Observation Sensitivity Calculations Using the Adjoint of the Gridpoint … · 2008-05-28 ·...

Observation Sensitivity Calculations Using the Adjoint of the Gridpoint StatisticalInterpolation (GSI) Analysis System

YANQIU ZHU* AND RONALD GELARO

Global Modeling and Assimilation Office, NASA Goddard Space Flight Center, Greenbelt, Maryland

(Manuscript received 6 October 2006, in final form 23 February 2007)

ABSTRACT

The adjoint of a data assimilation system provides an efficient way of estimating sensitivities of analysisor forecast measures with respect to observations. The NASA Global Modeling and Assimilation Office(GMAO) has developed an exact adjoint of the Gridpoint Statistical Interpolation (GSI) analysis schemedeveloped at the National Centers for Environmental Prediction (NCEP). The development approach isunique in that the adjoint is derived from a line-by-line tangent linear version of the GSI. Availability of thetangent linear scheme provides an explicit means of assessing not only the fidelity of the adjoint, but alsothe effects of nonlinear processes in the GSI itself. In this paper, the development of the tangent linear andadjoint versions of the GSI are discussed and observation sensitivity results for a near-operational versionof the system are shown. Results indicate that the GSI adjoint provides accurate assessments of thesensitivities with respect to observations of wind, temperature, satellite radiances, and, to a lesser extent,moisture. Sensitivities with respect to ozone observations are quite linear for the ozone fields themselves,but highly nonlinear for other variables. The sensitivity information provided by the adjoint is used toestimate the contribution, or impact, of various observing systems on locally defined response functionsbased on the analyzed increments of temperature and zonal wind. It is shown, for example, that satelliteradiances have the largest impact of all observing systems on the temperature increments over the easternNorth Pacific, while conventional observations from rawinsondes and aircraft dominate the impact on thezonal wind increments over the continental United States. The observation impact calculations also providean additional means of validating the observation sensitivities produced by the GSI adjoint.

1. Introduction

Modern atmospheric data assimilation systems ingestmillions of observations each day to produce initial con-ditions for weather and climate forecasts. The vast ma-jority of the observations are from satellites, the num-ber and variety of which will continue to increase sig-nificantly during the next decade. Because it is unlikelythat even next-generation data assimilation systems willbe able to accommodate all available observations,there is an increasing need to develop intelligent strat-egies for data selection and utilization. Even now, theneed for such strategies is made clear by the gross un-

derutilization of the current observation set. For ex-ample, operational forecast centers that assimilate At-mospheric Infrared Sounder (AIRS) data use roughlyone out of every several thousand of the available mea-surements provided by the instrument (Goldberg et al.2003). At the same time, these observations have in-creased the total number of assimilated observationssignificantly, while producing small to moderate gainsin forecast skill (Le Marshall et al. 2006). Even if muchof the data are redundant and could be compressed byseveral orders of magnitude, it is unlikely that currentdata selection strategies adequately capture the avail-able information. To make optimal use of the increas-ing volume of observations, flexible and efficient toolsfor quantifying the “value” of observations are re-quired.

A common method for assessing the observationvalue in the context of numerical weather prediction isto perform so-called observing system experiments(OSEs), in which selected subsets of observations areremoved from a data assimilation system. This is a di-

* Current affiliation: Science Applications International Corpo-ration, Beltsville, Maryland.

Corresponding author address: Dr. Ronald Gelaro, GlobalModeling and Assimilation Office, NASA Goddard Space FlightCenter, Greenbelt, MD 20771.E-mail: [email protected]

JANUARY 2008 Z H U A N D G E L A R O 335

DOI:10.1175/2007MWR2046.1

© 2008 American Meteorological Society

MWR3525

rect way to measure the value of such subsets on fore-casts and assimilation products. Meaningful compari-sons require an appreciable spinup period, followed bya long period over which resulting output statistics arecomputed. Interpretation of results in terms of analysisquality generally requires the production of short-termforecasts. OSEs are intermittently performed at opera-tional centers (e.g., Lord et al. 2004; Kelly et al. 2004;English et al. 2004) but, because of their expense, usu-ally involve relatively small numbers of independentexperiments, each considering variations in only largesubsets of observations. So, for example, it is prohibi-tive to investigate the impacts of all individual channelsfor a given satellite observing system. In addition, eachvariation of the observing system changes the gain, orrelative weights given to the observations, with respectto the original baseline experiment.

Recently, Baker and Daley (2000) have shown thatthe adjoint of a data assimilation system provides anefficient way to estimate the sensitivities of an analysisor forecast measure with respect to observations. Thesensitivities may be computed with respect to any or allof the observations simultaneously based on a singleexecution of the adjoint system. This permits arbitraryaggregation of the sensitivities, for example, by datatype, channel, or location. It also allows for the estima-tion of the impact of any subset of data on an analysisor forecast measure. This approach has been used todiagnose the effectiveness of specific targeted observa-tions (Doerenbecher and Bergot 2001; Fourrie et al.2002), as well as to perform comprehensive assessmentsof observing system impacts on short-range forecast er-rors (Langland and Baker 2004). Other, somewhat re-lated, methods for estimating observation sensitivity in-clude the second-order adjoint approach proposed byLe Dimet et al. (1995), the data resolution matrix(Menke 1984), the entropy reduction method (Rabieret al. 2002), and the influence matrix diagnostic (Car-dinali et al. 2004).

The key to the adjoint approach is to compute thetranspose of the gain matrix that determines theweights given to the observation-minus-background re-siduals, either explicitly or through a sequence of avail-able operators. For modern data assimilations, the sizeand complexity of these operators render this task non-trivial. There are, however, several possible approachesto producing the adjoint, the suitability of which de-pend on the design of the assimilation system and theacceptability of any inherent assumptions. To date, forpractical reasons, most implementations have relied onmodification of the existing (forward) analysis solver toproduce an approximate adjoint, as opposed to the de-

velopment of a line-by-line tangent linear model fromwhich an exact adjoint is derived.

