1994 THOMPSON AND TRIPPUTI 897NWP-Initialized Satellite Temperature Retrievals Using Statistical Regularization and Singular Value Decomposition Methods OVv-EN E. THOMPSON AND M_ARK T. TRIPPUTI *Department of Meteorology, University of Maryland, College Park, Maryland(Manuscript received 30 October 1992, in final form 14 September 1993)ABSTRACT Several research groups have announced plans to merge satellite profile retrieval methods and numericalweather prediction methods into "interactive" satellite retrieval approaches for both weather and climate-scaleendeavors. Satellite profile retrievals, produced from algorithms that depend on hydrodynamic weather predictionmodels for first-guess and conditioning data, may be expected to contain some influence of the numerical weatherprediction (NWP) model quite distinct from any influence of the satellite measurements. Research is describedin this paper in which possible adverse impacts of NWP-produced first-guess information on temperature profileretrievals appear to signal danger for interactive methodologies. Deep-layer, synoptically correlated NWP forecast errors influence satellite retrieval errors in such a way that systematic distortions of the hydrostatic andbaroclinic character of the resulting fields could lead to degradations of a subsequent forecast cycle rather thanimprovements. Two related temperature retrieval algorithms are examined and compared using initializing and conditioningdata derived from NMC T80 spectral model forecasts. The algorithms are the well-known statistical regularization method, also called the "minimum variance method," and a method derived from a singular valuedecomposition (SVD) of the radiative transfer operator with regularization accomplished by truncation ratherthan a priori statistics. The two algorithms allow for a rational distinction between the effects of "statistics"and "physics" on the results. The SVD method provides an opportunity to explicitly examine the adverse effectsof retrieval matrix instability and to infer how that may be influencing the statistical regularization algorithmfor which matrix instability is an implicit property of both the physics and statistics incorporated into thatalgorithm. Finally, the effect of lineafization of the retrieval problem on retrieval errors is examined. For systematic firstguess error fields such as those encountered in this study, the contribution to retrieval error attributable tolinearization is substantial. The retrieval algorithm based on SVD can be unambiguously iterated to reduce thissource of error.1. Introduction The problem of converting satellite measurements ofupwelling infrared and microwave spectral radianceinto vertical profiles of temperature and moisture--theso-called satellite profile retrieval problem--is an illposed problem. This generally means that there is nounique profile solution, for a given set of radiances, andthat two solutions can be very different. The term verydifferent is used subjectively here, but Thompson et al.(1985a) have shown that two temperature profiles withfinite-amplitude differences of rather large verticalscale may be indistinguishable to a High-ResolutionInfrared Sounder (HIRS) satellite sounding instrument. * Current affiliation: s. M. Systems and Research Corporation,Bowie, Maryland. Corresponding author address: Dr. Owen E. Thompson, Department of Meteorology, 2213 Computer & Space Science Bldg., University of Maryland at College Park, College Park, MD 20742-2425.Thus, a retrieval at radiance convergence could be verydifferent from the ambient condition producing the radiances. One is therefore required to have some a prioriinformation about the vertical profile in order to calculate useful soundings from satellite measurements.Often, such a priori data consists of statistical information about atmospheric structure partitioned latitudinally and seasonally. But more than 20 years ago, theidea of initializing the retrieval problem with numericalweather prediction (NWP) forecasts was introduced bySmith et al. (1972). While the National Oceanic andAtmospheric Administration (NOAA) operationalsounding methodology has taken other forms sincethen, there has been increasing interest in recent yearsin so-called interactive retrieval systems in which hydrodynamic models are used to provide a priori datafor initializing the retrieval problem, while resultingretrievals are assimilated into fields for initializing thenext forecast cycle (see Kalnay et al. 1990; Susskindand Pfaendtner 1989). Plans are evidently in progressto implement an interactive retrieval schYme for NOAAoperations (see Fleming et al. 1989).c 1994 American Meteorological Society898 MONTHLY WEATHER REVIEW VOLUME 122 The present authors have some concern about amethodology that initializes NWP models with retrievals that themselves have been initialized by the sameNWP model. Results to be presented in this paper suggest that there may be a danger of self-inflicted woundsif there are systematic NWP forecast errors that themselves distort the satellite retrievals in ways unrelatedto the satellite measurements. This fear has also beendiscussed in the context of the much longer rangeproblerfi of assessing climate and global change fromsatellite. The debate is further articulated in a recent National Aeronautics and Space Administration(NASA)-NOAA report (see TOVS Pathfinder 1991).Some argue that coupling hydrodynamic models to satellite retrieval algorithms would provide the optimummethod for assessing evidence of climate and globalchange from the historical record of satellite sounderradiance observations. Others argue that a search forevidence of climate and global change from the recordof satellite-based observational data should not bemade subject to the behavior of a particular hydrodynamic model of the system. The results presented inthis paper will offer some insight into the influence ofNWP model forecasts on interactive satellite retrievalsand may also offer a pathway to analyzing and possiblycontrolling some of that influence by using iterated singular value decomposition techniques. To study the interactive approach to satellite sounding, a hydrodynamic weather prediction model [theNational Meteorological Center's (NMC) T80 spectralmodel ] is used to produce grid-located first-guess profiles and statistics for retrieval algorithms. Two relatedtemperature profile retrieval algorithms will be intercompared, each implemented in a linear perturbationform such that dependent variables are deviations fromthe forecast first-guess condition. The first retrieval algorithm is an interactive adaptation of the well-knownstatistical regularization solution of Foster (1961),Strand and Westwater (1968), and Rodgers (1970),alternatively known as the minimum variance method.The second retrieval algorithm is one obtained by performing a singular value decomposition of the radiativetransfer operator and by regularizing the inverse problem by truncation rather than by use of a priori statistics. In a previous paper, Thompson (1992) has provided a derivation and preliminary analysis of this algorithm and has discussed its relationship to thestatistical regularization algorithm.2. The satellite profile retrievalsa. The direct problem In this section, the retrieval algorithms to be intercompared in an interactive mode are briefly summarized. The reader is referred to Thompson (1992) fora more complete