transform of these

increments back to the model's variables and sigma coordinates. The notation here follows

that of Cohn and Parrish (1991).

The forcing vector _k used to modify the model's tendencies during the IAU assimilation

from time tk - 3 to time tk + 3 is held constant during that 6-hour interval and is obtained

by transforming the pressure-level analysis increment back to the model's sigma coordinates

and variables (see Takacs et al. 1994 for details of the vertical interpolation). Thus _k is

defined by

{k - N_t S+ Kk(w_

"I

(2)

where N is the number of model time-steps of length At seconds in the 6-hour assimilation

update period centered on time tk. As mentioned above, S+ represents the pressure to

sigma transform for analysis increments. Since there are more model sigma levels than

analysis pressure levels) the transform S is not actually invertible. Hence, S + only denotes

an approximate inverse for S. It does, however) act in such a way that _k = 0 in the interior

of any region where Kk(w_ - Hk@[) = 0, i.e., where the analysis increment is zero because

there are no nearby data, the relevant components of _k will also be zero.

By minimizing the variance of the analysis error, g_ - S(w_ - w_), we are led to a system

of linear equations which depends on the expected error characteristics of the background

and observations and has the gain matrix Kk as its solution (cf. Jazwinski 1970, Example

7.5),

Kk = [P_-,Hk•+ + + +

Using Kk from (3) in (1), we also obtain an equation for the analysis error variance,

diag{/b_k } = diag{(I - KkHk)[9lk - KkjT}. (4)

6
In (3) and(4),/_k = (_¢_T> = Sp_ksT is the pressure-coordinate analysis error covariance

m tr x, -- = SPIS isthe r ur oo,dio te b k ro.nde.o 

m tr x, = theo serwtion erro m t ,x, -- the

pressure-coordinate cross-covariance matrix of background and observation errors. The

background error is defined by %"! - S(w_ wk/), and the observation error is defined by

e_ -- w_ - Hk_, where _ = ,.qw_ is the unknown true state projected onto the analysis

grid. We have made the assumption that both the background and observation errors are

unbiased, i.e., that (_k// = (e_)---- 0. The standard assumption will also be made that _k/

and e_ are uncorrelated, so that Jk = 0 (see Daley 1991, for the more general ease). Given

this last assumption (3) and (4) simplify to

Kk= [P/,Hk] -f T [H kP_Hk ~f T + Rk] -1 ' (5)

and

diag{/3_} = diag{(I- KkHk)Fdk}. (6)

The computational flow of the assimilation is as follows:

Step

Step

Step

Step

Step

4r / 1. The innovation vector, [w_ - Hk k], is computed, and the quality control of the

observations is performed. See section 5 for details.

2. The components of _ are partitioned into small geographically localized subsets

of related variables. For example, one of these subsets might contain the height and

wind components for the analysis grid-points in a small latitude-longitude-pressure

box. For each of these subsets of wk-a components, a local data selection is made from

the quality checked innovation vector. See section 6 for details of this partitioning

and the local data selection algorithm.

3. The innovation covariance matrix, [Hkf:'YkH T + Rk], is approximated locally using

the types and locations of the selected data. Equation (5) is then solved for those

observation weights in Kk that give the contributions of the selected data to the _k_

components being computed. Sections 7 and 8 contain details of the approximations

for Rk and HkP_"I H_,T respectively.

4. The weights from Kk computed in step 3 are also used in (6) to estimate the

analysis error variance components in diag{/3/_}.

5. The forcing vector (k is computed as in (2) and passed to the model for use in the

assimilating IAU integration, which starts at time tk -- 3 hours and extends to time

tk + 3 hours. At tk + 3 hours, _k is discarded and the integration is continued without

analysis forcing to tk+l to produce the background, wk/+x, for use in the next analysis.

The forecast error variances, diag{/M+l }, which are also needed for the next analysis,

are estimated using an error growth model together with the analysis error variance

estimates for time tk that were computed in step 4. See section 8 for details of the

forecast error growth model.

These steps are further described in the following sections.
5 Quality Control of Observations

5.1 Pre-analysis Data Checks

The GEOS-1 DAS ingests the global conventional observations and the temperature retrievals from the HIRS2/MSU/SSU sounders on the NOAA satellites. Either the NESDIS

retrieved temperature profiles or those created by the GLA physical retrieval system are

used. Conventional data in NMC's Office Note 29, format as well as the NESDIS format

retrieval data are unpacked and put into common format data sets of one day each. A day

corresponds to the four analysis times: 0, 6, 12 and 18Z. Thus, the observations in one file

will be from 2100Z of the previous day to 2059Z of the current day.

The unpacking process keeps all data that appear in the original data sets, except those

that do not have a realistic time stamp. Reports that are obviously in the wrong synoptic

time (late arriving data, for example) are moved to the correct file. These data sets form a

complete set of historical observations that are easily manageable on the computer system.

A second preprocessing step standardizes the observations for ingest into the OI. Only

observations of quantities to be analyzed are extracted for these data sets: sea level pressure

and wind, upper-air height, wind and moisture. The satellite temperature retrievals are

converted into thicknesses. The observations are stratified by type (e.g., surface land,

surface ship) and in some instances by location. This is necessary to assign the proper

observation errors to the data. More stringent checks on the data are performed:

1. Quality marks provided with the observations are used to eliminate bad data outright.

Data marked as being of suspicious quality are kept along with the quality flag.

2. Observations with grossly bad values are deleted.

3. A hydrostatic check is performed on rawinsonde data.

4. Satellite profiles are checked for completeness.

Before the observation data sets are ingested by the analysis scheme, a program is run to

detect gaps in the observation time series. This program categorizes each observation by

type, synoptic time and in the case of retrieved satellite temperature profiles, by the vertical

extent of the profile (stratospheric and tropospheric profiles in the case of NESDIS TOVS

data). These counts are then graphically displayed as a series of bar charts as in Fig. 3. In

this figure, the number of NESDIS TOVS path A (clear column) temperature retrievals for

each synoptic time in the month of August 1985 is shown. The program has identified a

significant gap in the TOVS data record from 11 to 17 August. In this case, it is likely that

the data were not received properly from the primary archive center. This particular gap

was not filled for the reanalysis, but usually such gaps have been filled with additional data

received through secondary sources. These charts also serve as a record of the observational

data that are available to the analysis at a particular time and are made available to the

users.
3OOO

2000

1000

Number of NESDIS TOYS PATH A reports for August 1985

I I I I I I

_::_ _:: _::_.: _::_ _._ _ ..._:: _ _ _

5 10

I

15

_t

_..'__ :-:_:_

iiitiiilIi!!

20

_ . .I!

, !_i_i!_,_i_,i t,I _I::_

,!I!I!._i/ti__itli_l

25

" _ i_i,_i_i _tiI

3O

Strot.

only

ialmm

Con3,pietY,

prortte

_,::,..,:¢?:_:_::.:_:_:_:_ -

Trqp. _

only

IIIIIIl_-

Dc ,s

Figure 3: Number of NESDIS TOVS path A (clear column) temperature retrievals for each

synoptic time in the month of August 1985. Stratospheric retrieval profiles are those that

only have levels above 100 hPa; tropospheric retrieval profiles are those that only have levels

below 100 hPa. Complete profiles report at all levels.