In this study, a tangent linear model and exact adjointof the National Centers for Environmental Prediction(NCEP) Gridpoint Statistical Interpolation (GSI)analysis scheme (Wu et al. 2002) are developed andtested in the context of the Goddard Earth ObservingSystem atmospheric data assimilation system (GEOSDAS; Rienecker et al. 2007). The GSI is expected tobecome the operational analysis scheme at both NCEPand the National Aeronautics and Space Administra-tion (NASA) Global Modeling and Assimilation Office(GMAO) in the near future. The choice to develop anexact adjoint is motivated by design aspects of the GSIalgorithm, as described in later sections. The currentstudy focuses on the development of the tangent linearand adjoint versions of the GSI, their validation, andpreliminary results.

Section 2 provides a brief description of the GSI al-gorithm as well as overviews of the theoretical andpractical aspects of developing the tangent linear andadjoint systems. In section 3 we examine the behaviorof the tangent linear model, and compare this with thebehavior of the full GSI in response to a range of per-turbations applied to the input innovations. In section 4we present observation sensitivity results produced bythe GSI adjoint, which in turn are used to estimate theimpact of various observing systems on selected mea-sures of the analyzed increments of temperature andzonal wind. As applied here, the observation impactcalculations serve primarily as a validation tool for theadjoint results. Concluding remarks and plans for fu-ture work are presented in section 5.

2. Problem formulation

a. The GSI algorithm

The GSI is a three-dimensional variational data as-similation (3DVAR) analysis scheme based on NCEP’scurrent operational Spectral Statistical Interpolation(SSI) system (Parrish and Derber 1992), but with thespectral definition of the background error covarianceoperator replaced by a gridpoint version based on re-cursive filters. The current implementation of GSI in-corporates a set of recursive filters that produce ap-proximately Gaussian smoothing kernels and isotropiccorrelation functions (Wu et al. 2002). However, bysuperpositioning Gaussian kernels with different lengthscales, it is possible to generate a large class of flow-dependent inhomogeneous background error covari-ance models as described by Purser et al. (2003a,b).

The analysis is obtained by minimizing the scalar costfunction:

336 M O N T H L Y W E A T H E R R E V I E W VOLUME 136

J �12

�x � xb�TB�1�x � xb�

�12

�h�x� � y�TR�1�h�x� � y� � Jq1 � Jq2 �1�

with respect to the control vector x(�, , T, q, oz, ln ps,Ts), where � is the streamfunction, is the unbalancedvelocity potential, T is the unbalanced virtual tempera-ture, q is the (scaled) specific humidity, oz is the ozonemixing ratio, ln ps is the logarithm of surface pressure,and Ts is the surface skin temperature. The vector xb

represents the background or prior estimate of x, and Bis its expected error covariance. The vector y containsthe available observations, the operator h(x) simulatesthese observations from x, and R is the expected co-variance of the instrument plus representativeness er-rors associated with the observations. The superscript Tdenotes the transpose operation.

The terms Jq1 and Jq2 are penalties for negative hu-midity and supersaturation, respectively, defined as

Jq1 � � 0 if q � 0

�1q2 if q � 0and �2�

Jq2 � � 0 if q � qs

�2�q � qs�2 if q � qs

, �3�

where qs is the saturation value of q, and �1 and �2 areparameters. To simplify the presentation, we omit theseadditional penalty terms from the development that fol-lows. These terms are however included in the actualtangent linear and adjoint versions of the GSI devel-oped for this study, and their impact on the sensitivitycalculations is examined in section 3.

Because h(x) is generally nonlinear, the most effi-cient means for minimizing J is through an incrementalapproach (Courtier et al. 1994) in which the problem isrepeatedly linearized about an updated reference solu-tion (the outer loop). A gradient-based iterative algo-rithm (the inner loop) is then used to minimize theresulting cost function:

Jk �12

��xk � �xb � xk��TB�1��xk � �xb � xk��

�12

�Hk�xk � dk�TRk� 1�Hk�xk � dk�, �4�

where k � 0, . . . , K is the outer loop index. The vari-ables

dk � yk � hk�xk�, �5�

and

�xk � xk�1 � xk, �6�

are the residual (or innovation) vector and increment,respectively. The matrix Hk is the Jacobian of h linear-ized about xk. In practice, a preconditioned conjugategradient descent algorithm with �100 inner iterationsand two outer loops is found to produce satisfactorilyconverged increments in most cases. The second outerloop accounts for changes in quality control (especiallyfor radiance data) and weak nonlinearities in some ob-servation operators (e.g., surface wind speed) but, gen-erally speaking, produces relatively small changes tothe analysis increments. In this paper, we focus on ob-servation sensitivities based on tangent linear and ad-joint versions of the GSI with a single outer loop itera-tion. Multiple outer loops will be addressed in a sequel.

b. Observation sensitivity

Setting Jk/ �xk � 0 in (4) and neglecting all but thefirst outer loop, we obtain the analytical form of theanalysis increment:

�x0 � �B�1 � H0TR0

� 1H0��1H0TR0

� 1d0, �7�

where x0 � xb. Substituting (5) and (6) into (7) andlinearizing about x0, we obtain the tangent linear analogof the analysis increment:

�x � K�y � Hxb�, �8�

where K is the gain matrix:

K � �B�1 � HTR�1H��1HTR�1. �9�

Here, the tildes denote tangent linear variables and thesubscript for the outer loop index has been dropped forconvenience. Following Baker and Daley (2000), wedefine the sensitivity of the analysis increment with re-spect to the observations as

��x�y

� KT, �10�

where KT is referred to as the analysis adjoint. By ap-plication of the chain rule, the sensitivity of any scalaraspect J, not to be confused with J in (1), of either theanalysis or forecast1 with respect to the observations isgiven by

�J

�y� KT

�J

�x. �11�

Note that KT maps a vector in physical space to a vectorin observation space, while the mapping by K is in theopposite sense. Moreover, (11) indicates that, for a