discussion of the algorithms. For thepurpose here, the retrieval problem may be stated asfollows: If the radiative transfer equation is renderedfor a clear atmospheric column--is linearized aboutsome known a priori first-guess temperature profile, ifthe object of the retrieval problem is to infer the ambient temperature profile, if surface emissivity in everychannel of a satellite sounding instrument is taken tobe unity, and if the integral equations are reduced byquadrature to a linear system, then the forward radiativetransfer problem may be written in linearized matrixform as R-R* =K'(T-T*)+~or (1) r =Ktt +~,where t is a column vector of deviations of ambienttemperature T from the known first-guess profile T*,r is a column vector of deviations between observedupwelling spectral radiances R and those accuratelycalculated from the first-guess profile R*, e is a columnvector of radiometric observational errors, and K' is an(I x J) matrix of radiative transfer kernel functions inits rows, which are evaluated over J vertical levels ofthe first-guess profile and are directly calculable fromatmospheric spectral transmittance models at each ofthe center frequencies of I radiometric channels in asatellite sounding instrument. The kernel matrix elements are defined here in terms of atmospheric spectraltransmittance r(v, z) and Planck function B[v, T(z)]: f dB(v,, zj)for levels away from the surface, orK.,, = [BO,,, z,)~'O,,, + [~(v,z,) z~)] I~.(~,, [ dr(z,) d,-(~,,at the earth's surface, j = J. Throughout this paper, itis important to keep in mind that this kernel matrix Kis defined for spectral transmittances and Planckfunctions evaluated for the first-guess temperatureprofile T*(z).b. The inverse problem The satellite temperature profile retrieval problem isto calculate the inverse solution of (1) to obtain anestimate t of the ambient profile condition T producingthe upwelling radiances R. Such an inverse solution isgenerally represented by a matrix equation of the form 'i7 - T* = I~(R - R*) or (2) i =The physics of atmospheric radiative transfer is suchthat the rows of a kernel matrix in (1) representing amodern satellite sounding instrument are not entirelyMAY 1994 THOMPSON AND TRIPPUTI 899independent of each other, which means that K is generally an ill-conditioned matrix. In short, this meansthat the satellite sounding problem does not have aunique, stable inverse solution matrix C. Thus, the retrieval problem becomes one of designing optimizedinverse solutions, each of which involves some additional constraint not involving the satellite measurements themselves.c. The statistical regularization solution The statistical regularization (STATREG) solutionof (1) is given by Foster (1961), Strand and Westwater(1968), and Rodgers (1970) as the following inversesolution, assumed unbiased: i = (KE-~K' + S-~)-~(KE-~)r. (3)Before implementing this solution in an interactivemode, it is useful to recall the original theory behindit. In the traditional implementation of this solution, thefirst-guess profile about which the problem is linearizedis an appropriate ensemble mean profile, the matrix $is a natural covariance matrix of atmospheric profilesabout the mean profile, and matrix l:: is a covariancematrix of radiance observational errors representing thenatural ensemble of ~ vectors. According to Rodgers(1970), the STATREG solution is the most probableinverse solution of (1) if the ambient situation falls intoa population represented by T*, $, and I:. For practicaluse, it is assumed that one may estimate the first-guessprofile and solution covariance from a priori historicaldata of "true" profiles. It is well known that this procedure will suffer if the a priori data is not statisticallyrepresentative of the ambient atmosphere that is beingsounded (Crosby and Weinreb 1974; Spankuch et al.1977). In an interactive retrieval approach, this solution isimplemented in a quite different way. The first-guessprofile within i is taken to be a location-dependentNWP forecast profile, and $ is taken to be an errorcovariance matrix of NWP forecasts. It should be further mentioned that there are pattern recognition methods of initialization, such as those given by Thompsonet al. (1985b), Chedin et al. (1985), or Goldberg et al.(1988). These methods also produce location-dependent first-guess profiles and associated a priori firstguess error covariance statistics in nontraditional ways.The interpretation of Rodger's theorem would be different for such nontraditional implementations of theSTATREG theory. The adaptation of the STATREGalgorithm to an NWP initialization no longer has theproperty that the solution is the most probable onegiven what is known about the natural atmosphere.However, an interactive STATREG solution may bethe most probable solution consistent with how theNWP forecast model works. This is our principal concern about interactive retrieval methodologies.d. The truncated, physical singular value decomposition solution Thompson (~992) calculated an optimized inverseof (1) based on singular value decomposition (SVD)of the radiative transfer kernel information. An abbreviated derivation of this solution is as follows. The radiance deviations in equation (1) may be scaled byinstrument noise standard deviations leading to a scaledform of (1): Ftr = F'K't + F~,where FF' = E-~. The scaled radiative transfer operator(FtKt) can then be diagonalized to obtain singular values and singular vectors: (KF)t = UFVt.In this notation, U and V are the left and right singularvectors, respectively, and F is a diagonal matrix of singular values. The straightforward inverse of the scaledproblem is then given by i = (Vr-'U')(F'r).It is easy to show that this solution is equivalent to theso-called Penrose-Moore pseudoinverse: i = (KE-~K')-~(KE-')r.This solution generally has the problem that there aresmall singular values in F that make F -t unstable. Thisis equivalent to stating that matrix KI::-~Kt has smalleigenvalues and is therefore ill-conditioned. One maysee that the STATREG solution regularizes the inverseby adding in the covariance matrix $ to statistically"inflate" the small eigenvalues. Thompson (1992)provides extended discussion of the relationship between SVD and STATREG solutions and shows thatthe inverse problem can also be regularized by truncation of the SVD representation of matrix (K--~Kt)-~as a clear, nonstatistical alternative. A truncated, physical singular value decomposition solution, herein denoted (TP-SVD), may be obtained by retaining onlythe M most significant singular vectors of the above(~j, ~r ), which leads to the approximations (KI::-XKt) -1~ [~r(f,t~)-~t], (KE-~) ~ (~f, tO,)Ft and finallythe resulting solution, assumed unbiased: i = (~ f'-'OtFt)r, (4)where ~' is square with only the M most significantsingular values retained. It is to be noted that mathematical stability of this solution is obtained by the truncation such that f'-~ becomes a well-conditioned inverse.e. Interactive implementation of the inverse solutions In the study to be conducted, the influence of a prioriNWP model data on the STATREG and TP-SVD solutions should be clearly understood. Each algorithm900 MONTHLY WEATHER REVIEW VOLUME122~' ~ol ihJ I im !hJ ~ool $o~ loo~ HIRS 15 MICRON1 I It, I~ N. ~ - x ~t~ - X ' ITS,\ ~lx,.. , Its\ ~ ~ ~ ?,,'.x---.?< I'.". X. ~ ~ --.'~2 :"" "-~3 1 ~.~,~. E:>..4..~'-~ ~1 ? ,~..~""~, ~ SCALED VALUE 1HIRS 4.3 MICRONSCALED VALUEMSU 5O GHZ I [ , , ISCALED VALUEF~c,. 