9
5.2 Analysis Data Checks

The data quality control is an important part of any data analysis scheme, as demonstrated

in Shaw et al. (1986). More recent studies performed with the GEOS-1 DAS confirm

that changes in quality control and subsequent data selection can have a major impact on

the assimilation. The quality control technique currently employed in the GEOS-1 DAS

consists of two major steps: a gross error check and a buddy check (Seablom, 1990). The

gross error check is defined to be:

A_ < ((no? + (aS)_) ,, (7/

where A is the difference between an observation and the interpolated background firstguess value, (a°) 2 and (aS) 2 are the observation and forecast error variances, respectively,

and r is a subjectively defined tolerance value which varies with quantity, latitude and

height.

The tolerance value is somewhat reduced for all quantities in the tropics and increased

slightly for the winds near jet level in the middle latitudes. Those data that fail to satisfy

(7) are marked as suspect. The buddy check involves performing a single pass successivecorrection analysis of the data that passed the gross check to the locations of the suspect

data. The difference between the interpolated value and the suspect value is then compared

to the error statistics as in (7) and a decision is then made to reaccept the observation or

to reject it. Typical rejection rates are between 5 and 10%.

6 Localization of the Analysis Problem

z

The GEOS-1 DAS global analyses are performed as a series of localized analyses on smaller

regions referred to as mini-volumes. These mini-volumes are a set of non-overlapping groups

of analysis grid-points. Associated with each mini-volume is an approximately cylindrical

search region, having 3200 km diameter, from which the data are selected for the generation

of the analysis in the mini-volume. The GEOS-1 DAS analysis has three distinct types of

mini-volumes, each containing a different number of horizontal grid-points. In the tropics and low latitudes, each mini-volume contains six horizontal grid-points, rectangularly

arranged. The mid-latitude region, between 30 degrees and 82 degrees latitude, contains

mini-volumes with eight horizontal grid-points, while the polar regions place an entire latitude band of grid-points into each mini-volume. Additionally, each mini-volume contains

two vertical layers of grid-points. The total analysis contains nearly 12,000 mini-volumes.

A data search within each search region selects the 75 observations closest to the volume

midpoint, with at most 60% of these from rawinsondes. The covariance matrix is then

formed and the linear system of equations in (5) is solved for the weights of the observations

Kk. With the current configuration of mini-volumes, neighboring search regions have an

approximate 85% overlap. Because only 75 observations are selected, however, the actual

overlap could be substantially less.

The mini-volume approach to performing the global analysis is preferable to a single gridpoint approach for two reasons. First, solving a local problem centered on a region rather

10
thanona particulargrid-pointsignificantlyreducesthe number of covariance matrices that

need be set up and solved. The data search, also computationally expensive, is minimized

by avoiding redundant calculations. Second, the independent nature of the mini-volumes

allows the GEOS-1 DAS analyses to operate well in a parallel computing environment. The

major drawback is that the limitation of 75 observations per mini-volume reduces the search

region overlap, particularly in areas of high data density, leading to the adverse effect of

boziness in the analysis increments.

7 Specification of Observation Error Covariances

The elements of Rk = R = [rw] are the observation-observation error covariances, i.e.

re, 3 (e°e_ T) The time index k will now be omitted, and subscripts will be

used to denote vector and matrix elements. We write rz,j -_a_ooo a_p,,_, where as°2 =

%°:= e_ , and #,,_

°

is the observation-observation error correlation. In the GEOS-1 DAS

implementation ag and a_ are obtained from tabulated values which vary with instrument

type and level. Vertical interpolation using the tabulated values is done as needed. The

observation error standard deviations for the instrument types being used in the DAO's

baseline five-year assimilation are shown in Table 1.

The error standard deviations shown in Table 1 are in general larger than the standard errors

in the measurements. The tabulated values contain an error of representativeness in addition

to the instrument error. If w°,(true) is the theoretical result of a perfect measurement

corresponding to the real measurement w,°, then [w,° -w_(true)] is the measurement or

instrument error. For the analysis, we have defined the corresponding observation error by

e ° -- w_° - (H@_), = [w_° - w_(true)] q- [w°(true) - (H@t),] .

(8)

The last term in brackets in (8) is what is called the error of representativeness (Lorenc

1986). It depends on such things as grid resolution, subgrid-scale variability, footprint

size for satellite instruments, and importantly on errors in the formulation of the "forward

problem" operator H. These effects are not explicitly modeled in our system. The values in

Table 1 have simply been inflated to account in a crude way for the representativeness error

term. It should also be noted that observation error correlations for satellite soundings are

probably in large part a reflection of the correlation between the representativeness errors

for the observations.

In general, the correlation, p o, between e,o and ejo is set to _,_, the Kronecker delta, so that

errors from distinct observations are assumed uncorrelated. However, in two situations ps°j

is modeled as the product of an isotropic horizontal correlation, #°'h(s_), which depends

only on the horizontal distance, s,_, separating the observations, and a vertical correlation,

v °,', which depends only on their vertical separation. The first case occurs when the data

are two height observations from the same radiosonde ascent. Here #°'h(s,j) = 1 and z/°'v

is similar to the function from L6nnberg and Hoilingsworth (1986). Our current v °'' has

been extended in its vertical extent and its eigenstructure has been slightly adjusted, as

shown in Table 2. The second case occurs when the data are two height observations from

11
Q;

q_

0

Q;

"_ b0

\

0 o

a;

u.)

o0

°_D--

o o0

°_c'OI

b0 i c',_

06

p-,

. llJ

o11 

co

_6 I _ 1_ t __°

u"s b- O_ O0

o o o

0 0

0

IIIIII1111111111

1 I I

I

I

I

I

I

1

I

.--,,,

12
Table 2: The vertical correlation of observation errors for two geopotential height observations from

the same radiosonde ascent.

Pressure level (hPa)

level 20 30 50 70 100 150 200 250 300 400 500 700 850 1000

20 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

30 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

50 0.00 0.00 1.00 0.89 0.88 0.84 0.81 0.73 0.70 0.53 0.43 0.24 0.01 0.00

70 0.00 0.00 0.89 1.00 0.89 0.86 0.83 0.75 0.71 0.56 0.46 0.30 0.04 0.01

I00 0.00 0.00 0.88 0.89 1.00 0.88 0.86 0.78 0.73 0.59 0.51 0.36 0.10 0.03

150 0.00 0.03 0.84 0.86 0.88 1.00 0.88 0.81 0.76 0.62 0.54 0.45 0.23 0.07

200 0.00 0.00 0.81 0.83 0.86 0.88 1.00 0.84 0.76 0.61 0.53 0.46 0.24 0.08

250 0.00 0.00 0.73 0.75 0.78 0.81 0.84 1.00 0.84 0.71 0.57 0.40 0.14 0.03

300 0.00 0.00 0.70 0.71 0.73 0.76 0.76 0.84 1.00 0.78 0.61 0.39 0.13 0.05

400 0.00 0.00 0.53 0.56 0.59 0.62 0.61 0.71 0.78 1.00 0.78 0.54 0.14 0.09

500 0.00 0.00 0.43 0.46 0.51 0.54 0.53 0.57 0.61 0.78 1.00 0.70 0.24 0.15

700 0.00 0.00 0.24 0.30 0.36 0.45 0.46 0.40 0.39 0.54 0.70 1.00 0.64 0.36

850 0.00 0.00 0.01 0.04 0.10 0.23 0.24 0.14 0.13 0.14 0.24 0.64 1.00 0.48

I000 0.00 0.00 0.00 0.01 0.03 0.07 0.08 0.03 0.05 0.09 0.15 0.36 0.48 1.00

the same TOVS sounding or from different soundings of the same type. In this case, the

vertical correlation (see Table 3) is also a slight modification of that used in B87. The

horizontal observation error correlation between NESDIS TOVS soundings of the same

type is modeled by #°'h(s,j) = exp [--(s,j/150.0)2], where s,j is the separation in km. This

function is considerably sharper than the functions which appeared in B87. If the soundings

are of different types, then #,_o = 5,3 is used.