1 If J is based on a model forecast, then the calculation of J/ xin (11) will generally require the model adjoint. For the purposesof this discussion we need only assume that J/ x exists.


given J, the sensitivity can be computed with respect toany or all observations simultaneously with a single ex-ecution of the adjoint system. This permits arbitraryaggregation of the results (e.g., by data type, location,channel, etc.). From (9), we see that the sensitivity de-pends on the characteristics of the assimilation systemand on attributes of the observations such as their lo-cations and assumed errors, but not on the observedvalues themselves. In contrast, the “impact” or ex-pected change in J produced by assimilating the obser-vations will depend on both the sensitivities and ob-served values. We examine measures of the observationimpact in section 4.

c. Development of the adjoint system

For modern data assimilation systems that includecomplex observation operators and large numbers ofobservations, estimation of KT is nontrivial. There is no“best” approach, but rather several possible ap-proaches, the suitability of which depend on the designof the system (e.g., is it formulated in physical or ob-servation space?) and careful consideration of the un-derlying approximations and assumptions.

For example, in the observation space–based dataassimilation system developed at the Naval ResearchLaboratory (NAVDAS; Daley and Barker 2001), theanalysis increment is obtained by solving an alternativeform of (7) given by

�x � BHTz, �12�

where

z � �HBHT � R��1d. �13�

An iterative algorithm is used to obtain the vec-tor z with d as input, where the gain matrix is K �BHT(HBHT � R)�1. Because (HBHT � R)�1 is self-adjoint and the operators H and B are explicitly avail-able in this formulation, the sensitivity with respect toobservations can be calculated using nearly the samealgorithm, but with the columns of HB (or the vectorHB J/ x) replacing d as input. The obvious appeal ofthis approach is that it requires only minor modificationof the existing forward analysis code. A caveat is thatthe modified algorithm solves a different minimizationproblem than that used to obtain the analysis incre-ment. Thus, care should be taken when choosing theconvergence or stopping criterion since, strictly speak-ing, the correct adjoint is obtained only when the solu-tions are completely converged. This caveat notwith-standing, this approach has been used effectively atNRL (e.g., Langland and Baker 2004).

In the GSI, (4) is minimized directly, so there is noanalog of (13) which can be easily manipulated to ob-

tain KT � R�1H(B�1 � HTR�1H)�1. While it is pos-sible, in principle, to reformulate the cost function toobtain an adjoint, this would essentially amount tobuilding another data assimilation system in observa-tion space having little in common with the GSI. Inparticular, the preconditioning and handling of qualitycontrol operations and observation error assignmentperformed in the inner and outer loops of the GSIwould differ substantially in the reformulated system. Itis unclear how inconsistencies in these operations withrespect to their counterparts in the forward algorithmwould affect the accuracy and interpretability of theadjoint results.

For these and other reasons it was decided to developan exact adjoint of the GSI, that is, based on an exacttranspose of a line-by-line tangent linear model (TLM)of the forward minimization algorithm. The approach isanalogous to that generally used to derive the adjoint ofa numerical forecast model. In this case, the iterationsof the inner loop, including the intermediate analysisincrements and the sequence of conjugate gradient di-rections, serve as the trajectory for the TLM and ad-joint systems. The correctness of the adjoint is evalu-ated using the equality �x, Gy� � �GTx, y�, where G isthe TLM of the GSI; GT is the adjoint; x and y areperturbation vectors of the analysis increments and ob-servations, respectively; and �, � denotes the Euclid-ian inner product. Availability of the TLM provides adirect means of assessing not only the mathematicalcorrectness of the adjoint, but also its usefulness fordescribing the behavior of the GSI with respect to awide range of perturbations. This is examined in detailin sections 3 and 4. Another benefit of this approach isthat it can be extended in the future (e.g., to examinesensitivities with respect to observation error variancesand other parameters in the GSI).

Development of an exact adjoint generally requires asignificant initial development effort. In particular, thedevelopment of the tangent linear and adjoint versionsof the conjugate gradient descent algorithm requirescareful consideration of the nonlinear procedure forgenerating the sequence of conjugate directions andstep sizes used in the minimization. Other sources ofnonlinearity in the inner loop include the observationoperator for wind speed and precipitation rate (al-though we do not assimilate precipitation observationsin the current study). Also, as we show in section 3, themoisture penalty terms Jq1 and Jq2 act like switches thatmay introduce strong nonlinearity in the GSI.

3. Tangent linear experiments

The first step in evaluating the usefulness of the GSIadjoint is to determine whether the TLM accurately


describes the behavior of the GSI in response to mean-ingful perturbations. The accuracy of the TLM may beassessed by comparing the TLM responses, �x, corre-sponding to a given set of perturbed innovations withthe differences between the increments, �x, producedby the GSI with and without these perturbations. Theprimary measures of agreement used here are the ratioof the root-mean-squared (RMS) values of �x and �x,and the correlation between �x and �x. If the GSI be-haves approximately linearly, then both the amplitudeand structure of the TLM response and perturbed GSIdifferences should agree well (i.e., both the ratio andcorrelation measures should be close to 1). Exact agree-ment is not expected owing to the various sources ofnonlinearity in the GSI described in section 2.

The perturbed innovations have the form d� � (1 ��)d, where the parameter � determines the perturba-tion amplitude. Because different analysis variablesmay exhibit valid tangent linear behavior for differentperturbation amplitudes and perturbed innovationtypes, the choice of � requires careful consideration. Inthis study, we examined results for values of � rangingfrom 10�6 to 1, applied to innovations of individualobservation types as well as all observation types col-lectively. We focus most of the discussion in this sectionon results for � � 0.1 because these are sufficientlyrepresentative of the results obtained for the range ofvalues tested. Statistics produced by the GEOS DAS(not shown) indicate that while global RMS values ofthe innovations for most observation types vary byroughly 10%–50% from day to day, individual innova-tions can exhibit much larger variations. The responseto actual innovations is examined in the context of theobservation impact results presented in section 4.