1. Transmittance weighting functions Ov(v, p)/O In(p) for 15 selected channels of HIRS/MSU applicable to temperature sounding.2423depends explicitly on a T80 forecast first guess~ of theprofile embedded in the perturbation variables t =- T*) and r = (R - R*). Each algorithm uses a kernelmatrix K that depends at each grMpoint on the forecastfirst-guess profile for its evaluation, although the TPSVD algorithm restructures and truncates this information in terms of the singular vectors. Additionally,the STATREG algorithm depends on a priori statisticsof NMC T80 model-generated forecast errors throughthe error covariance matrix $, whereas the TP-SVDalgorithm does not depend on such forecast error .covariance information. Finally, both algorithms dependon a priori estimates of radiometer observational errorssummarized in matrix E.3. Data and methodology The two satellite temperature retrieval algorithms described above were intercomparcd in a "semi-interactive'' mode using the NMC TS0 spectral forecastmodel to gener.ate a prior! information. The term"semi-interacti~,e" is used to represent that retrievalsare calculated depending on forecast first-guess fieldsbut that those retrievals are not used to produce the nextcycle of the forecast. The NMC verification analysis isregarded as ground truth for this study and incorporatesinformation from the National Environmental SatelliteData and Information Service (NESDIS) operationalsatellite soundings. NMC T80 spectral model temperature profile forecasts, and verification fields from post analyses, wereobtained from NOAA/NMC for seven consecutivedays (27 January-2 February 1990), at each of fouranalysis times~(0600, 1200, 1800, 2400 UTC), and ona pressure grid of 1000, 850, 700, 500, 400, 300, 250,200, 150, 100, 70, and 50 mb. Spectral data were transformed to gridded vertical profile data on an equilateral162 x 82 global grid. Transmittance weighting runetions O~-(v, p)/O lnp for 15 HIRS-Microwave Sounding Unit (MSU) channels shown in Fig. i were computed using the fine-vertical-scale method of Susskindet al. (1983) applied to profiles interpolated to a higherresolution vertical .grid assuming temperature to dependlogarithmically on pressure between the coarser NMClevels identified above. 'Coarse-grid transmittanceweighting functions, and associated radiative transferkernel functions, were computed by an alg'6rithm thatassures that upwelling radiances calculated from coarsetemperature profiles and coarse-grid transmittances areidentical to those calculated from interpolated finergrid profiles (vith the finer-grid transmittance algorithm. Absorption effects of water vapor, ozone, andother variable trace gases were not simulated. Cloudfree simulated observations were computed from thefull radiative transfer equation using the NMC verification analysis profiles with random errors added to thecalculated radiance values. These random radiometerchannel errors were drawn from Gaussian populationswl~ose standard deviations are given by the instrumentnoise levels incorporated into matrix E. It should benoted that synthesizing radiance measurements usingNMC thermal analysis fields in the forward radiativetransfe, r equations provides synthetic observations thatare absolutely consistent with the NMC analysis subject only to those random errors associated with nominal instrument noise levels. This is, of course, a muchmore perfect representation Of the relationship betweenanalysis model and nature and of sounding methodology than one can achieve in practice. Cloud-free conditions, unit emissivity, homogene6us scene, perfectknowledge of the radiative transfer equations, absoluteconsistency of atmospheric analysis and upwellingspectral radiance, etc., are simplifications that are nottrue in reality. For this reason, results shown in thissimulation study are more accurate than those achievedin analysis of real data. Nevertheless, the simplificaI~IAY 1994 THOMPSON AND TRIPPUTI 901tions stated above are permitted here because it is thebehavior of the inverse problem, not the forward problem, that is being studied. First-guess and retrieval radiances were calculated from NMC forecast profilesand retrieved profile solutions, respectively. Study datawere divided into a dependent set (27, 29, and 31 January and 2 February) for the purpose of computingstatistical forecast error covariance matrices for theSTATREG retrieval algorithm. Retrieval intercomparisons were done for the independent set (28 and 30January and 1 February). Note that the dependent apriori data are interspersed with the independent datain this study so as to provide the most highly representative statistics for the STATREG algorithm. In the retrieval calculations, the spectral transmittances, kernelfunctions, and SVD decompositions were computed ateach grid location using the forecast first-guess profileat that grid. The same forecast profiles were used explicitly in the retrieval equations (3) and (4) in theconstruction of t, i, and r. Unless otherwise stated,results to be shown are for a single application of eachalgorithm--that is, no iteration. In simulation mode,one knows everything at each step. One is thereforeable to also monitor the retrieved temperature profileerrors and retrieval brightness temperature errors ateach step.4. Gross statistical performance of retrieval algorithms Table I contains statistics for forecasts and retrievalsperformed on the three days of independent global dataat four different verification times, a total of 155 544cases. Ensemble root-mean2square errors for equivalentbrightness temperature and temperature profile areshown. The rms temperature profile errors were computed both over the entire vertical extent (1000-50mb) as well as over only the lower-tropospheric layer(1000-500 rob). Table I also shows values of meanbias error at the 850-rob level for the independent data,a statistic that will become relevant to discussions laterin this paper. The identifier "TP-SVD5" refers to theTP-SVD algorithm truncated to five singular vectors.The identifier "TP-SVD5-plus" refers to an iteratedversion of this algorithm, which will be defined anddiscussed in section 11. Table 1 shows that the NMCT80 forecast model provides, in a gross average sense,first-guess profiles within 2 K rms of the analysis, andradiometer brightness temperature signals of about1.4 K rms. The forecast-initialized temperature retrieval algorithms thus operate on rather small brightness temperature signals in seeking improvement of theprofile errors. The profile retrieval algorithms shown inTable 1 reduce the brightness error by a factor of 712 in producing a corresponding improvement of profiling of 0.7-0.8 K in an rms sense. An interesting feature shown in Table I is that the TP-SVD method performs better in the lower troposphere as compared withthe total atmospheric depth while STATREG behavesoppositely. The lower-tropospheric forecast errors aresomewhat larger than upper-atmospheric forecast error,and these appear to be corrected more effectively bythe TP-SVD approach. This result is consistent witharguments given by Thompson (1992), which hold thatthe TP-SVD algorithm can more faithfully retrievetemperature in the lower atmosphere since the HIRSMSU instrument provides substantial amounts of measurement information about the lower troposphere ascompared with the upper levels. The forecast error covariance matrix