8 Specification of Background Error Covariances

The central feature of Kalman filter theory (Jazwinski 1970, Daley 1991, Cohn 1993, and

references therein) is its inclusion of a prediction equation for forecast error covariances,

p[ = Ak-1P_-IAk-1 o r +Qk-1, (9)

where Ak-i represents a linearization of the atmospheric prediction model and Qk-1 is the

covariance matrix of model errors, i.e., Qk-1 = (e__leIT1) with w_ = Ak_lwi_ 1 + etk_l .

Optimum interpolation can be viewed as an approximation to the Kalman filter in which

(9) is replaced by an empirical model for the evolution of Pk/ (see Todling and Cohn 1994).

Note that (9) gives some guidance in the design of such empirical models in that it shows

that P[ depends upon: (1) the recent history (contained in P_k-1) of the distribution and

quality of assimilated observations; (2) the model error covariance (contained in Qk-1);

and (3) the dynamical evolution of existing errors (governed by Ak-1). In the GEMS-1

DAS implementation of OI, the last of these three factors is ignored and the first two are

accounted for only in a crude way. Writing the elements of Fk/ = [151_] as 151_= a !, al.!J,_,j, we

separately model the error standard deviations, a[ and af, for the _th and 3th components

of the background error vector, and the correlation, #_, between these error components.

13
Table 3: Vertical correlations of observation errors for two geopotential height observations from

NESDIS TOVS retrievals of the same type. The correlation function from the upper triangular part

of the table is used for type A (clear) and type B (partially cloudy) retrievals. The function from

the lower triangular part of the table is used for the type C (cloudy) microwave retrievals.

Pressure level (hPa)

level 20 30 50 70 100 150 200 250 300 400 500 700 850 1000

20 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

30 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

50 0.00 0.00 1.00 0.87 0.82 0.67 0.57 0.45 0.31 0.28 0.26 0.22 0.12 0.00

70 0.00 0.00 0.87 1.00 0.83 0.70 0.58 0.45 0.30 0.25 0.23 0.17 0.06 0.00

100 0.00 0.00 0.81 0.85 1.00 0.83 0.70 0.57 0.43 0.38 0.35 0.25 0.16 0.00

150 0.00 0.00 0.68 0.73 0.83 1.00 0.81 0.71 0.59 0.53 0.48 0.33 0.22 0.00

200 0.00 0.00 0.56 0.61 0.73 0.84 1.00 0.85 0.77 0.65 0.55 0.28 0.15 0.00

250 0.00 0.00 0.46 0.50 0.63 0.75 0.85 1.00 0.86 0.77 0.65 0.32 0.16 0.00

300 0.00 0.00 0.35 0.39 0.53 0.65 0.79 0.86 1.00 0.84 0.73 0.37 0.19 0.00

400 0.00 0.00 0.34 0.36 0.47 0.55 0.67 0.77 0.84 1.00 0.85 0.56 0.36 0.00

500 0.00 0.00 0.29 0.31 0.50 0.47 0.55 0.64 0.72 0.85 1.00 0.72 0.52 0.00

700 0.00 0.00 0.24 0.24 0.30 0.31 0.32 0.38 0.47 0.67 0.81 1.00 0.82 0.00

850 0.00 0.00 0.16 0.12 0.19 0.20 0.17 0.20 0.29 0.51 0.67 0.83 1.00 0.00

I000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00

8.1 Background Error Variances

The background error standard deviations a[ for geopotential height, sea level pressure,

and water vapor mixing ratio are estimated in a two-step process. First, the analysis error

standard deviations, a_, for these quantities at time tk-1 (square roots of elements along

the diagonal of/5__1) are computed using a localized version of (6). A growth term is then

added to simulate the effect of Qk-I on the diagonal of/5/ (Pfaendtner and Sivakumaran,

1991). The equation used for a[ at time tk can be written

a[ = a_ +[growth termJ = a,_ + a{1 - a'_ (10)

flj"

The parameters c_ and fl used in the definition of the growth term, which determine the

growth rate and saturation values for the errors, respectively, depend on the pressure level.

For geopotential height and sea level pressure there is an additional crude dependence on

latitude. This latitudinal dependence is defined using tropical, c_tr and fltr, and extratropical, c_xt and flxt, parameter values with a smooth transition between 15 ° and 45 ° north

and south latitude. For example, for fl we have:

fl,,, < 15°

= - 1⁄2(D', _ 20 It), 15° < I_1 < 45° (11)

fix,, 45° >_ kol •

The form of this functional dependence on latitude was motivated by an examination of the

growth of zonally averaged RMS errors from an ensemble of short-term (to 48 hr.) forecasts.

The parameters (see Table 4) have been empirically adjusted by comparing the mean square

innovation vector components, <[w, °- (H@/),12>, to the OI estimate <a,°)z + <(Hal),> 2.

14
An example of the effectiveness of the error growth model is shown in Fig. 4. The figure

shows a time series of 500 hPa of forecast error standard deviation estimated from the

innovations and observation error standard deviations by the formula

_S _ ((w o - HwS)_ _ (ao)2) '/2 ' (12)

compared to the forecast error standard deviation produced using the growth model, eq.

(10). This figure shows modelled and observed aS for a box in the South Pacific (165W105W,60S-36S), from July 7 to August 4, 1979. Although the magnitude of the modelled a !

is slightly larger than observed, both curves are highly correlated after July 14. Therefore,

the empirical growth model seems to capture the main characteristics of the temporal

variations of the forecast error standard deviations.

80.0 i _ r

'°°I............................... i........................... _i................................ i-:................... _II

°°°I....7........................ i;;;::_; ........... /_;i....... ;_............... _i_"-",. .............. /-tl _o.oli.._ ............ "....... i,"__',:;--; _"....:.i_-t ...... :_'/I_;- ...... ,,<i_"_; ...... :I..... W

7-7-79

400

"li!!i!i!t!i!i!!lil!i! i!ii!i!ili! ::i!!!!i!i!!f!i!i!!! !!!ii!i ._ 300

Ill

,_ 200

0

EL

"6 100

d

z 0

7-7

7-14-79 7-21-79

7-14 7-21

Date

7-28-79 8-4-79

7-28 8-4

Figure 4: Upper panel: Observed (solid line) and modelled (dashed line) 500 hPa forecast error

standard deviations for a box in the South Pacific (165W-105W, 60S-36S) from July 7 to August 4,

1979. Lower panel: the number of observations used to estimate the observed forecast error standard

deviations.

Outside of the tropics the standard deviations of the u and v components of the wind errors,

a,u and a_, used in the multivariate upper air height-wind analysis are determined using

the geostrophic approximation as in B87 (see also Appendix A),

g2(az)2 Ol#z':(s) "_, (13) (,,;,)2= @,;)_= 17T_ { o82 ,=oJ

where g is the gravitational constant, f the Coriolis parameter, a_z the forecast height error

standard deviation from (10), and pz,z (s) the isotropic separation dependent model function

15
Table4: Parametersusedin thebackgrounderrorvariancegrowthmodelfor geopotential height,

watervapormixingratioandsealevelpressure.