Results are presented for August 2004 based onanalyses produced at 0000 UTC. For practical reasons,we use a relatively low horizontal resolution version ofGSI corresponding to 1.875° in latitude and longitude,with 64 vertical levels defined on � surfaces. The back-ground forecast is provided by the GEOS-5 model(Rienecker et al. 2007). Analyses are produced using a6-h assimilation cycle that includes all conventional ob-servations and satellite radiances assimilated operation-ally during the study period. Conventional observationsare assimilated from rawinsondes, aircraft, surfaceships, and land stations. Additional conventional datatypes include winds from geostationary satellites, pro-filers, and scatterometers, wind speeds from the SpecialSensor Microwave Imager (SSM/I), and ozone obser-vations from the Solar Backscatter Ultraviolet (SBUV/2) instrument. Satellite radiances are assimilated fromthe Television and Infrared Observation Satellite(TIROS) Operational Vertical Sounder (TOVS) Ad-

vanced Microwave Sounding Unit (AMSU-A andAMSU-B), the High-Resolution Infrared RadiationSounder (HIRS-2 and HIRS-3), the Geostationary Op-erational Environmental Satellites (GOES), and theMicrowave Sounding Unit (MSU). In accordance withthe development in section 2, both the GSI and TLMare run with a single outer loop.

Figure 1 shows daily time series of the RMS ratios ofthe TLM response and perturbed GSI differences forseveral variables at analysis level 25 (approximately 500hPa) for experiments in which the innovations for allobservation types have been perturbed by 10%. Theresults are representative of those at other vertical lev-els. The ratios are close to 1 for all variables on mostdays except 2 and 29 August. On these days, the specifichumidity response of the TLM is roughly 3 times largerthan the perturbed GSI differences, although the ratiosfor the remaining variables are much closer to 1. Thedisagreement in the humidity responses for these casesis examined in more detail below. Ratios slightly largerthan 1 also occur on a few other days for some vari-ables, but the responses are generally in good agree-ment.

An example of the response to perturbed innovationsof an individual observation type is shown in Fig. 2 forthe case of 6 August. This example is representative ofthe good agreement between the TLM and perturbedGSI differences observed in the vast majority of cases.The figure shows the TLM response (Fig. 2a) and GSIdifferences (Fig. 2b) in terms of zonal wind at level 30(approximately 300 hPa) when all satellite radiancesare perturbed by 10%. In this example it can be seenthat the perturbed radiances have a much larger impactin the Southern Hemisphere, reflecting the greater in-

FIG. 1. The ratio between the RMS TLM response and GSIdifference at analysis level 25 during August 2004 when all of theobservation innovations are perturbed by 10%.


fluence of satellite data where conventional observa-tions are more sparse.

The moisture penalty terms in (2) and (3), althoughstrongly nonlinear, are continuous and differentiable innature, with continuous first derivatives. In most cases,these terms make adjustments that are therefore wellmodeled by the TLM. However, in 2 of the 31 casesexamined, these terms exhibited strong nonlinearity toa degree that degraded the TLM solution noticeably.Figure 3a shows the TLM response (contours) and per-turbed GSI differences (shaded) in terms of specifichumidity at approximately 500 hPa for the poorly mod-eled case of 29 August noted earlier. The GSI differ-ences are distributed across all longitudes with maxi-

mum variance in the tropics and subtropics. The TLMresponse bears no likeness to the GSI differences, withtwo isolated extrema over the Gulf of Alaska and theequatorial Atlantic Ocean.

Figure 3b is similar to Fig. 3a, except that the mois-ture penalty terms have been excluded from both theTLM and GSI for this single analysis cycle. In this case,the TLM response and perturbed GSI differences agreewell. Moreover, it can be seen that the GSI differenceswith and without the penalty terms are very similar,indicating that the penalty terms have not changed theGSI increments themselves significantly during oneanalysis cycle. Direct comparison of the GSI incre-ments with and without the moisture penalty terms for

FIG. 2. The zonal wind at analysis level 30 at 0000 UTC 6 Aug for the (a) TLM response and(b) GSI difference when all radiance innovations are perturbed by 10%. The contour intervalis 0.08 m s�1. Negative contours are dashed and the zero contour is omitted.


this analysis cycle (not shown) reveals that their differ-ences are approximately an order of magnitude smallerthan the increments themselves.2 Because the TLMwithout the penalty terms is capable of representing thebehavior of the GSI without the penalty terms, andbecause the results from the GSI with and without thepenalty terms for a single analysis are quite similar, the

TLM (or adjoint) can be run without the penalty termswhen necessary and still produce a reasonable estimateof the sensitivity. In such cases, it is not necessary toremove the penalty terms from the GSI analysis itself.In more recent versions of the GSI than the one used inthis study, the control variable for humidity has beenredefined in a way that reduces the need for these pen-alty terms substantially (J. C. Derber 2006, personalcommunication).

The average correlations between the TLM re-sponses and GSI differences at approximately 500 hPa

2 If the GSI is run with two outer loops, the differences canbecome somewhat larger, but are still significantly smaller thanthe values of the increments in most locations.

FIG. 3. The specific humidity at analysis level 25 at 0000 UTC 29 Aug for the TLM response (contours) and GSIdifference (shaded) when all innovations are perturbed by 10%. The moisture penalty terms in the TLM and GSIare (a) included and (b) excluded. The values have been multiplied by 103, with negative contours dashed and thezero contour omitted.


Fig 3 live 4/C

for the month of August 2004 are summarized in Table 1.The top row lists the perturbed innovation types, whichcorrespond to satellite radiances (rad), followed byconventional observations of virtual temperature (T),zonal wind (u), meridional wind (), specific humidity(q), ozone mixing ratio (oz), surface pressure (ps), andnear-surface wind speed (spd), plus all observationtypes collectively (all). The leftmost column lists theanalysis increment variables. The numerical valuesshow the average correlations between the TLM re-sponse and GSI differences for each analysis incrementvariable on the far left in response to the perturbedinnovation type at the top. Except for results in therightmost column (all), only the specified innovationtype has been perturbed for each set of experiments,while the remaining innovation types are left unper-turbed.