in the STATREG algorithm statistically modifies the solution in these lower levels when,apparently, there is no need to do so. The opposite argument holds in the upper levels where HIRS-MSUinformation is "weaker" and there is a greater need toextract additional information from. the a priori data.The more physical TP-SVD solution does not performas well in the upper levels since it does not draw any~uch information from statistics. The mean bias errorsat 850 mb also reveal that STATREG tends to carrythe tropospheric forecast bias through its solution. TheTP-SVD does less of this and even shows a tendencyfor overcorrection in the gross ensemble sense. Thesedifferences in bias and layer response will be furtherdiscussed in subsequent sections of this paper wheresmaller subensembles of data are considered. Another interesting feature of the data in Table 1relates to the values of rms brightness temperature error. The TP-SVD solution filters the radiance measurement information through, the truncated set of orthogonal left singular vectors U, which range over radiancespace [see (4)]. It might be feared that this filteringprocess would remove a certain amount of valuable TABLE 1. Ensemble root-mean-square error and mean bias at 850 mb for NMC T80 spectral model forecasts and STATREG, TP-SVD5,and TP-SVD-plus retrievals. Statistics are for global soundings at 0000, 0600, 1200,. and 1800 UTC for 28 January, 30 January, 1 February1990 (155 544 cases).rms error brightness rms error temperature rms error temperature Mean bias temperature(15 channels) (1000-50 mb) (1000-500 mb) (850 mb)Forecast 1.369 1.92 2.26 0.49STATREG 0.186 1.12 1.27 0.43TP-SVD5 0.192 1.16 1.08 -0.20TP-SVD5-plus 0.107 1.11 0.97 -0.38902 MONTHLY WEATHER REVIEW VOLUME 122satellite radiometer signal information. If that were thecase, then a TP-SVD solution should never producebrightness errors that were smaller than an optimummethod since less than all of the measurement information is incorporated into TP-SVD. On the otherhand, if the TP-SVD filtering process removed onlymeasurement errors, instead of signal information, thengross statistics on retrieval brightness temperature errorshould remain competitive, possibly even superior, inthe TP-SVD approach. While the rms brightness errorfor a single iteration of TP-SVD5 slightly exceeds thecorresponding value for STATREG, it seems apparentthat the SVD filtering of the radiance measurementsthrough only five retained singular vectors has no majoradverse effect on the information content of the measurements. The influence of measurement errors onprofile retrieval results depends, of course, on the stability characteristics of the retrieval operator. This willbe further analyzed later in this paper. That the STATREG algorithm shows smaller valuesof brightness temperature errors than the single iteration of TP-SVD5 raises an important practical issue.From the operational point of view, one would acceptthe STATREG retrievals over the TP-SVD5 retrievalsbecause of their apparent superiority as assessed in radiance space. The 1000-50-mb profile error valuewould seem to support this decision but the 1000-500mb results would not. It would appear that part of therole of a priori statistics in the STATREG algorithm isto add finescale structure in the retrieval profiles thatimproves the radiance fit, in an ensemble mean sense,even though it has an adverse effect on the accuracy ofestimating the lower-tropospheric thermal structure.5. Retrieval errors in profile ' In addition to the gross error statistics shown in thepreceding section, it is useful to inspect the errors inprofile. Figure 2 shows ensemble mean profiles of forecast first-guess error, and STATREG, TP-SVD5, andTP-SVD5rplus retrieval error for the independent ensemble represented in Table 1. The TP-SVD5 methodshows a consistent pattern of retrieval improvementover STATREG in .the lower atmosphere, from 1000to 400 rob, while showing clearly inferior behavior between 300 and 50 mb. An interpretation of this behavior was given by Thompson (1992), in which it wasshown that the truncated, physical SVD approach issuch as to produce an algorithm that draws more heavily on radiance measurements for retrievals in thislower layer than does the STATREG algorithm. Thatis,' TP-SVD is more "physical" in the lower atmosphere than STATREG. TP-SVD depends more heavily on the first-guess profile in the.higher levels thandoes STATREG, although. STATREG also incorporates information extracted from the error covarianceStatistics. The'superiority of error structure of theSTATREG retrievals over the TP-SVD retrievals5O 100V 50028,30 JAN, ! FEB - 1990155,544 CASES ~ . / GUESS ~\ RMS-50 = 1.92 RMS-500= 2.26 , ,'/ ~,t ~ RMS-50 = 1.12 RMS-500= 1.27 ......... ~ .... ii I I ~-s~5 ~ RM~-50 = ~.~6 RMS-500= 1.08 ?~~~ ~-S~-Plus RYS-50 = 1.11 RNS-500= 0.97 ~ ~~~ ~ ~.~'- ~ 1000 0 1 2 3 4 (K) Fro. 2. Ensemble mean profiles of NMC T80 forecast first-guesserror, and STATREG, TP-SVD5, and TP-SVD5-plus ietrieval errors.Results are for all independent cases: global soundings at 0000, 0600,1200, and 1800 UTC for 28 January, 30 January, and 1 February1990 (155 544 cases).above 300 mb would therefore seem to be attributableto the a priori covariance statistics and not to any morephysical way of deriving information from the radiancemeasurements. The performance difference in low and high layersis also shown by the layer rms error values. The rmslayer error from 1000 to 500 mb is significantly lessfor the TP-SVD5 algorithm than for STATREG. Although not yet defined in this paper, the TP-SVD5-plusalgorithm results shown in Fig. 2 give hint to the potential advantage of iterating the TP-SVD algorithm, aprocess that is unambiguous since one does not haveto consider the question of whether covariance matrixis appropriate for successive iterations.6. Retrieval errors in map view A major focus of this study is to examine the influence of forecast first-guess profiles, and forecast errorcovariance statistics, on patterns of satellite retrievalerror as viewed in weather-map form. This is particularly important in the context of interactive retrievalapproaches where forecast first-guess fields have synoptic structure, and the resulting retrievals are introduced into the next hydrodynamic forecast cycle. Onefear, of course, is that there may be systematic modelgenerated errors that may induce similar errors in theretrievals that, in turn, would influence the next forecastM~- 1994 TROMPSON AND TRIPPUTI 903cycle. Reuter et al. (1988) analyzed HIRS-MSU deeplayer thickness retrievals initialized by an NWP forecast model for the ALPEX (Alpine Experiment) regionfor two successive synoptic periods and argued thatthere was little impact of the forecast first-guess fieldon 1000-500-mb thickness retrievals, and only modestimpact for the 300-100-mb layer. In an evaluation offorecast-initialized VISSR (Visible-Infrared SpinScan Radiometer) Atmospheric Sounder (VAS) satellite retrievals, Fuelberg and Olson (1991) show temperature retrieval errors at 200 mb that appear significantly correlated with the forecast first-guess errors,while correlation is weak to nonexistent at 700 rob.However, Fuelberg and Olson's data also showed thatwhen the forecast first-guess field overestimated thehorizontal temperature gradient in the troposphere-even the horizontal gradient in 1000-500-mb thickness--then so also did the forecast-initialized VAS retrievals, in