Geopotential height(m) Mixingratio(g/ks)

Tropics Extra-Trop. Tropics Extra-Trop.

level (3_tr dltxt Btr _xt _ B

10 125.00 70.00 147.00 112.50 -- --

30 110.00 59.00 132.00 109.50 -- --

50 95.00 48.00 117.00 106.50 -- --

70 80.00 37.00 105.00 103.00 -- --

100 60.00 35.00 96.00 98.50 -- --

150 25.00 32.00 76.00 95.00 -- --

200 21.00 30.00 56.00 91.50 -- --

250 17.00 28.00 37.00 88.50 -- --

300 14.00 26.50 29.00 85.00 0.05 0.40

400 13.00 23.00 26.90 78.00 0.15 1.20

500 12.00 20.00 24.50 71.00 0.20 1.60

700 10.50 15.00 20.00 58.00 0.65 5.20

850 9.00 12.50 20.00 56.50 0.90 7.20

1000 9.00 10.50 20.00 55.00 1.20 9.60

SLP 1.13 1.31 2.50 6.88 -- --

for the horizontal correlation of forecast height errors, which is discussed in the next section.

In the tropical band 15°S < _ <: 15°N, a u and a,_ are assigned the pressure level dependent

value atropU'Vgiven in Table 5. At points in the subtropical bands, 25°S _< _o< 15°S and 15°N

< _o < 25°N, a_ and a,_ are defined by

°9s2 s=o

where f2s = 2f_sin(25°) •

Outside of the tropical band, 20°S _< _0 _< 20°N, the standard deviations of the uj and vs

components of the near surface wind errors over the oceans are defined as in B87 by

= -- a2 xvs 6982 8=0 '

(15)

where a,p is the background sea level pressure error standard deviation from (10), a the

Earth's radius, C, and F, the coefficients for the surface wind model derived in B87, and

#P,P(s) the isotropic separation dependent model function for the horizontal correlation of

background sea level pressure errors discussed in the next section. In the tropical band,

10°S _< _o < 10°N, a,_" and a,v' are set to 5 ms -1, while in the subtropical bands, 200N

_< _o< 10°S and 10°N < _o _<20°N, they are defined by

(a_") 2 :(a:*) 2 :[20_0[_°] - ] ×(5)2 -k []q°ll010] - (a'P)2 o,/C,,2+F2)( 02/_"I'(s) o: l._-o}'

16
Table 5: Tropical upper air wind forecast error standard deviations.

Tropical

Pressure interval _u and _ (m s-l)

p < 20 7.1

20 _< p < 40 7.1

40 _< p < 60 8.0

60 < p < 85 7.3

85 _< p < 125 7.3

125 _< p < 175 7.3

175 _<p < 225 7.4

225 _< p < 275 7.0

275 < p < 350 6.2

350 _< p < 450 5.2

450 _< p < 600 4.5

600 < p < 775 4.4

775 _< p < 925 4.4

925 < p 4.3

8.2 Background Error Correlations

For all permutation pairs, [a, b], of variables for the upper-air multivariate analysis [geopotential height (z), east-west wind component (u), and north-south wind component (v)] the

forecast error correlations, #_, are assumed separable i.e. they are written as the product

of a horizontal, pa'b(s,3), and a vertical, v a,b ,a , error correlation. As in B87, the horizontal

height-height error correlations are modeled using a damped cosine function (Thiebaux

1975),

1

_z'z(8) _-- Cl"""_"1- [cl COS(C28) + c3][1 + (c4s)2] -c5 . (17)

Notice that this function has two length scales, c2 and c4. The fitting parameters have been

recomputed since B87, and a single function is used globally in contrast to the regionally

dependent functions described in B87. The current parameter values (cl = 0.0129928,

c2 = 0.00389265 km -1, c3 = 0.694005, c4 = 0.00105123 km -1 and c5 = 1.20815) were

obtained by a fit using 500 hPa radiosonde height data from North America for January

and February of 1979. The graph of (Cl + c3)#z'z(s), illustrating its fit to the data, is

shown in Fig. 5. The vertical correlation of background height errors, v,_,z, is obtained by

interpolating from the values appearing in the upper triangular part of Table 6. In GEOS-1,

the chord length approximation is used to computed distances (see Appendix 9).

For the correlation of forecast height errors with forecast wind errors as well as the windwind forecast error correlations an approximation to the geostrophic assumption is used.

Starting with the height-height error covariance model, ¢,E_/ ~g "Z _ __- _,aj/z,_ Z Z g,Z v,3Z_Z , and the

t

geostrophic approximations for the wind component errors,

- g 0_z g Ogz

and U = -- (18)

a f c%p a f cos _o0A'

17
Table 6: Vertical correlation of background errors for geopotential height (shown in upper triangular

part of table), and for water vapor mixing ratio (shown in lower triangular part of table).

Pressure level (h Pa)

level 10 30 50 70 I00 150 200 250 300 400 500 700 850 I000

20 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

30 1.00 0,00 0.00 0,00 0,00 0,00 0,00 0.00 0,00 0,00 0,00 0,00 0,00

50 1.00 0.87 0.71 0.29 0.19 0.06 0.00 0.06 0.08 0.10 0.14 0.09

70 1.00 0.75 0.31 0.19 0.05 0.00 0.05 0.06 0.09 0.12 0.09

100 1.00 0.60 0.46 0.30 0,21 0.22 0.21 0.15 0.18 0.12

150 1.00 0.82 0.66 0.56 0.47 0.43 0.25 0.19 0.10

200 1.00 0.81 0.72 0.61 0.52 0.32 0.24 0.14

200 1.00 0.86 0.77 0.71 0.49 0.40 0.25

300 1.00 0.85 0.80 0.59 0.48 0.30

400 0.94 1.00 0.87 0.70 0.59 0.37

500 0.60 0.71 1.00 0.77 0.68 0.43

700 0.07 0.12 0.33 1.00 0.83 0.57

850 0.02 0.04 0.13 0.50 1.00 0.66

I000 0.04 0.05 0.04 -0.01 0.27 1.00

we get:

=

el, \,9_,/,q= a/,

a g \e_ \OAJJ = a/_cos:_ O)_j f. 7cos qoj

a/, cos _, \ \ OA 1, _ - af , cos _, OA,

,:T,:,t_:,to: j, = a_:,_:--_,

,_2y,Lcos_, , \ \ o_,), \ o_,) :/ = -a_A/,cos_, 0_,0_,

a2f,/_cos_, \ kO)_), kO_)ff = -a2X/_cos_, 0),,0:_

(19)

(20)

(21)

(22)

(23)

(24)

(25)

:

a_f,f_ cos _p,cos _ O,_,O,_

In computing the partial derivatives of (_;) in (19)-(25)we make the additional assump18
0.8

0.6

8 0.4

oL

0.2

0.0

0

I ......... ! .........

o

lOO0 2000 3OO0

Separation (km)

Figure 5: Fit of scaled damped cosine correlation function to radiosonde minus background

500 hPa height difference correlations using North American radiosondes for January and

February 1979.

tion of local homogeneity for the height error variances,

(90 "z _0 .z

o. (27)

Consequences of this homogeneity assumption are discussed in Cohn and Morone (1984) and

Morone and Cohn (1985). Derivations of effective computational equations for the heightwind and wind-wind error correlations including modifications for the tropics are given in

Appendix C. Note that separate methods have been used to decouple tropical wind-wind

variances (eq. 14) and tropical wind-mass correlations (eqs. 118,120-123).