Overall, the results show that the GSI behavior iswell represented by the TLM, consistent with the RMSratios in Fig. 1. The results shown here are representa-tive of those at other vertical levels. The vast majorityof the correlations are close to or greater than 0.9, withsome notable exceptions involving the responses ofsome variables to perturbed innovations of ozone and,to a lesser extent, specific humidity. For example, whenthe ozone innovations are perturbed, the average re-sponse correlation is greater than 0.9 for ozone itself,but substantially lower for most other variables. Thisoccurs because, while the inner-loop problem in theGSI is linear (except for certain observation operatorsand the moisture penalty terms), the minimization al-gorithm used to solve this problem is nonlinear. Ozoneand moisture are not strongly coupled to other vari-ables through the background error covariance opera-tor, B, and so cross responses involving these variablesmay be significantly affected by the nonlinear nature ofthe minimization algorithm. In contrast, cross responsesbetween wind, temperature, and pressure are stronglycoupled through B and therefore not significantly af-

fected by nonlinearity. The direct responses of all vari-ables to perturbations of the same observation variabletend to be strong and linear as a result of the diagonalelements of B. Because cross responses tend to beweaker than direct responses in general, the correla-tions for all variables are high when all the innovationsare perturbed simultaneously (rightmost column).

4. Adjoint experiments

Having demonstrated the ability of the TLM to rep-resent the general behavior of the GSI, we now exam-ine results produced by the adjoint. In contrast with theTLM, which takes a perturbation �d in observationspace as input and produces a perturbation �x in analy-sis space as output, the adjoint takes a gradient J/ x ofa response function J in analysis space as input andproduces a gradient (or sensitivity) J/ y in observationspace as output. The response function J can be anydifferentiable scalar measure of interest defined glo-bally or for a particular region of interest.

In this study, we examine results for four responsefunctions of the following form:

J �12 ��x, S�x�, �14�

where S is a projection operator that selects only asubset of the total analysis increment �x. The first pairof measures, denoted JTNP

and JUNP, include only incre-

ments of either temperature or zonal wind, respec-tively, at all vertical levels in a 23° � 60° box centeredover the eastern North Pacific. The second pair, de-noted JTUS

and JUUS, are similar to the first, but for a box

centered over the continental United States (see Figs. 4and 6). The measures, while not necessarily of meteo-rological significance, allow us to examine sensitivitiesof the wind and temperature increments in regionswhere the mixture of satellite and conventional obser-vations differs markedly. The quadratic forms of these

TABLE 1. Average correlations between the TLM responses and GSI differences at analysis level 25 for the month of August 2004when the innovations have been perturbed by 10%. See text for details.

Incrementvariable

Perturbed innovation type

rad T u q oz ps spd all

T 0.967 0.991 0.931 0.897 0.973 0.787 0.914 0.881 0.971u 0.966 0.973 0.998 0.970 0.947 0.781 0.984 0.965 0.989 0.969 0.964 0.995 0.984 0.943 0.799 0.983 0.955 0.987� 0.958 0.972 0.993 0.972 0.944 0.784 0.985 0.975 0.982 0.877 0.946 0.999 0.992 0.944 0.764 0.959 0.896 0.994q 0.944 0.877 0.680 0.720 0.999 0.630 0.749 0.726 0.925oz 0.897 0.816 0.458 0.478 0.935 0.999 0.762 0.505 0.998ln ps 0.890 0.890 0.989 0.969 0.910 0.728 0.999 0.986 0.986Ts 0.996 0.996 0.978 0.947 0.998 0.909 0.981 0.972 0.994


measures prevent cancelation due to increments of op-posite sign and have the added convenience that theirgradients with respect to the analysis, required as inputto the adjoint, are equivalent to the increments them-selves where the response functions are defined.

a. Observation sensitivity results

Observation sensitivities were computed once eachday for each response function based on the 0000 UTCanalyses for the month of August 2004. The observingsystem and resolution of the GSI adjoint are the sameas those used for the TLM experiments in section 3.Figures 4–6 show examples of JTNP

/ y and JUUS/ y for

selected observing systems on 5 August. The results forthis case are representative of those throughout thestudy period. They are presented here to highlight basiccharacteristics of the sensitivities including their depen-dence on the type and location of the observations, andon the density of surrounding observations. In fact, thesensitivities depend in a complex way on all aspects ofthe gain matrix K in (9). The reader is referred to Baker(2000) and Baker and Daley (2000) for more detaileddescriptions of these dependencies.

Figure 4a shows the sensitivity of JTNPto rawinsonde

temperature observations at 500 hPa. Note that the sen-sitivities have units of the gradient JTNP

/ y that, fortemperature observations, are K2/K � K. To highlightobservations with the greatest sensitivity, values withinone contour interval of 0 are shaded gray. The box overthe eastern North Pacific outlines the area where JTNP

isdefined; for convenience we refer to this as the targetarea in the discussion that follows. There are no rawin-sonde observations located within the target area inFig. 4a. However, JTNP

is sensitive to several rawinsondetemperature observations to the northeast of the targetarea, along the west coast of North America. In par-ticular, the sensitivity with respect to the observationover Vancouver Island is close to 6.5 K, implying that,to first-order accuracy, a 1-K increase (decrease) in thetemperature of this observation would increase (de-crease) JTNP

by approximately 6.5 K2. Observationsalong the U.S. west coast, while equally close to thetarget area, have much smaller sensitivity values. This ismost likely due to the greater density of observationsover the western United States compared with westernCanada, which tends to reduce the sensitivity to indi-vidual observations. Because of the global nature of the3DVAR solution, observations far removed from thetarget area exhibit small, but nonzero, sensitivity val-ues.

Figure 4b shows the sensitivity of JTNPto channel-5

brightness temperatures on NOAA-16 AMSU-A.