general. The database for the present studyprovides for much more analysis than journal spaceallows, so only selected, representative results will beshown. In what is to follow in this section, we concentrate on midlatitude forecast and retrieval results.a. Time evolution of forecast errors in standard pressure surfaces Figures 3a and 3b show the Northern and SouthernHemisphere midlatitude temperature forecast errors at6-h intervals on 28 January 1990 at the 500-, 700-, and850-rob levels. Contours are shown at intervals of 2 Kwith the zero contour deleted for clarity. The arrangement of the data in the format of Fig. 3 provides theopportunity to detect what appears to be synopticallyrelated systematic errors in the forecast. Indeed, at eachof the three levels, there appears to be a wave train inthe Northern Hemisphere error field, from about 120-Eeastward to about 30-E, in which various positive andnegative error centers propagate eastward, presumablyin lockstep with moving weather patterns. The amplitude of the systematic error centers, and consequenterrors in horizontal and vertical temperature gradient,are quite significant. Similar wave train error behavioris apparent in the southern midlatitudes.b. Time evolution of STATREG retrieval errors in standard pressure surfaces Figures 4a,b show similar depictions of STATREGtemperature retrieval errors for the Northern and Southern Hemisphere midlatitudes. At each of the three levels, the STATREG retrieval errors show horizontallycoherent error centers that also appear to propagateeastward, presumably reflecting the influence of thepropagating forecast first-guess error field. However,the retrieval errors are not very simply distributed inthe vertical. In the Northern Hemisphere midlatitudes,the 500-mb STATREG retrieval errors appear to havea significant negative bias (retrievals generally toocool). The STATREG retrieval errors at 850 mb appearto have a significant positive bias in the same latitudebelt (retrievals too warm). Similar bias in the lowerlevel has evidently also been noted by NOAA/NESDISduring internal evaluations of retrievals using a similarmatrix inverse algorithm not initialized by NWP forecasts (see Dey et al. 1989). A number of negative firstguess errors appear to be systematically overcorrectedby the STATREG algorithm at 850 mb leading to significantly positive retrieval errors (e.g., see the regionaround 170-W). These bias and overcorrection featuresare not so apparent in the Southern Hemisphere midlatitudes (Fig. 4b) where the retrieval algorithm seemsquite efficient in reducing forecast first-guess error.Even with such offsets, the general pattern correlationbetween retrieval errors and forecast first-guess errorsis very evident at the three levels.c. Time evolution of TP-SVD retrieval errors in standard pressure surfaces Figures 5a,b show depictions of the TP-SVD5 temperature retrieval errors similar to those in Figs. 4a,bfor the STATREG algorithm. TP-SVD error centersalso appear to be moving in phase with forecast firstguess errors. The forecast error reduction appears morebalanced at all three levels in the Northern Hemisphere,in comparison with STATREG, and the TP-SVD5 algorithm shows a more balanced distribution of positiveand negative errors. The major exception to this is theresponse to the negative forecast error center in easternNorth America. In the Southern Hemisphere midlatitudes, the TP-SVD algorithm shows some tendency fornegative biasing at 850 mb and positive biasing at700 rob. This behavior will be further examined in section 9.d. Errors in characterizing the l O00-500-mb layer Associated with the Northern Hemisphere midlatitude results, Table 2 shows mean bias errors at the500-, 700-, and 850-rob levels corresponding to Figs.3a, 4a, and 5a. The forecast bias errors show that theNMC T80 spectral model estimates more static stabilityof the 850-700-mb layer and more static instability ofthe 700-500-rob layer compared with verifying analysis. This seems to be reflected in the TP-SVD5 biaserrors. However, the STATREG algorithm producessignificant positive temperature bias at 850 mb and significant negative temperature bias at 500 mb. This jointerror behavior leads to a systematic destabilization ofthe entire 850-500-mb layer compared to verificationanalyses. To further illustrate the deep layer aspects of systematic forecast influence on retrievals, layer thickness errors were computed over the 1000-500-rob layer forthe data represented in Figs. 3, 4, and 5. Figures 6, 7,904 MONTHLY WEATHER REVIEW VOLUME 122 500rob .~/ ,. 7/ ...... .. :;?.'~v. ./- ,'~o'[ :'.:' "'d,~--"~;~ g'?,.:,.-.,,..;: r .... ..... ?~'~..~).~.~: ,~ .~Iioo~ "-J t,:',..~ C"~ . g ;?j ,"hJ. (~...:?~ ...... '::5<'; ~ .. ~.,, ~ -~: . ~ :~-_.>. ;.,~ ;~ ~ ..~:i ..... -- ,~'~ : , :,I .',,,, ,:: ~, "-:-". ......... ',:.~:..~'=':,.N._~ .'-~ '~ ;'<: ?':--*." "~i' " 06Z~o- ...... ~,,.,,.~ ~.~ ~,,,: ,. ,.. ,', ~;-~, ~' ' :"")' ~ "C'-: vc,t'j ....... W " .... '"-"- -40- , ~_/ ~, ,%; .~ %; ..... ~ - :.... ......... ~,, ~ - w"'. v ::, -."-~/;' ..::._~~ -' %~-%-:%.~-. ':. ," '~:~J~,-N~".~.~~-xc' ..."7-'~ ~-. .. --: , ,q.-// ,~ ~ ', . .,, .. ~ .--~ '.-'; ~.,,,(.~ ! /-/_ ............ , .... . ....... . .......... ....'~',<'~.:[..,;~ .... . -.. .,, .. .~ .... ,,-I~.. ! ?'. ......~o .......;.~.- ~; .......~ "~7,',~.~ ...... ~" ' " ...... \" ,"T ; ...... ' ................... ',4,-~-~'~",JG_' .... 60E 120E 180 120W 60W 0 60E6050.4050405O4O5O403- 60-700rnb~ ,i; .'?:?:?~:~i-,Z .~.~.~.: .... o~....?'0~,-:~'...,.. o~~;~, ',?,;~ ,,',? ~..-. :,~:.:.;.;;g.;;..~. - ~<-.~, ~:~,'.:2'-::~'.',~;":' ,5."..~ 120-OOZ06Z12Z18Z60E~o 850rob .;<.;<'' ~ : ';-', ' ',:-;" % "M '",~,,'.:, :,,'",;,,~:) -4'-1 'r~~,~'~' ,, ~ - /C.,/ . ,.. ~.:.., t.----' Z ~--;~' (J" ,. I'~ _~b--~')~VIW'~-i~w/~~. -. o ,...' .... YiE--?9, ~/ . ..'.$ .......5 .... t.'~'/ .4 ........... ~. % ?; .;-'~ - ' :;.~,; ', ',.,:s.;,i/,? ......,,. ,. ' -&':."-~-.; " ' :~:/'~,40 I.~'~ ~r'9~ '"'/~.,~ fi-~ ,:,'uu ..... >:2-,~%,' ......'.-~ .~ ............. '~-t~ - ' - ;~.;'a - - -,.,.:. - ' . ,4'4 c~- - ~4-',: - '~A"-..,:'~~ .t~...'..-'-. ,-,,':L. ;,e- ..,~,.~ ,. 5"~._..~~ ..,.--'.->-'...5'"'~. - -.'.<40. '~-,..~ - :'--.---~ ....... .......... ~: ~ .. s~'-~'-~'-~'-~'-~'-~.' ....... ~. '.'--~> [, / ...... ~'~; =':"':'>"~ ,-~ ".':~Z=~..~_? ' '~" ,.,M ,":',,o If'~..~..t.30 I """ ",' ",. '"'., _ . ..."'>='~T$'C.-' 60E 120E 1~0 120W 60W Fro. 3. NMC T80 spectral model temperature forecast errors at 500-, 700-, and 850-mb levels at verification times of 0000,.0600, 1200, and 1800 UTC 28 January 1990. Contours of T*(NMC T80) - T are drawn at 2-K intervals with the zero contouromitted for clarity: (a) result~ for Northern Hemisphere midlatitudes, and (b) results for Southern Hemisphere midlatitudes.M~Y1994 THOMPSON AND TRIPPUTI 90560504050405040504030OOmb ,:-, ') ..~/ ~ ~-.~.. ,,: ~ t >' , "~':"~ ~' '~' ' ?':' ~) !'::":"i: .... ~:~": ..... , ' ' '~,~:":~ ~ :b "~'. "' ~ 'o ~ ~"~;::' ~" /~ 2'"~ .... ~. ~/ ~' '~ ~ ' " - ~,.. . / .z./' : fi ,~-~ /-,, ~,::,.~ ~%~,:~,, . .,::, ..........-~ . ~ '" ~--'>-:'-::-:~i/' '::~:~'~ "*~-:~/~ '~ ~ ,- ...... ........) ~ .... ~:. ~ ;,~~' .'q . --..... ~.~ ,:--..~_ . .... . -.--..~ , s .:.::---._. '. , ~~'~ .... ~ / I' '. ' ~[".- - ' ~ ~ ' '~:*'.', '~ "*~. ', .... '~-' ~', : ~ /-'"'-- '*-.-, ~-,... ~' ~ ' I;~'-~ . ', ', ....~....'-~ ..... ........ .., ,~-,~.-~ , , .,. .. ~ .... .,' ..... ',.' ..... c*, ~:~ ,: ..... ...:- ...... .. ~" -~ ..... /L.?.,~/i.