9 Current Development Status of GEOS DAS

In the time since GEOS-1 DAS was created to perform the 5 year reanalysis, one new

version of the analysis component has been made operational (OI 1.5) and another version,

with many more significant changes, is about to become operational (OI 2.0). The most

important change in version OI 1.5 is the correction applied to the wind-wind forecast

error correlations in the deep tropics. The geostrophically derived correlations are not used

across the equator because sign changes introduce spurious divergence in the wind analysis;

no changes were introduced in the extratropics (defined as those latitudes poleward of 15°).

For OI 2.0, the code has been generalized to allow up to 300 observations (one observation equals one observed variable) to contribute to the analysis of the grid points in

each mini-volume. We have also added the ability to assimilate the SSM/I surface wind

speeds, SSM/I total precipitable water retrievals, and have experimented with assimilation

19
of the microwavelimb soundertemperaturesfrom the UpperAtmosphereResearchSatellite (UARS/MLS). Thecodehasbeengeneralizedto acceptnewconventionaldata types,

suchas the ARINC Communication Addressing and Reporting System (ACARS) aircraft

data, which are winds reported by wide-bodied US commercial aircraft. The large minivolumes at the poles, which had been a collection of all pole points, have been broken up

into mid-latitude sized mini-volumes. This eliminated some discontinuities associated with

such large mini-volumes. The observation increment history data set now contains a history

of the quality control decisions, and we are carrying the time of each observation through

the analysis to begin experimenting with the decoupling of satellite temperature retrievals

from different orbits. Work is in progress to retune observation and forecast error statistics

consistent with recent improvements in the system.

Since GEOS-1 DAS, there have also been several improvements made to the GEOS GCM.

The first improvement entails the development of a fourth-order version of the Sadourney

scheme for the momentum equation. The "fourth-order Sadourney" is derived in Suarez

and Takacs (1995). This scheme conserves total energy and potential enstrophy for nondivergent flow, and is fourth-order in the sense that the advection of the second-order

vorticity reduces to that governed by the fourth-order Arakawa (1966) Jacobian. This

scheme was chosen rather than the Arakawa-Lamb (1981) scheme modified to fourth-order

by Takano-Wurtele (1982) due to severe polar noise problems generated in three-dimensional

GCM calculations from the terms required for potential enstrophy conservation for general

flow. Together with the "fourth-order Sadourney" scheme for the momentum equations,

the current GEOS GCM also uses fourth-order horizontal advection in the thermodynamic

and moisture equations. It uses a scheme developed by Arakawa which has been used for

many years in the UCLA GCM.

The dynamical climatology of the GEOS GCM has been thoroughly examined by Takacs

and Suarez (1995). They have shown that increasing the resolution and/or the order-ofaccuracy in the GEOS GCM has a very significant impact on the zonal mean flow. Running

the second-order GEOS-1 GCM at both 4° x 5° and 2° x 2.5 ° horizontal resolution, an

examination of the first and second moments from a five year simulation revealed a strong

systematic bias for most fields between the two resolutions, particularly in the southern

hemisphere. Most of these biases are related to the simulation of the transient eddies, with

increased accuracy enhancing transient transports of heat and momentum and decreasing

transports of moisture. These systematic biases are removed, however, when fourth-order

accuracy is used. The higher-order and higher-resolution experiments are in closer agreement to the GEOS-1 DAS analysis in the southern hemisphere where the transports are

primarily determined by the transient flow.

Another major enhancement to the GEOS GCM is the ability to run both simulations

and the GEOS DAS assimilation using a rotated coordinate. In the current system, the

geographic placement of the computational pole is arbitrary. We have found that due to

variations of the scheme near the poles required from conservation constraints, the computational instability discussed by Hollingsworth et al. (1983) for the nearest grid-point to the

pole gives rise to polar noise when confronted with strong cross-polar flow. This instability

is proportional to the mean zonal wind speed and the Coriolis parameter. By rotating the

computational grid to the geographic equator, however, the instability near the computational pole is removed due to the vanishing Coriolis term. In addition, the geographic pole

20
now using the transformed grid is also free of noise.

Acknowledgments

We would like to thank Larry Takacs for assistance with the model description and update

on the current status of development of the GEOS GCM. The development and research

documented in this report were supported by the EOS Interdisciplinary Science Program

and by the NASA Research and Applications Program. Computer resources and funding

were provided by the EOS Project through the Scientific Computational Facility of the

Data Assimilation Office.

21

Appendix A: Chord-length/Great Circle Distance Approximation

The background and observation error correlation models used in the analysis are expressed

as isotropic functions of the separation distance between points on the sphere. Given two

points P_ = (Ai, _oi) and P/- (Aj, _oi) on the sphere, we can write their position vectors as:

pi = aqi = a [cos _oicos hi, cos _ sin )q, sin _ol] (28)

and

pj = aqj = a [cos _oj cos ;kj, cos_oj sin Ay, sin _oj], (29)

where a is the Earth's radius. The true shortest distance arc length along the great circle

connecting Pi and Pj is given by

sij = a arccos(qi, qj). (30)

On the other hand, the square of the length of the chord connecting P_ and Pj is:

%.2 = (p_ _ pj). (p, _ pj)

= 2a2(1- qi.qj). (31)

In GEOS-1 DAS the chord length approximation (31) is used to compute distances. For

the sake of completeness, this section will derive equations based on the great circle formula

(30) and the chord-length formula (31).

We will need the first and second partial derivative of % and _ij with respect to _0i, hi, _oj,

and )U" Note that if 7 and y are any two (possibly the same) of these four independent

variables we have

-_70%'sm(_s_J) = _O_iJ(_)= - aO(q_qJ)07 " (32)

and

Oyc9----"_ a sin(_-) 02slj + cos(_) Osij Osijcoy CO7 = 0_-_ sijco2sij ^ + cos'JcOy_Ijco7-- -a202(qiqJ)COYCO')' " (33)

In particular, we need the following partial derivatives for the height-wind and wind-wind

error covariance computations.

• For (zu),

• For (uz),

• For (zv),

co_Oj co_oj --aciy.

(34)

bOs,J

O_oi O_oi -a%j. (35)

b 0% oCO_ij

= =

(36)

I::-S'sE'3i?_t: i'L.:*_:!:?_t./¢.fg r,i_:T FI!.ME_

23

PA_E iNTENIIONALL.Y_;.,_NK
• For (uz),

• For (uu),

• For (uv),

• For (uu),

• For (vv),

b OSlj _.08ij

-_, = b-_-;- - a_s.

bz 02siJ _ _a(b2cuy + cijciucvj)

O_o_O_j

_3 0281J =

b 3- 028ij -_- -a(b2cvx "4- CijCyjCix)

_3 028ij __ a(b2cyx 31_ CyjCix).

O_iOAj

ba_D2so _ -a(b2c_u + cocxjciu)

O)tiO_j

_3 028ij = _a(_2cxy .31_ CxjCiy).