These observations provide temperature information ina deep vertical layer that peaks in the midtroposphere.In this case, observations from three orbits of the sat-ellite lie within the target area, producing the largestsensitivity values of all observation types examined forthis case. The sensitivities exhibit a complex pattern ofpositive and negative values, reflecting small-scale fea-tures in the increment field at various vertical levels. Amore detailed view of this dense pattern in the vicinityof the target area is shown in Fig. 5, along with thesquared analysis increments of temperature at approxi-mately 550 hPa used to compute the response functionJTNP

. This level is close to where channel 5 on AMSU-Ahas its peak response. There is a clear correspondencebetween the extrema in the increments and large valuesof observation sensitivity. An exact correspondence isnot expected because the sensitivities also depend onthe increments at other vertical levels, as well as onvarious aspects of K as noted earlier.

Figure 4c shows the sensitivity of JTNPto GOES in-

frared cloud drift observations of zonal wind at 500 hPa.There are several observations within or near the targetarea that exhibit the largest sensitivity values for thisobservation type. These values are one to two orders ofmagnitude smaller than the maximum sensitivity valuesfor the AMSU-A radiances or rawinsonde tempera-tures in Figs. 4a,b (although, strictly speaking, it is dif-ficult to make a quantitative comparison between sen-sitivities with respect to wind and temperature). Windobservations at this level affect the analyzed tempera-tures above and below this level indirectly through bal-ance conditions imposed by the background error co-variance. The results show that the sensitivity of JTNP

tothese wind observations is relatively small.

Moving to results for JUUS/ y, we show in Fig. 6a the

sensitivity with respect to rawinsonde zonal wind ob-servations at 500 hPa. The largest sensitivity values oc-cur along the edges of the target area where the obser-vation density is lowest. While the results are notstrictly comparable to those in Fig. 4a, the magnitudesof the responses in these figures suggest that tempera-ture and zonal wind increments are each similarly sen-sitive to rawinsonde observations of the same variables.Conversely, comparing the sensitivities with respect to(the same) cloud drift wind observations in Figs. 6c and4c, and comparing their magnitudes with those of theother observing systems in each figure, it is clear thatcross sensitivities may be considerably weaker. How-ever, the relatively large sensitivity of JUUS

with respectto channel-5 AMSU-A brightness temperatures shownin Fig. 6b indicates that cross sensitivities are not alwayssmall.


FIG. 4. The sensitivity of JTNPat 0000 UTC 5 Aug with respect to (a) rawinsonde temperature (K)

observations at 500 hPa, (b) channel-5 brightness temperatures (K) on NOAA-16 AMSU-A, and (c)GOES IR cloud drift zonal wind (K2 m�1 s) observations at 500 hPa. The box outlines the area in whichJTNP

is defined.


Fig 4 live 4/C

b. Observation impact

The sensitivity information produced by the GSI ad-joint can be used effectively to estimate the impact ofobservations on the response function J. As appliedhere, this provides not only a powerful diagnostic toolfor data assimilation, but also a means of verifying theaccuracy of the observation sensitivities themselves.

From (7) and (9), we can express the analysis incre-ment as

�x � Kd. �15�

Combining (15) and (11), and using the definition of anadjoint, we obtain

��J/�x, �x� � ��J/�y, d�. �16�

For quadratic measures of the form (14), we have J/ x � S�x that, when substituted into (16), yields

J �12 ��J��y, d�. �17�

Equation (17) is an estimate of J computed in ob-servation space, based on the inner product betweenthe observation sensitivities and the innovations. Theimpact of any or all observations on J is thereforeeasily computed by summing only the elements cor-

FIG. 5. Limited-area view of (a) the squared analysis increments of temperature at approximately 550 hPa(contour interval is 0.2 K2) and (b) the sensitivity of JTNP

with respect to channel-5 brightness temperatures (K) onNOAA-16 AMSU-A at 0000 UTC 5 Aug. The box outlines the area in which JTNP

is defined.


Fig 5 live 4/C

FIG. 6. The sensitivity of JUUSat 0000 UTC 5 Aug with respect to (a) rawinsonde zonal wind (m s�1)

observations at 500 hPa, (b) channel-5 brightness temperatures (K�1 m2 s�2) on NOAA-15 AMSU-A,and (c) GOES IR cloud drift zonal wind (m s�1) observations at 500 hPa. The box outlines the area inwhich JUUS

is defined.


Fig 6 live 4/C

responding to a selected set of observations. If allthe observations are included in (17), then the resultmay be compared with (14) as a measure of the accu-racy of the observation space estimate of J (and thusthe observation sensitivities). Figure 7 shows daily timeseries of this comparison for each of the response func-tions examined in this study. The values agree ex-tremely well in all cases and, in many cases, the curvesare nearly indiscernible. The results demonstrate theaccuracy of the adjoint response to realistic perturba-tions.

The accuracy of the observation space estimate of J,in turn, allows meaningful aggregation of the resultsaccording to observation type, location, channel, etc.Figure 8 shows a basic application of this capability. Inthis case, the total impact of the observations has beenseparated into the contributions from the satellite radi-ances and conventional observations for each of the

response functions examined in this study. For thetemperature increments over the North Pacific (Fig.8a), the abundant satellite radiances in this region(cf. Fig. 4b) affect the analysis of temperature sig-nificantly. The radiances account for most of the im-pact on all days except 13 August, when much of thepolar-orbiting satellite data were missing for technicalreasons. As might be expected, the situation is reversedfor the zonal wind increments over the North Pacific(Fig. 8b). For these increments, conventional windobservations (primarily from GOES-10 and commer-cial aircraft, not shown) dominate over the indirect im-pact of the radiances. Over the U.S. region (Figs. 8c,d),conventional observations have the dominant impacton both the temperature and zonal wind increments.This is not surprising given the abundance of observa-tions from both rawinsondes and aircraft over thecontinent, which are given significant weight in the

FIG. 7. Time series of (a) JTNP(K2), (b) JUNP

(m2 s�2), (c) JTUS(K2), and (d) JUUS

(m2 s�2) during August 2004calculated from the analysis increments directly (solid) and from the total observation impact estimate (dashed).All values have been multiplied by 10�3.


analysis. For the zonal wind increments in particular,there is almost no impact from satellite radiances de-spite the coverage by, for example, AMSU-A radiancesshown in Fig. 6b.