~ ~--:e., ~, ':,:~ , /"" ~ ~ /% ~ ...... ~'~ e~ ~9'~ ~ s~ '" / ~ ,", - ~ _:/* ~-,.:: ~.. --~ ~.'~g~ ~ ~ (- v/'-../ ~ ...... ~ "~ '-" ~';:~. '- 3-~' ";~.~,-,~, ~_ '.: c-., ......... ~:.. ?f. ;.~ ?~.' . ,..~ ? ~':~:,~ ....... ~,::~.,2~,. ......... <.,.;:~,:. -: -~..:~)(f~.; ;.--:~ ..... >~ '~,:-'::-~ ..... - ~ .., ~ ,~ ~(( ~'"~'~ ...... ~ ~ . ' ' "'~':':'., ...,':'...-:~:-%.:~ ~:~:",. '~.?,~.., ~ ~, ~X., o' ~ :,> ,-. __ ~ .... ~' ........... q'5:b-. ..... ~ ~:~-.; ..... =:;U V ~ ::: J) l~ : ~ ..~*.. ~ ~"-~ ~::. / c. ~ '-<-.-,. ~. ~.. ,..._.'~ ,..~[ ........ ........... q'::i-' Wo-'~ ....... ~-'~; ~x ,:.- ...... ~)(:-.- ............ ~,:::'.. ~ ~c, .~..v::.,...~ ..... c.:*~ .c?. ..... '" ~ " ........ ~ %'"0 ~'"?' '% "~ "~'%~', ~ "-"% ' "% ~ , .' ...... :.:.~.. ~ ~: t '~..~0 , ..~ '0 ....... ~ .... ~ . z ...... ~ ..... ~.~-~ ......... .:,~.,,, ...... .llO].,.~,.,, .......... .~.,~..,., .,~,~...: ..... n .... O .......... o '~*"' o """~;%: ""' ))V~ ', ---. 1[~~..:l ~ /'c,.'/ % ~: -*, ~ ~ -~... ---:;-., ~.. ',,. ~ - .:~..,, ,, ;. ~ ~ .-,, .. ,. ....,:. , .,~.~ , . .... o ,,,, ,~ :~. ..... ,.., ,,120~ 60~ 0 60E 120E 180OOZ06Z12Z18Z120W 700mb0' (% - ,..--,,~ _~,o' .; ',.'. ,,~ .,o ~ ~.~--.:... ~::,'2'~.....,o ~ t ~ :::::, " ': ~. :?... ' ,, ~ - .~..~40.%-.- --~30 120WOOZ06Z12Z18Z120W850mb~o <~ I~ ,~/~:__~:,~ ~' o..~~o .'" '~ ,[.F~ ,':'-.. '. :::",' ....... ~ ~.~ -:.~ ..... :'~ ~iz-.' ~ .... ]/'"~..~ ~-;'-::::',.D ' r;~;~:: ' ~?,~i FY.4o [ ~d q~.:--- ,z...<.:-~;d/~)/ ~-- ':'tx~ ~ "~%);~ (~'? ~ ~' ' - ......... ~ /~ .... ~t~ L:'.~ "o .. ..~., ~ . ~ ~ ' ~ ~ ~ ~ ~ ~ , . . ~ " ~ ~ ~.~ t"".. ' ~:~:. ,~. ~ '-:..-~, ~. -.. '., , .. ..... ~ "'~. =~.50 ~ ~.-~ v u .---M% ~ "., ,?', %'&~. ~ h'~ '~ '.40 - . , ...... '-. - - ' ~' ....... t .. ~ ~ ~' ' '. .... x':~ ,~/ ..':,"~.',~ ~ ~:~:~> , ';~:" '~ ,-, c'> ..... ~ v '% ') ~1 ':"~ ~ ~ ~ ~v. _, _ ~ .... %, ~1 e - .~.~ ~ ~-. _--~_, ~ -. ~ 'J ~ .-, '. ', ," '-, ',.,', t ~, ,., '~o ........ ............ ~ ................ ,., ~,.:. ............ ~,:~.~'- -~~,,, ~ :., -.. ~ ~ v ... (", ~ ~ ~ ~, "~'q't', ~ ....... ~' ' ~ ~ '~ ' ~ ...... ;4 ...... '~ .... ~1~ t',, ~ '~.. ~.,, ~ ~ ~ ~ ~, ~ .%.~'.~, . ~ ~ ~ . ~ .. ~,o ' ' ~ ~ ."-' ~ - ", .~,.. ',-. ~)]] , ::-)X~ ~ . t~,;'~ . ~...:' ~ .'~ ~ de> - ~. '" ~: ~ ," ", ~ '~'~ ~"- ~'" ';-'I "' , _, . ,.,,, ...... .~, . ..~q ,.., ~ ..,~.,,~o. ',7'~ ~ ~-~'::'- :::::2..~A-" ,- ' ",' . ,.-,,: .......... :h -:::;-: : ..... ....... ~ 20~ 60~ 0 60E ~ 20EOOZ06Z12Z18Z180 120WFIG. 3. (Continued)906 MONTHLY WEATHER REVIEW VOLUME122~o 500mb /.-..] ,"~;;~3:. ~'5'-o.~, --/'? *~.*..,m_ '- ~:,, ~ ~ ~ ~ ~ .~-~ .~~~ ~ ..-. .... v-~ ...... '~ -. ~ -. ..... ~ ........ '/ ~"~ :~":" - : '~ ~. ~', ~ ' '. 0 ,' ~ ...........~ .~-:---' ,"~' ~~o ..... ~-.~.~,,~.~... ~e~ ..-,, ~ ~-.~----~'-, ~- ~:~ ~,~<a~ ~~-. ~.-.~ [~> ..~5~-o'~=L;~, ~....-..~ ~;~' ~ t:.;. ~..;,~-?,~'" ~:,~ ~'~;'~-' ~.: 7.. ?~t~., t~- . ~. 4 ~. ~ t Z ...... ~ .... ~,~-r~'~.. ~ ~ V~~ X~~. ' '~ .~*~,~.'-. ,~'~At:-.., ~. ~ ..... e~ ~'~40 - " .. _' )m '~-~'~ '~ .... --" o ' ~'~:~:~ /' ~ ,~ ' . G ..'~::~:, :~:t <.., ~ ~. .,'-, ~...~ ~'c~-~:, .:~ ~~- :=--::~~0 -., ...... ~ :~~. :..-. ~s~..., .~. c~, .~ c~:~-? ~>~-.,' 5~ - .(.~,'. ~ ~' . I '. ~ *~, ~ k '~ V I g - ~*~' [ ~ ~ J ~ ~'. ~,tt:'". "~" '- t '6 ' .~ ~' ' ~P ~ ~'~,~~ ................ ~ '~%'~ ......... ~ - ~,~, ~ ........... , . .,j~ ~ ~ ~':,~, , ~~.": .... . ~~.~ ) ,~ -.;.~% ~ .~. ( : ~ ~:~' ~.. "-" ~ ~ ~' ~ ~ ~ "' ~'~N- ' ~ ' ~ ' ~.~..~. .~.3. - - ~ ~-~, . ~ , .-. .. .. .... . ~-.. / ~ 0 -- t' . t t ..... ~ , ,~ :.-..~ --~ : -. ~ ~ ~--,. .... :: ...... . ~ : -,,~.:~,,~2 %~..,,., (~.~: .--. ,, ~- ,....~0 ...... .-, ~; .......... . ' "~'"" .......... .. [-~ ..... ''-~'~'-~' ...... ~' ~2~'~ ...... ~.~',~*. ...... - ..~...~, ............. -.'~','''W~? .... . ~ ..... :: .... ~/~....:. ~, v .~.~ .~'m~ ,. _, ~, ~ ~.~o ........:'- ' ~'~ ~N~/' ' '/?' .......... ~ ' ~* "F ~) ..... ; ~ '~ "~ _ ~ ~ '___~ ~-"r9 ~ . / F ~ ~. ~ - ~...- ._ ~. z~ ~ .. ~-.. ~ -. ~- ~ ~ '~.~ ~ ~ ~ .~ ~ .-~ ,-. / ....... ~ ~ ..,~ ~,~ ....... . ........ ,.......~o ..... ,- ......... ~.: ~':'......5 . .'.::. ~ ::'~:... ,:':,:, . ~. ...X. .~::~e.'.~ (~,~L.:?.'.'...~,..~'..~ ..... ,.::c... ~ ~ ~ .' ~'"-'~ .' '~ ~ ~ ~ ~ ~ -' ~. ~ .~ ~ ~ ,4.,' ~ ~,~ . . ,--, ~_. ~ ~ ~ .... ~ ~ '~.. ,~o ....... ~' ......... ~ ~ ~.~,, ,.. ......~ .................. ~ ....... '.~ ................. ,.~ ...~.~ ,.~.~o z,:-:' ~ r~,,. ,:~,:~ , ' % ~' :. o ~g~:~ ,: ~. 60E 120E 180 120W 60W 0ooz06Z12Z i18z60~ 700mb60 I .,-,,'~.~=-,,,-~l. ~'/~' .....~ ~ ~ ,. ~,.~,~ .~ ~\ "' . ~ ~ ~r~ .~.~\~501: :..?'%%"~ . . ?::c,:: .' . ,,. %,:, ,o ''~' I,~:..,~CA ::~ ',;':5'h-(, '7~..2~: ,., 4' ~.....','.r:-:-.', .. ,2-" -.'; y '~ "'-~ .'.'.-"' ~. ._ . ' ' "-' ,.. t'.....~,;~:?,~.,, -..,,-..,..~.4 ,...:.. ,-,. ~ ~. ~.".F:~ . . P;~ ,~::.~ :.~.~.~ ' ",.'L"~f:.~n~'?;Y-'~ ~_k ~", ?-:' ' ' "' Y ','" '-' ..... f_ ? ",~,-:~'~.~--~' ~: | ' ':;LLg?~.' ~ %--'2 \",~'.'. - '%._ _ / - '~/ : .~ ~-~J." c: -- ~ z- ~ - ~ t ~ ~ v N ~' ~-'~ .: ~ 4~..-~ -'% ,,, ~'~,"~ ~. ,-, r~o%2 .504 ...... ~ . NJ .-. , - - ,:.: .. :r--. - ~: ~ .... ~::'~ ~ ,':,,'~'~ .... ,;77,; * - :->-' ,'~':_'~ ~ '- '~ /.-" "' ~ ~,~' ~'"~-:""o <~::-::" J~h ::~&L /"~ ~:,o-: - :,:..?..::..,,.~..:...,-:, ,-~ ..... k - '~' '~ff ....... /,.,2~ -~_:-~'>'~:~&: -'.'-~ ' , q ~'ffc'V'\%-q /'1 % _ / 6::/ '--~-.--.:, ",Z ' ,.-,---'~',,/--~/;' '~" '",,. - - , "---. ~1~ ..~\ '- ,, ~ ~'~.,~.~ \.,,/~ . ,.~,..,,- ,~ ,-, ~,~. ,;-..'~, % . . .,~ ............. c,Z,. . .~ ............... ~,. o.. ~ .g.-~ .......... ~'d='~. ........ :.,..PFL:~i .......... ; ....50 , ,,- ~_j [' . '. ;'.;) " ~ , t ..:.-;~ ,', <, '-' ,,0 ~ ..~:',:, .o~/~_' . -- / .%",~Y ----*,/.'b~.. ~.~ ~. - "--' ~._C~-~-. ,-t:.'~t ';;t-~ .... ~ % .... i ; r /..L~_:__Z~wN~_~ 'C2 " . "~:~;' S ~ 'x]&' ,":,"','~ % _ / L:/:'-' q'~.~ .... ' ~ ,,~;=~:: O' . ~..~,- ~,~ ' :-' ~-~.:& c:' '~. ~'-~-~'~50 . . ~ 7. ~. N,~. ~. : ' - ~ .. ,.~ \%',%... .~ - ~,% ~-'~r:~t;. '.._u~.'~,. ~, >~-.~'/. .. ....................~ ."'-. C./ ,..~ -..~/..~..F ......... . '"..)~ '..~, ' NJ ~..~... / ~ ,~o ~.~z.~..~ (~ '.':.:?-'-';' .... ,~ .q.*-.,. ~ ,,..,,..~- .~ , .............. ..................: , *,:. -- . .,. ,, - ,i , r, ,-... - . *. -', - - IE 120E 180 t20W 60W 0ooz06Z12Z18Z60E Fi~. 4. Same as Fig. 3 except for STATREG retrieval errors, t(STATREG) - T:(a) results for Northern Hemisphere midlatitudes, and (b) results for Southern Hemisphere midlatitudes.MA-1994 THOMPSON AND TRIPPUTI 907OOZ06Z12Z18Z120W '700rob504o- '~.~ ,~ ../:?5040.5040.5040.30 ~ ,,,,:, ~ ~'% - ,..,. ~ . '-:::~-~:.:,. ~ '~'~"'" '~ .' ~"' .57' :.' '% - ~ - ~ .~ ,~ .~ ~ ~ ' ~ ~ ~ ~' . ~'~ ' "~o ?-'~, . ~........ .~...>[:~. ...... :.:~...~... ........ ~ ............ ~.:,~:....~:~:~:~ .... .~ ......... ~- : .)/~,'. - ~~ ~- ~ ~,::~, '~ ~. ~.: ~ ~ ~" ,~:~_ /- , ' ~ ,'"~ ) ~ ~ ~ 9 "'~ i ~ ~ ........ ,~. . . .;,:~ ......... ~ ..... ~ .......... ~. %~ ............. ~. ....... ~ ..... ~ .. .... ~ ........ ~ ~ _ ~. I~ ~"'. ~, V ~' ~ .... '.' ...... 5'" ...... : .................. 2) ~' :~ .... :"~ :~ "~ .................... , ..........120~ 60~ 0 60E 120E 180)OZ)6Z12Z18Z120W605040850robOOZ06Z12Z18Z120WFIG. 4. (Continued)and 8 show the 1000-500-mb layer thickness errors,for NMC T80 forecasts, STATREG, and TP-SVD5 retrievals, at the four verification times on 28 January1990, for both Northern and Southern Hemisphere midlatitudes. Contours are drawn at 5-dam intervals for theNMC forecast with the zero contour deleted for clarity.908 MONTHLY WEATHER REVIEW VOLUME 122 500mb -' ~ ~ (2 ,':?