O,_Ocpj

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(4s)

Where we have used,

= ¢7

^

_} --_ (-_-)= V/2(1- co) ,

cij = qi • qj

= + cos _i cos Ai cos _j cos ,_j + cos _Pisin ,ki cos cpj sin _j + sin _Pisin cpj,

Oqj

cix : qi" O_j

= -cos_icos)_icos_jsinAj+cos_isin)_icos_pjcosAj,

Oqj

Ciy = qi " _j

= - cos cpl cos )q sin q_j cos Aj -cos_isin)qsinqajsinA./+sin_icos_pj,

0qi

c_ = 0)_-S"q_

(46)

(47)

(48)

(49)

(50)

24
and

= -cos_isinAicos_jcosAj + cos _i cos Ai cos _0jsin A/,

Oq_

cuJ -- O_oi " qj

= - sin _oi cos ,_i cos _j cos ,_j - sin _'i sin Ai cos _j sin ,_j + cos _i sin _j,

Oq_ O_

c_ = OA_ "OA--'_-

= + cos_oisinAicos_j sinAj + cos_oicosAicos_0jcosAj,

Oq__._i.Oqj

Cxy :

OA_ 0_oi

= -t-cos_pisinAisin_jcosAj -cos_icosAisin_ojsinAj,

Oqi . Oq....._

Cy:v --" 0_o_ OAj

= + sin (Pi cos Ai cos (pj sin Aj - sin _oisin Ai cos _0j cos Aj,

Cyy

Oqi Oqj

= + sin _oicos Ai sin _oj cos Aj + sin _i sin Ai sin tPj sin Aj + cos _i cos _./.

(51)

(52)

(53)

(54)

(55)

(56)

Appendix B: Damped Cosine Correlation Model

As in Baker et ai. (1987), the horizontal height-height error correlations, as well as the

sea level pressure-sea level pressure error correlations, are modeled using a damped cosine

function (Thiebaux 1975),

# = #ZZ(s) = #PP(s) = 1----_[c, cos(c2s) + c3][1 + c42s2]-c5. (57)

C1 -_- C3

The fitting parameters have been recomputed since Baker et al. (1987), and a single function

is used globally in contrast to the regionally dependent functions described in that paper.

The current parameter values,

cl = 0.0129928, (58)

c2 = 0.00389265km -1, (59)

c3 = 0.694005, (60)

c4 = 0.00105123 km -1, (61)

cs = 1.20815, (62)

were obtained by N. S. Sivakumaran using a fit to 500 hPa radiosonde height data from

North America for January and February of 1979.

If we define

6"I -- Cl , (63)

cl + c3

c3 - c3 = i- cl, (64)

cl + c3

25
wecan write

_2 = e,c_ + 2c_c5,

3' = _2 ,

2c2c5

= /32 -1-7,

A = (1 +c_s2) -I,

A _ = -2c]A2s,

B = A c5,

B' = -qfl2ABs,

sin(c2s)

C =

(c_s) '

c' = (1/s)[cos(c_) - el,

sA' sB'

D = 2c_As2=- A =-c'_

(65)

(66)

(67)

(68)

(69)

(70)

(71)

(72)

(73)

(74)

# = [clcos(c2s) +_3]B. (75)

Differentiating we get

#! = (#/B)B'- _lc2sin(c2s)B

: -q/32At.t8 -- _lc]CBs

= -fl2s[r/A# q- 7BC], (76)

and differentiating again gives

p" = -/32(_--_2s) -/32s[q(A#' + A'tt) + 7(BC' + B'C)]

= -_2{71[A # + sm#'+ sm'#] + 7[BC+ sBC' + sB'C]}

= -/32{m#_/[1 - 2c_As2(1 + c5)] + BT[cos(c2s)- 4c2c5ACs2]}

= -fl2{A#_/[1- (1 + c5)D] + BT[cos(c2s)- 2c5CD]}. (77)

At the origin we have: #(0) = 1, lims__o (_--q_ = __2, and #"(0) = _f12. The graphs of

the functions #, -tt'/(sfl2), and -#"/fl_ are shown in Fig. 6.

Appendix C: Height-Wind and Wind-Wind Background Error Covariances with Damped Cosine Model Function

i

In this appendix we give the explicit formulas for the correlation functions used in the

multivariate wind-height analysis.

26
0.4-

0.2-

0-

-0.2 -_

-0.4

0

Damped Cosine and its Derivatives

0.8- ___., .........._................................ I

°' ....'i2i\i I.......",el

......... _ ........... i........................ :---"--'-- 7.:.T_eo'_-'-,._, ,_'-.,,._

r, o_"

i

1000 2000 3000 4000

S(kin)

Figure 6: The functions p, -IZ'/(sfl2), and _#,,/f12 calculated using the parameter values

given in the text.

C.1 Correlation modeling assumptions for background errors

For all permutation pairs, ab, of the three upper-air analysis variables [geopotential height

(z), east-west wind component (u), and north-south wind component iv)] the correlation

between background errors in variable a at location Pi = ()_i, qoi, Pi) and background errors

in variable b at location Pj = (hj,_pj,pj) is defined by

a b

(78)

a i aj

Our first modeling assumption, often referred to as the separability assumption, is that each

of the/z(e?e_)'s can be written as the product of a horizontal, #iajb = Iz_b(gij), and a vertical,

v_ b -- t/_b($ij), error correlation, where gij = g(hl, ¢Pi, hi, qaj) is a measure of the horizontal

distance between points Pi and Pj, and _ij = $(pi, pj) is a measure their vertical separation,

i.e.,

a b t_,,b(gij)v,b(_ij) ,b a_, (79)

The second modeling assumption is that the background wind component errors can be related geostrophically to the background height errors, i.e. we will make use of the geostrophic

approximations for the background wind errors,

g 0_ _ g 0e_ (80)

e_ _-, afi Oqoi and ei _ afi cos <Pi O,ki"

The geostrophic assumption implies that we can express all of the (e i*_j)b,s in terms of (_i_ei)_

and its partial derivatives with respect to hi, _i, hj and _j:

2: Z Z g ZZ ZZ

27
g ,o&_ z, 9 0<_) (83)

('i cj) _ afj cos_i

g ,o&_ ,, g on<_;) (82)

g2 2 z z

,,,, g2 _on4on_)= (86) <E'_t) _ a2f_ft'O_=on_j a2fd_ on,p_on_j

_ 92 O&_OnE_ g2 _ = •

on (q_t) (87) <E_'t)= a_fjtcos_t<oo_b--_ )= _/d_cos_,oo_,on)_

2 z z. g 2 2 z z

_ g <O&_0_ (88)

v v _2 (on_ z OEj ..02 0 2 (Q_j) z z (89)

(Q tJ ) _ a2fifj cos Wi cos qoj onAi-_j) -- a2fifj cos _oi cos 9t OnAionAJ"

The third modeling assumption is that the height error standard deviation is horizontally

homogeneous. This homogeneity assumption (see Morone and Cohn 1985) is expressed

through the approximations,

ona_ Oa •

_ _ o, (90)

and will be used to simplify the partial derivatives of (_e_) which appeared in (82) through

(89).