The results in Fig. 8 can be further separated into thecontributions from individual observing systems. Figure9 shows daily time series of the impact of several dif-ferent observing systems on JTNP

. It can be seen that theNOAA-16 AMSU-A radiances have the largest impactson average (note the different ordinate scalings), andaccount for most of the large impact from satellite ra-diances on JTNP

observed in Fig. 8a. Interestingly, on 13August, when much of the radiance data are missing,the impacts of other observing systems (e.g., rawin-sondes and surface marine observations) increase sig-nificantly compared with their average values for theperiod. This illustrates the relative nature of the im-pacts of different observing systems, and possible re-

dundancies between them. Other observing systems, in-cluding rawinsondes, HIRS-3 radiances, and aircraftobservations from the Meteorological Data Collectionand Reporting System (MDCRS; Moninger et al. 2003)have significant but smaller impacts on JTNP

thanAMSU-A throughout the period. Also note that whilethe total impact of all observations on JTNP

must bepositive, the contributions from individual componentsof the observing system may be negative.

Finally, the impacts of these same observing sys-tems on JUUS

are presented in Fig. 10. In this case, theimpacts of rawinsondes and MDCRS aircraft observa-tions are larger by nearly an order of magnitude, onaverage, than those of the other observing systemsshown. The results are consistent with those in Fig. 8d,which show that conventional observations account forvirtually all of the impact on JUUS

throughout the studyperiod.

FIG. 8. The contributions of the satellite radiances (dashed) and conventional observations (dotted) to the totalobservation impact (solid) for (a) JTNP

(K2), (b) JUNP(m2 s�2), (c) JTUS

(K2), and (d) JUUS(m2 s�2) during August

2004. All values have been multiplied by 10�3.


5. Conclusions

An exact adjoint of the GSI analysis scheme was de-veloped and tested in the context of the GMAOGEOS-5 atmospheric data assimilation system. Devel-opment of an exact adjoint was deemed an appropriatestrategy given the formulation of the existing nonlinearminimization problem, including the quality controlprocedures and observation error assignment per-formed in the inner and outer loops of the GSI. As aprerequisite, a line-by-line TLM of the GSI, includingthe conjugate gradient descent algorithm used in theminimization, was developed and thoroughly testedwith realistic-sized perturbations of the input innova-tions. Comparison of the TLM responses to perturbedinnovations with differences between the GSI incre-ments with and without these perturbations reveal thatthe overall behavior of the GSI is well represented bythe TLM. Perturbed innovations of wind, temperature,moisture, and satellite radiances produce highly linear

responses for most of the analyzed variables exceptozone and, to a lesser extent, specific humidity. Theresponse to perturbed ozone innovations is linear forthe ozone itself, but nonlinear for other variables. Be-cause direct responses—that is, the response of onevariable to perturbed innovations of the same observa-tion variable—tend to be large and highly linear, thetangent linear assumption holds strongly for all vari-ables when all the innovations are perturbed simulta-neously. Penalty terms for supersaturation and negativehumidity in the GSI cost function may cause intermit-tent, but severe, nonlinearities, which cannot be mod-eled by the TLM. However, these terms can be omittedor reduced in amplitude when computing the observa-tion sensitivity in such cases without altering the resultssignificantly.

In accordance with the TLM results, the GSI adjointproduces accurate estimates of the sensitivities with re-spect to observations. In a series of experiments usinglocally defined response functions based on the analysis

FIG. 9. The impact of various observing systems on JTNP(K2) during August 2004. All values have been

multiplied by 10�3.


increments of temperature and zonal wind as input tothe adjoint, the sensitivities are found to be in goodagreement with Baker (2000) and Baker and Daley(2000) in terms of their magnitudes and dependence onthe type, distribution, and density of surrounding ob-servations. Larger sensitivities are observed with re-spect to observations close to where the response func-tion is defined, while much smaller sensitivities are ob-served with respect to observations elsewhere. Forobservations of the same type and in the same generallocation, the sensitivity is largest when the density ofsurrounding observations is low, and vice versa.

A powerful application of the observation sensitivityinformation is in estimating the impact of a set of ob-servations on a given response function. For the re-sponse functions studied here, the impact is easily com-puted from the inner product between the observationsensitivities and the corresponding innovations. It wasfound, for example, that AMSU-A radiances have thelargest impact of all observing systems on the tempera-ture increments over the eastern North Pacific, while

conventional observations from rawinsondes and air-craft dominate the impact on the zonal wind incrementsover the continental United States. The combined im-pact of all observations provides an observation space–based estimate of the total response function, whichmay be compared with the response function computedfrom the analysis increments directly. The observationand analysis space values were found to be in extremelyclose agreement in all cases examined, confirming theaccuracy of the observation sensitivities. As shown byLangland and Baker (2004), the combined use of sen-sitivity information from the analysis and forecastmodel adjoints can be used effectively to estimate theimpact of observations on short-range forecasts. Ex-periments combining the GSI and GEOS-5 model ad-joints are in progress at the GMAO and will be re-ported on in a future study.

The adjoint results presented in this study were pro-duced using a single outer loop of the minimizationalgorithm, while the GSI itself is usually run with mul-tiple (usually two) outer loops to accommodate small

FIG. 10. Same as in Fig. 9, but for JUUS(m2 s�2). All values have been multiplied by 10�3.


nonlinear effects from observation operators such asfor wind speed and precipitation. This capability is cur-rently being developed for the adjoint. However, assuccessive outer loops tend to make only small changesto the increments, we do not anticipate significantqualitative differences with the adjoint results pre-sented here. In addition, more recent versions of theGSI include variational quality control procedures anda redefined control variable for humidity that reducesthe need for the highly nonlinear moisture penaltyterms in the present version. These features are cur-rently being incorporated into the adjoint and their im-pacts will also be reported in future studies.