~ ~ ' ~ .:~ _ "a ...... < ~3o/ ~ ] ~ ~ ,.~'-'. - - ~~. ' ../ . c.:,~:: ~ ,.q ~o~ ~2o~ ~o ~2ow ~ow o ~o~ooz06Z12Z18Z60Eooz06Z12Z18Z60E FIG. 5. Same as Fig. 3 except for TP-SVD5 retrieval errors, ~(TP-SVD) - T:(a) results for Northern Hemisphere midlatitudes, and (b) results for Southern Hemisphere midlatitudes.M,~v 1994 THOMPSON AND TRIPPUTI 909 500rob60 '' - ,~::.> ,...C2'%..: !...;; ~;..,..~ . . ..~o .. - O. o ,'v.~.;~ o" -~:'"'-:::,.,,4-0 ..bO- - - -'"', ' o x'~ ~:',~ - ' r-'-~ .... ; ' ':!;': .:::/JFJ/:: re'":':':'.'." ....... "'"' --':' ...... :' '"'"40 ..... ~ ~ ' '"=- ,O /"-.. - ,,'~ ......... , .. ~",~' L/' .?.-, U--/'.--... ~ b.,.':,---;~. / , - ..... - -" ~'Z2 '~'-;' '~"' ',"-. ' -',-',-'~.-,.~,-, ~"~.50 ......... '.~' ' 7 ':~',.; ' ' ',~.:. .......... ?' , ' ' ' ;-~, ........... ,' ";, ......... o'.' ',,~' ' 'C."-' - ' ';;~ ,,' '~ .... ~,v ....... .... ~, -:.,Y '"::~-' .. 0 ;...;-.,'d: 0 ,;" "",..:-... "".:?? r- -'~, - ,', '~:":~i" / '" - ' ' -' "'"- ~],/')"'~ ; '-'; r'l '''~'' '"' "J"". '-~o. o ~ .... ' ; t.' ,'";';.*~ '~-.; ' ' .~:.'/,/-.' o H " ~ ~~ ; ....so ............ ,::: -'. ..... .:;- -?:Do..::., ....... 3-~..,., .~c:~.,.a i"'",, ............. "?'x?:..-' ~.,.~'~-,:,,:.,~-.-.,...b ...... . ~':,-,,::',., ......~0..: ......... :~..~ ..... ~,~:~::.., ....... ~.~, ..... ;;rlj.,.--,::. ........... 0..-~..:,..::~> ..... ,::. .....................30 ' '" ~'""Y/-' , -,." - "~'2". , '"';120W 6~W O 60- ~20- 180OOZ06Z12Z18Z120W 700mb6O ,,,c,,~.50-0-- ~50,40OOZ06Z12Z18Z120W~o 850rob50- ' ~( ;?' . ~ t"l' r'r~;'40-- /;' '~' '" .5040504050405(]120W %. ., .. ~w v~ ~ '~ :'-.,-~: ' ,.'-C-"K~:., "..'""5 ~"~: ~'' .... Y',,' -~% '" (-; ~ '"'~ .::,,i .,, ,D - "-,,-) ,"'"' ' *:'-,' :'","', . - ;' (- ,~ ,,-"v '~ '~,t.-,,~:;~ ; ~ ., . i ~ ~; .-'- : ~' ~' ,~ . ~, ,.:.-?, ~;~-.-,-; ~ .. , ~ ,~'~ . ,.,, ~:' ... "~-',~/.,-...~ ,;"::' ~ ,:,; . , :,'~ ', k:, (: "-~t'. ~ ""' .-? "'~,.,:-' ~ - ,.,, .;.~ .-; ~ ':."-,. ,. ',~ ....-,,-,~ ~ - // - ' .~. ~ '- ' '% ~-' ~'~' -- " ~,* - - ~ '~~- eo ~' ' ' ~ "'J 0 t - "=~' "' "'"' ~ ~,; -,t ,, .......- - ~ ~' .... ~,-~/., ..., .. " U4..,c~,,.:,,r~ , ~'.............. tt'f,.~'. ............... , .......f'.';. :-, ...... t"t-,_?.:Y-? .'.%~., '~" .~ ,~ :,' ~? !"* ",.::- '% '";'. ......~ .....2? ".'" f;':' ........... - ~'2 . .'-"J /21 ':,"': (':t- r-, ~. ,t "-. ~ ,~ '... [ , , - .o.~ ~ - . ....... ~. - <.':., I~/ ,, '" '~;" d., ,' ~'-" - i - I,.I ' ~ I~ '~ ? -' '.-.; ,['( ,~' ~ ?.--. o, ",7-"~-~'~0' ~? . - ,............ '..i.V--- ............................ t: ?.it ................ )....%-.-::,.~%~L . , .~ . . f .,!['~ . .............. f~ .................. ,~:::-.':,'.' ....... n..';'..,:,, ............ !(7'~]]:'... :,'::~ ...... -..'.. ',,.':',..' ........, ~ o . ~. u' . ' '-.'~. .. ! '., ,,2:? ,.~ .... ":,."..i ...... , , -~' , ' ,~ '~ .~ ./ 60W 0 60E 120E '!80 '120WFIG. 5. (Continued)OOZ06Z12Z18ZThe retrieval errors are significantly reduced in amplitude and are drawn at 1-dam intervals, also with thezero contour deleted for clarity. The forecast first-guesstemperature errors certainly do not average out overthis deep layer. The traveling forecast error centers areclearly illustrated from eastern Asia across the Pacific,910 MONTHLY WEATHER REVIEW VOLUME 122TABLE 2. Mean bias errors for forecast and retrievals on 28 January 1990 in northern midlatitudes.Forecast STATREG TP-SVD5 TP-SVD5-plus0000 UTC500 -0.318 -0.730 -0.180 -0.145700 0.49~ 0.052 0.464 0.309850 0.088 1.089 -0.120 -0.3210600 UTC500 -0.290 -0.408 0.007 0.008'700 0.347 0.107 0.405 0.310850 -0.1116 0.493 -0.355 -0.4751200 UTC500 -0.363 -0.439 -0.031 -0.011700 0.214 0.117 0.430 0.338850 -0.288 0.528 -0.339 -0.4521800 UTC500 -0.289 -0.522 -0.025 0.021700 0.230 0.150 0.430 0.361850 -0.4y7 0.517 -0.422 -0.528the North American continent, and the Atlantic in theNorthern Hem!sphere, and from 0- to 180- in the Southern Hemisphere,~ Both the STATREG and TP-SVD5retrieval results produce thickness error fields in lockwith the forecast first-guess field. Such pattern ~correlations were not evident in the results shown by, Reuteret al. (1988). The horizontal gradient in the thicknesserror patterns corresponding to the retrievals shownhere would appear to be significantly detrimental fordynamical processes if these 'retrievals were introducedinto the next forecast cycle. While both retrieval algorithms undesirably exhibitprominent features of the first-guess .field when implemented in an interactive mode, they may be comparedin the magnitude of their response. Figure 9 shows thedifference in absolute Value of 1000-500-mb thicknesserrors: the absolute value of the thickness error corresponding to TP-SVD5 retrievals is subtracted from thatcorresponding to STATREG retrievals and plotted at5-m contour intervals. Positive values of'this differencecorrespond to superior performance of TP-SVD5 retrievals as compared with STATREG retrievals. Theerror reduction by the TP-SVD5 algorithm is strongover substantial portions of the northern and southernmidlatitudes, ranging as high as 40 m or so in particularNorthern Hemisphere areas. STAT ~REG Outperformed_TP-SVD5 in the middle Atlantic at 0000 UTC by asmuch as 25-30 m.7. Correlation of retrieval errors with forecast first guess errors It is generally appreciated by remote sensing scientists that because the ill-posed satellite retrieval problem must be regularized by some a priori' information,that same information must be manifest in the resultingderived products. Results in the previous section showthat forecast first-guess errors appear to be carriedthrough to satellite retrievals in very systematic waysthat involve both hydrostatic and baroclinic distortionsin the retrieval fields, although the errors are reducedsignificantly. This behavior can be examined further bylooking for error correlations. The forecast first-guess errors and retrieval errors forthe independent ensemble of 0600 UTC 1 February1990 were used to investigate statistical correlations.Figure 10 shows scatter diagrams of STATREG retrieval errors versus NMC forecast first-guess errors atsix vertical levels within Northern HemiSphere midlatitudes 30--60-N at this analysis time. Numbers printedin each quadrant of these scatterplots are the percent ofthe total number of cases (2268) that fall within thatquadrant. Outside of some small circular region nearthe origin, the STATREG algorithm produces retrievalerrors at the upper levels (700-100 mb) th~tt are clearlycorrelated with the first-guess error. At 8~0 rob, theSTATREG algorithm shows a significant "loosening"of the correlation and an apparent bias toward retrievalsthat are too w~arm, some two-thirds of the cases havingpositive retrieval errors at 850 mb'. The portion of 850mb cases.in quadr~int IV (30.2%) seems notably largerthan at the higher altitudes (9.7% to 16.2%). Similarly,the portion of 850-mb cases in quadrant III (20.5%)seems notably smaller than at the higher levels (32.2%to 43,3%)..These results suggest that the STATREGalgorithm tends to overcompensate for first-guess errors that are too cold at 850 mb tending to ."move"cases from quadrant III to quadrant IV in comparisonto the distribtition of results at higher altitudes. At 1000rob, the STATREG retrieval errors are relatively smalland the scatterplot takes on a peculiar shape quite distinct from the gcatterplots at other levels. At the 1000and 850-mb levels, more