C.2 Derivation of wind component error variances

In order to derive expressions for a _ and a _ which are consistent with our correlation

modeling assumptions, we will first examine (e_'e)') and (e_e_) in detail. From the separability

assumption we have:

Using our geostrophic approximation and separability a second time, this becomes

g 2 on2(e_t) z z

(7i aj ]_Lit Vii _._ a2f_f_ on_ion_t

2 Z Z ZZ ZZ

9 2 on (a i at #ij vii ) (92)

With two applications of the homogeneity assumption, we arrive at

u u uu uu g2 0 r , _onzi7 _,

aia;#ij _,j _ a2f, fton_,_iaj-_--_j_j_it )

2 on2.. zz

,_ _ -a_a_[ _' f'ij "_v_f (93)

a fifj k O_ion_t

28
Wenow usethechainruleto rewritethe expressionin bracketsto get

2 02/z_] 0_ 0_ =z 02_

ai at #ij vii _ _ai aj _ O_2 0_i O_t + O_ O_iO_j (94)

Taking the limit as j -_ i, or equivalently as _ --+ 0, we have

(o'U) 2 - g2(a_)2 _ limit' a2. _ijzz O_ O_ O"_ijzz 02g

a2fi 2 j-4i( _ O_iOcFt -}- O_ }" (95) 0_o_0_.,t

If #izj_ is modeled with the damped cosine function described in the previous section, we can

use equations (75), (76), (34) and (35) to simplify and get

(a_') 2 - g2/_2(a----'---_)2 lim[ a2ci-ycvj _, (96)

a2 fi 2 j-*iL b2 J

where f12 depends on the damped cosine fit parameters as given in (58)-(62); clu and cut

are defined in (50) and (52); and/_ = b from (46) if _ is the true arc length, or b = b from

(47) if g is the chord length distance approximation. In any case, the limit in (96) is equal

to -a 2 and we have:

(o )2 _

fi 2 (97)

as appeared in Baker et al. (1987). This clearly needs to be modified near the equator. These modifications, the so-called decoupling of the geostrophic approximation in the

tropics, will be discussed in subsection C.4 below.

The analog of (94) for (e_'e_') is

v v vv vv g 2 02. zz. zz

O'iO't[£ij Vij "_ a2fifjcos_Picos_Pj "7[" 082 O)_iO_j "_- 08 O,_iO_ j

Taking the limit as j -+ i and using equations (75), (76), (36) and (37) we get:

(a_) 2 - g_/_2(a_)2 ljm.__ a_ci=c=J }, (99)

a2fi 2 _-+,t b2 cos _oicos _t

where cix and cri are defined in (49) and (50). The limit in (99) is also equal to -a 2.

Hence, (or) 2 is the same as (a_) 2 as given in (97). Notice that a _ equals a v only under the

assumption of homogeneous variances expressed by (90).

C.3 Formulas for computing background error correlations for the upperair analysis

Now that we have formulas for the background wind error variance, we can substitute

them into our separable background error covariance models to get effective formulas for

computing the various error cross-correlation functions. We start with #_.

29
Substitutinga_' and a_ as given by (97) into (94), and noting that v_ u must equal viiz, we

get:

uu i)sig _u _2.Pijzz 08 08 Opij zz 02_

Pij sign (qaa_n (_J) + (lO0) = t" _ 0_o_ O_oj 0,_ O_o_O_oj_"

Substitution using (73) and (74) allows us to rewrite this as

• ._[l'L_'(-_)l o_ 1 r_,,7"(_)I

Pi_u =sign(q°i)slgn(_J){-_-Tt _¢_2g jO_oiO_oi g2L_--2-_-jc_,cv_} • (101)

If we are using the chord length approximation for horizontal separation, i.e., g = _, we can

substitute using (39) to obtain,

{ [#_y(g) ] 1 [ [._' (_)1 [ #_J' (_)1 ] } p_j'_ = sign(qoi)sign(_i) t __2g jc_ + (g/a)2 [t __2g J - t __2 jjclucuJ

= sign(_i)sign(_j){/)(_)%y + T(._)c,y%j}, (102)

where

ZZl ~

[ __/_2g J'

(103)

and

_(_) _ 1 ( [#_"(_) ] "t, (_/a)_ _(_)- t -_ JJ" (104)

On the other hand, if we are using the true arc length for horizontal separation, i.e., g = s,

we can substitute using (38) to obtain,

( 8/a zz!

sign(_°i)sign(_J) / sin('s-/a)[Pijt-:?_jc,, (S)]

f#_;"(_)_l

+sin2(s/a)i sin(s/a) s/a [#C/'(S)lcos(s/a [ -_s J )_ t_J]

sign(,.pi) sign(qai){R(s)%_ + r(s)ci_%j},

CiyCyj }

(105)

where

s/_ [_,fi'(S)l (106) R(s)- sin(4a_t-Z_ J'

and

T(s) -- 1 [_,_7"(s) 11

sin_(s/a){R(s)cos(s/a)- t---Z_--._._" (107)

The computational algorithms for all of the various correlations will require the exact

computation of gii = (sij or hii) and then table look-ups for _zz(_q), /_(sij), :F(sq)]

or _ZZ(sq), R(s0), T(sij)] as appropriate. Note that in Appendix B we showed that

lim_-_o #_" (s) = limi-_0 p_ (_) = 1, and also that lim_-_0 R(s) = lim_-,0/_(_) = 1. We can

also show that lims__o T(s) = lim_o T(_) = 2(ac4)_r/+ (ac_)27 + (a_)_r/(1 + 32) where _, 7

3O
and r/are definedin (65)- (67). This limit at _ = 0 can be used when tabulating T(s) or

a2 ^ __zzlI[ 7,'I

{ f'_J _°Jll lim T(_) --- lim n(_) -

.i-.-_O .i-.}O _ L ___2 J J'

which according to (75) and (76) can be written

a 2

--- _-_01im_-f_TBC_ + A#rl(1 + cs)D - B7 cos c2_ + 27c5BCD ).

Using (73) and (74) this can be written

a 2

= _-_01im2a2c24A_Aprl(l_ + c5)+ 27c5BC} - _-_olim-_TBC',

and, since lim_o[C'/_] = - 51c2, 2 we get

"Ya2 c 2

lim_--+0 T(_) = 2a_c_[_(1+ c5)+ 27c5]+ _ 2

7 (108)

_-- 2(ac4)2r/q- (ac2)2"y -{- (a_)2T/(1 q- _).

For the damped cosine parameters given in (58) - (62), we get lim___o T(k) _ 219.2. Note,

however, that in (102) and (105) limj_,i ciucuj = 0 and limj_,/cuu = 1, so that limj__i/_j_ =

1.

In a manner completely analogous to our treatment of #_._, we obtain the following formulas

from (81) - (89) for the background error correlations:

zz u_j = #-(_)

#ij•" = ( ) _k(_ ) -sign _oj a ij civ

#i_z = - sign (_oi)aj3 k( gij ) c_j

z.'O

#ij --

T.IZ

#ij -- -]-

#i_" = +

#_" = -

tttl

#ij = --

#i_" = +

(109)

(110)

(111)

[ ciz ] (112) + sign (_'.i)al3R(gij) tcos _j

[c_j ] (113) sign (_oi)a/31_(gij) Lcos qoi"

sign (_i_j) {R(gij)cy_ + T(sij)CiyCyj ) (114)

sign(_i_j){R(sij)[_] q- T(gij)cyj[_]) (115)

sign(qai_j){R(gij)[_] + T(s/J)Itcos c=j , 1c'u$ _. (116)

sign(_i_j){R(gij)[: czr ]+ T(gii)[_][ cix ]_. (117) cos qai cos _j tCOS_pj _

C.4 Decoupllng of the geostrophic assumption in the tropics

So far we have made uniform use of the geostrophic assumption as expressed in (80). This

assumption is clearly not valid near the equator, since it requires a division by f = 2f_ sin _.

This leads to the ratio of wind to height errors, a"/a _ = g_/[f[, in (97) becoming unbounded

31
neartheequator,and to discontinuities at the equator in the definitions of the correlation

functions given in (110) - (117).