Acknowledgments. The authors thank Ricardo Tod-ling and Ron Errico of the Global Modeling and As-similation Office (GMAO) for their helpful discussionof this work and review of the manuscript. We alsothank John Derber, Russ Treadon, and Daryl Kleist ofthe National Centers for Environmental Prediction(NCEP) for their discussion and guidance on variousdevelopment aspects of this work. The comments oftwo anonymous reviewers were very helpful in improv-ing the original manuscript. This work was supportedby the Atmospheric Data Assimilation Developmentcomponent of the NASA Modeling, Analysis and Pre-diction program (MAP/04-0000-0080).

REFERENCES

Baker, N., 2000: Observation adjoint sensitivity and the adaptiveobservation-targeting problem. Ph.D. thesis, Naval Post-graduate School, Monterey, CA, 332 pp.

——, and R. Daley, 2000: Observation and background adjointsensitivity in the adaptive observation-targeting problem.Quart. J. Roy. Meteor. Soc., 126, 1431–1454.

Cardinali, C., S. Pezzulli, and E. Andersson, 2004: Influence-matrix diagnostic of a data assimilation system. Quart. J. Roy.Meteor. Soc., 130, 2767–2786.

Courtier, P., J.-N. Thepaut, and A. Hollingsworth, 1994: A strat-egy operational implementation of 4-D VAR using an incre-mental approach. Quart. J. Roy. Meteor. Soc., 120, 1367–1387.

Daley, R., and E. Barker, 2001: NAVDAS: Formulation and di-agnostics. Mon. Wea. Rev., 129, 869–883.

Doerenbecher, A., and T. Bergot, 2001: Sensitivity to observa-tions applied to FASTEX cases. Nonlinear Processes Geo-phys., 8, 467–481.

English, S., R. Saunders, B. Candy, M. Forsythe, and A. Collard,2004: Met Office satellite data OSEs. Proc. Third WMOWorkshop on the Impact of Various Observing Systems onNumerical Weather Prediction, WMO/Tech. Doc. 1228, Alp-bach, Austria, WMO, 146–156.

Fourrie, N., A. Doerenbecher, T. Bergot, and A. Joly, 2002: Ad-joint sensitivity of the forecast to TOVS observations. Quart.J. Roy. Meteor. Soc., 128, 2759–2777.

Goldberg, M. D., Y. Qu, L. M. McMillin, W. Wolf, L. Zhou, andM. Divakarla, 2003: AIRS near-real-time products and algo-rithms in support of numerical weather prediction. IEEETrans. Geosci. Remote Sens., 41 (2), 379–389.

Kelly, G., T. McNally, J.-N. Thepaut, and M. Szyndel, 2004: OSEsof all main data types in the ECMWF operation system. Proc.Third WMO Workshop on the Impact of Various ObservingSystems on Numerical Weather Prediction, WMO/Tech. Doc.1228, Alpbach, Austria, WMO, 63–94.

Langland, R. H., and N. Baker, 2004: Estimation of observationimpact using the NRL atmospheric variational data assimila-tion adjoint system. Tellus, 56, 189–201.

Le Dimet, F.-X., H. E. Ngodock, and J. Vernon, 1995: Sensitivityanalysis in variational data assimilation. Proc. Second Int.Symp. on Assimilation of Observations in Meteorology andOceanography, WMO/Tech. Doc. 651, Vol. 1, Tokyo, Japan,WMO, 99–104.

Le Marshall, J., and Coauthors, 2006: Improving global analysisand forecasting with AIRS. Bull. Amer. Meteor. Soc., 87, 891–894.

Lord, S., T. Zapotocny, and J. Jung, 2004: Observing system ex-periments with NCEP’s global forecast system. Third WMOWorkshop on the Impact of Various Observing Systems onNumerical Weather Prediction, WMO/Tech. Doc. 1228, Alp-bach, Austria, WMO, 56–62.

Menke, W., 1984: Geophysical Data Analysis: Discrete InverseTheory. Academic Press, 289 pp.

Moninger, W. R., R. D. Mamrosh, and P. M. Pauley, 2003: Auto-mated meteorological reports from commercial aircraft. Bull.Amer. Meteor. Soc., 84, 203–216.

Parrish, D. F., and J. C. Derber, 1992: The National Meteorologi-cal Center’s spectral statistical-interpolation analysis system.Mon. Wea. Rev., 120, 1747–1763.

Purser, R. J., W. Wu, D. F. Parrish, and N. M. Roberts, 2003a:Numerical aspects of the application of recursive filters tovariational statistical analysis. Part I: Spatially homogeneousand isotropic Gaussian covariances. Mon. Wea. Rev., 131,1524–1535.

——, ——, ——, and ——, 2003b: Numerical aspects of the ap-plication of recursive filters to variational statistical analysis.Part II: Spatially inhomogeneous and anisotropic Gaussiancovariances. Mon. Wea. Rev., 131, 1536–1548.

Rabier, F., N. Fourrie, D. Chafai, and P. Prunet, 2002: Channelselection methods for infrared atmospheric sounding inter-ferometer radiances. Quart. J. Roy. Meteor. Soc., 128, 1011–1027.

Rienecker, M. M., and Coauthors, 2007: The GEOS-5 data assimi-lation system—Documentation of version 5.0.1 and 5.1.0.Technical Report Series on Global Modeling and Data As-similation, Vol. 27, NASA Tech. Memo. 104606.

Wu, W., R. J. Purser, and D. F. Parrish, 2002: Three dimensionalvariational analysis with spatially inhomogeneous covari-ances. Mon. Wea. Rev., 130, 2905–2916.


Observation Sensitivity Calculations Using the Adjoint of the Gridpoint … · 2008-05-28 ·...

Documents

Transcript of Observation Sensitivity Calculations Using the Adjoint of the Gridpoint … · 2008-05-28 ·...