In equations (110) - (117) for the height-wind error correlations, the discontinuity of the

factor sign(qo) at the equator is removed, as in Baker et al. (1987), by making the approximation

(aft) sign(_)_ H(q0)--(a/3)sign(qo){1- exp(-1_/9 0 }. (118)

For the wind-wind correlations in (114) - (117) the factor sign(qoi_j) is replaced with 1, i.e.,

the value it has when the two points are in the same hemisphere. Our final equations for

the height-height, height-wind, and wind-wind error correlations for the upper-air analysis

in the tropics (15S _< _o< 15N) are 1

ZZ #ij = I_zz(gi.i)

i_i7 = -H (qoj)R(gij)ciy

_ijzo = +H(_j)/_(g iS)[_1

#'7 = +H(_pi)fft(gij)[ c_j ]

t COS _i J

.,7 - 1 ,.1 t COS _0i J t COS (_i J -J

tcos _Picos q0j tcos _ojJ }

(119)

(120)

(121)

(122)

(123)

(124)

(125)

(126)

(127)

Note that if we set hi = _* and _i = _a* = -90 ° or 90 °, and let (_S,)_S) -+ (_*,'k_) we

obtain the following limiting values #izj_ = 1, #iS_- = #ij_'_ = I_ijz_ = #ij_'_ = O,/_u = I_ij_ =

cos(,_- )_), #i_v = sign(q0*)sin(,_- _) and #_ju = _ sign(qa*)sin(,_'- ,_). Contour plots

of the correlation functions defined in (120) - (127) for ,_ = 0 ° and _ E {10 °, 40 °, 90 °}

are shown as functions of _S and _j in figures 7--12.

The wind component error standard deviations given by equation (97) are modified in the

tropics as follows. In the tropical band, 15°S <_ qo < 15°N, a_ and a_ are assigned the

value at,.o_,'_'vgiven in Table 5. At points in the subtropical bands, 25°S _< _o < 15°S and

15°N < _o < 25°N, a_' and a_ are defined by

(128)

where f2s = 2ftsin(25°) •

Inadvertently, the frozen system that produced the five-year reanalysis did not remove the discontinuity

across the equator associated with the factor sign(q_) in the wind-wind correlations. This problem has been

corrected in subsequent versions of the system (see section 9).

32
_W

_W

ZV

................ i mm

VZ

......... i

i !iii--iii-iiiii-i-,

m_

mw

Figure 7: Graphs of the functions Pijzu, #ij'_z_ Pijz_',and #ij_*for _oi = 100 and ),i = 00.

33
UV

qbm, W

uk,ml U

Figure 8: Graphs of the functions/_u, #ijuv,/tij_, and llijvv for 9_i = 10° and hi = 0 °.

34
ZU

_m

ZY

O_*W

UZ

D

mU

YZ

......... i......... i......... i....

Figure 9: Graphs of the functions/zij, ._ #ij,/zij, _z zv and #ij_Zfor_oi=40 ° and.ki=0 °.

35
UU UV

enl_m

II

i

i

ii

dg t D 6

¥U

e.

_m

mM

L

-m

E

j

mW

V¥

uu uv "_ nnd _Vforq_i=40o and hi=0 °. Figure 10: Graphs of the functions #ij,/_ij, _ij , _ij

36
ZU UZ

......... I/

: ,,ir'""

mW mW

Z¥

v ....... _i....i  ̧__v

_ ! i_:i._ _,

VZ

m_ mW

Figure 11: Graphs of the functions #i_", #i_z,/_i_ ", and #i__ for _oi = 90 ° and Ai = 0 °.

37
till

_ <!i_ • _;ii?: _ __

U¥

!

_m

mat_

¥1J

¥¥

mW

_W

Figure 12: Graphs of the functions/t_ _, #_j_, #_'_', and #i_j" for _i = 90 ° and ,_ = 0 °.

38
Appendix D: List of Acronyms

ACARS

AGCM

AIDS

ARINC

ASDAR

CFL

DAAC

DAO

DAS

EOS

GCM

GEOS-1 DAS

GEOS-1 GCM

GLA

HIRS

IAU

LAPACK

MLS

MSU

NAVAIDS

NESDIS

NMC

NOAA

NRC

OI

PBL

RAS

RMS

SSM/I

SSU

STRATAN

TIROS

TOVS

TOVS A

TOVS B

TOVS C

UARS

UTC

ARINC Communications Addressing and Reporting System

Atmospheric General Circulation Model

Aircraft Integrated Data System

Aeronautical Radio, Incorporated

Aircraft to Satellite Data Relay

Courant-Friedrichs-Levy (commonly used computational

stability condition)

Distributed Active Archive Center

Data Assimilation Office

Data Assimilation System

Earth Observing System

General Circulation Model

Goddard EOS-Version 1 Data Assimilation System

Goddard EOS-Version 1 General Circulation Model

Goddard Laboratory for Atmospheres

High Resolution Infrared Sounder

Incremental Analysis Update

Linear Algebra Package

Microwave Limb Sounder

Microwave Sounding Unit

Navigation Aids (in Table 1 this refers to a special class of pilot balloons)

National Environmental Satellite, Data, and Information

Service

National Meteorological Center

National Oceanic and Atmospheric Administration

National Research Council

Optimal Interpolation

Planetary Boundary Layer

Relaxed Arakawa-Schubert

Root Mean Square

Special Sensor Microwave Imager

Stratospheric Sounding Unit

Stratospheric Analysis system

Television Infrared Observing Satellite

TIROS Operational Vertical Sounder

TOVS clear sky retrieval

TOVS partly cloudy sky retrieval

TOVS cloudy sky retrieval

Upper Atmospheric Research Satellite

Universal Time Coordinated

39

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44


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January 1995 Technical Memorandum

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Technical Report Series on Global Modeling and Data Assimilation

Volume 4 - Documentation of Goddard Earth Observing System

(GEOS) Data Assimilation System - Version I C - NAS5-32332

6. AUTHOR(S)

James Pfaendtner, Stephen Bloom, David Lamich, Michael Seablom,

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Laboratory for Atmospheres

Data Assimilation Office

Goddard Space Flight Center

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11. SUPPLEMENTARYNOTES

Pfaendtner and da Silva: Goddard Space Flight Center, Greenbelt, Maryland;

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13. ABSTRACT (Maximum 200 words)

This report describes the analysis component of the Goddard Earth Observing System, Data Assimilation System,

Version 1 (GEOS- 1 DAS). The general features of the data assimilation system are outlined, followed by a thorough

description of the statistical interpolation algorithm, including specification of error covariances and quality control of

observations. We conclude with a discussion of the current status of development of the GEOS data assimilation system.

The main components of GEOS- I DAS are an atmospheric general circulation model and an Optimal Interpolation

algorithm. The system is cycled using the Incremental Analysis Update (IAU) technique in which analysis increments

are introduced as time independent forcing terms in a forecast model integration. The system is capable of producing

dynamically balanced states without the explicit use of initialization, as well as a time-continuous representation of nonobservables such as precipitation and radiational fluxes. This version of the data assimilation system was used in the

five-year reanalysis project completed in April 1994 by Goddard's Data Assimilation Office (DAO) (Schubert et al.,

1993) Data from this reanalysis are available from the Goddard Distributed Active Center (DAAC), which is part of

NASA's Earth Observing System Data and Information System (EOSDIS). For information on how to obtain these data

sets, contact the Goddard DAAC at (301) 286-3209, EMAIL daac@ gsfc.nasa.gov, or consult DAO's Mosaic Home Page

URL http://hera.gsfc.nasa.govldao.home_page.html.

This document is available e!ectronically via anonymous tip from dao.gsfc.nasa.gov, directory pub/tech_memos, file

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14.SUBJECTTERMS

Optimum interpolation, Data assimilation, Analysis, EOS, Statistical

interpolation, Observations, GEOS

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