than 60% of the retrievals aretoo warm, while nearly 60% are too cold at the 700and 500-mb levels..These results suggest a rather complicated reaction to first-guess errors'within the STATREG retrieval algorithm. Corresponding results forthe TP-SVD approach will be shown later in this paper. Before proceeding, it is worthwhile to consider whattraditional wisdom suggests for such scatterplots as Fig.10. In a traditional retrieval approach, such as the original Rodgers solution cited earlier, the first-guess profile would be a standard atmosphere or large ensemblemean profile, likely a rather poor estimate itself of anyindividual profile in an independent set of data. Withsuch a relatively poor first-guess profile, the perturbation radiance vector r has substantial magnitude, andthe operation (Or) would make a substantial correctionto the standard profile. The radiance measurementswould therefore have a large influence on the retrievalin comparison with the mean first-guess profile, Now,on the other hand, NWP m6dels, as well as carefullydesigned pattern recggnition procedures, can producefirst-guess profiles that yield calculated radiances thatMAY 1994 THOMPSON AND TR1PPUTI 91160504050405040 50 40. NORTHERN HEM I SPHERE MIDDLE LAT I TUDES . ~ ~ ~ . ~ ~.~ ,~,~ _~ ~ ~ . ._~ ... ~. ~,~ ~;~~,---::~j~,~.~,.~ ~ :~ ..... ' ~o' ~ "' ' ':~" ' ('?"~'~ ~" ~0 kj':~~",,i'~i~Z~;~ )~ ~ ~~ ~.~ '~.~' ... (?;~. ~ ~, ~ ..... ~ ,~?-. ~ ~~ .~: ~ _~/ t~Y~ , ~ ~:?- ._/ '~ ~,., ~ ~ .... ,~ --.~ -, - [~:;.~ ,7~-~--. - - ' ..... ~ ...... ~ .... [.;(_ . _ ,, t --~- ~,,~: ~,~, ~,I, ' o ... .. - . .....[ .... ,~ )~ ~ . ~ -.I .._ f ~ , ~ . / - ~- -~ ~ ~ ~ -- % ,--c-.- % ~-~ , ~ ~ , .?~ ~.~':~'-, ,,, .~, ~ ,-. ............. ~ ........ ~...~...-... ~,~. ?~ ...... ,~ .~.~?~... ~:~ .~ ~ ~ ./~...~ ...., ~ ~ ~,~t, , -' P ', .~--~,/ ~ -' ,~ ~ ~' ,',-~ "~ ~,?~,:c.~. [%~ _~~i "~ .... ~?)~ .~ k~ ' ~C ....... (~ ~-'~ c, .~., 3 ~ C~ ~ / [I/ ~ ,-, ":' ' ~' ~-~ ~ ~-k ~,/~"~' ~:?~" o '?, ~ .... ...... ~ ..... ~ ........ .~ .... ~...- ...... ~,~..~?~...~..~,-~.~..~.,,::-~ ~.~ ........ . ....~ ,~ , , ~ ,- ~:c~ ~-~ ~/:~ ~ ~,E'.,(~ ~ (~ ~, ,~:...~5~,.. ,,::, -~-~--~?:i ...................... ~-;~.~ "~ ............. 5~ .~- ..... ~ ...... ~-: ......... ~v,'~ ~,,~ ~~. < ~ . . . ~...... ,.~:, '~.~ ,. ~ _ ~ ~'~,,'~ .:,~OOZ06Z 12Z 18Z18o 12ow 6ow 60EOOZ06Z12Z18Z120W FIG. 6. NMC T80 spectral model 1000-500-mb thickness forecast error for Northern and Southern Hemisphere midlatitudes(300-60- lat) at verification times of 0000, 0600, 1200, and 1800 UTC 28 January 1990. Contours of forecast thickness minustree thickness are drawn at 5-dam intervals with the zero contour omitted for clarity.are very close to the upwelling radiance observations.In such cases, the perturbation radiance vectors are verysmall, and the measurements therefore have relativelyweak impact on the retrieval in comparison with theinfluence of the first-guess profile on the retrieval. Inthe limit, if one draws a first-guess profile that producescalculated radiances that agree with observations towithin the instrument noise level, then [ r[ ~ I e [ andthe radiance observations would have no impact on theretrieval since i = Cr ~ 0 ~ 'i~ ~ T*. In such approaches, there would generally be high correlation between first-guess error T* - T and retrieval error t- T. To complicate matters, such correlation behaviorwould be degraded or highly distorted if the retrievalmatrix G was the inverse of an ill-conditioned matrix.In that case, matrix (3 would have very large elementsand would dramatically amplify small elements of r. Itfollows that the more accurate the methodology forproducing first-guess radiances, the stronger will bethe correlation between retrieved and first-guess proflies provided the retrieval algorithm is well conditioned. But producing good first-guess radiances is not thesame as producing good first-guess profiles of temperature and moisture. One of the most important andproblematic aspects of the ill-posed retrieval problemis that agreement of two satellite measured radiancevectors is not tantamount to agreement of the atmospheric vertical structure producing those radiance vectors through radiative transfer. Thompson et al. (1986)have discussed this in more detail and have shown naturally occurring examples of very different atmospheric profiles that nevertheless produce essentiallyindistinguishable upwelling sounder radiance vectors.Placed in the context of the arguments above, therewould appear to be a serious potential pitfall for inter912 MONTHLY WEATHER REVIEW VOLUME 12260504050 ~40504050.40.30 60E NORTHERN HEMI SPHERE M I DDLE LAT I TUDES - " ~-:--..z- . ' ~ -,~ ~ ~ ~ "' -'"~' " " ~ ~- ,.~ - I... ...... ~. !~, -,,. d~v~ ...~ ,, ~ ~ ~ ~ ~ %? ,--. ..,,-. - .%. ~ - - /' ? ':vq ~ ~ ;~ :~ ~ ~ . ~ - -: ~ ~:,~.~, -.,-~, - ..... ~--, ~ 2' ~' _'~- ~. x~ ~t ~- 1)C'.," _ ~ d ~ ~-~ ~, ~ ~ 6~'-0~' -~ ,~ ~'-'" '~ ~": ~~ ~- c............ ~.~ ..... ~..: ,':~ c. . ~,.', -, ~.~ ,--' ..... ~ ~, o~z . --' ~.~.~'-,~ ' ~ r '~'~~:, ~ - u~ ~ '~ ~5 ~:' , ~- ~ ~ ~_~ ~ ~ ~ ~ ~/~' .-~ ~. ~ ~'~ ~'~ ,-. ~ ~ ~- . ' ~ , , ~ ~ - ~---:=~ .~ .-. '~ ~-, X Ik~ -;~ .'?................... ~...:~ ........... ~ .-o.?,~.~ ,~.. ~...~-~:~. ,~ t~ ~ ....... ,'., ....... ~- ~ ~ -'~ [ ~ ',"'~' "~ ~ ~ '~' ,*;:;'", ' '" I Z~~ ~..~,... ~~ ~:,:9~ .: ..... ~ '~:"~:~':~:~'. J~2 ~~ ~ ~ ' ' '' ::j ~ '~ ~ :;l:t ~ ~ ~ ----XI t 'y ~C~ I ~ ~ :~ /. - ~..~ "~ -~ ~ ,. ~,. - : ., ~ ,,. ,~ ~~< '-.~:~:,:~.,~,, .. ~,,:-.. ..:. ~... ,~ /~ ..l ll~. .......... ' ..... .... .... ' ~ ~'~ ','~. F ~ ~ ~ ~ F- ~(*~52 J~ ~ /~?~C U ' ;~? ' ~s,'- ...... ~' '~:~' ':' ~.~z ~ ............. ~ ......... *-0---~ .............. "~". rU- ~.~,-~' .~./ (/,' ..... l . ~ ~ '-',, ~XW ' ~3 -~ , ..... X 5 : ~ '" . ' '--' r 5 ...... : , ''Z ''. ~--~ .... l :20E 1g0 120~ 60~ 60E SOUTHERN HEMI SPHERE MIDDLE LAT I TUDES6o ) ~../ ~ L.~ ~ ./.. ~ ~,,~ ~ - -.. 2'~-Y' .~~ 0 ......... lj . L,~ ; ' I ...... ' .... ~ ......... . ~ ~7 ~C~ ' I ' ~ , I , I ~ ...... '~' : :' :' l~ l~ '':'' (: ' I __. ~ 'H ' ....,o ...... .~,~, l< ........... ..... :l.;~ ; ~ ,~ ~~--~ ~so ...... IX: ...... ~ :.~ .. ~g;:;~ >; l. ..... ,~, l ~' x-'" /_ ~ ~,2~ '[ ./ ~,~ ~ ~,o ..... ~..~..~ ....... ~ ........ ~j l l I Vii. .... ~ .... ~, *~] J '~ : ~ I / U ':~ o ~ ~: ~SO ............. -~'-~ .................... 0 ........ "~ ................ ~ ..... ~' ''(X'''~''. ...............4 0 ..... ,-~l-., ' . ' ] l'': ~ ') :"::~ ' .2 - '',- ::-~ .... ~~o ............. ~..'~ .................... ~..... ................. ...~~ ~,. .................. ,?..- .... ................... . .......... . .......... :::::30 , 120W 60W 0 60E 120E 1~0OOZ06Z12Z18Z120WF[a, 7. Same as Fig. 6 except for STATREG retrievals and for/-dam contour interval.active retrieval systems. An NWP-produced first-guessprofile could be very different from the ambient atmospheric profile and yet produce calculated radiancesindistinguishable from the satellite observations. Thevanishing radiance perturbation would assure that thefirst-guess profile is accepted as the "satellite retrieval"--remaining very different from the ambientstate in this null process--even though such a satelliteretrieval is little influenced by the satellite measurement. Following this line of thought, a quantitative theoryof the correlation between first-guess errors and retrieval errors will be presented in the next section,which wil! help to explain the behavior of the scatterplots in Fig. 10.8. Theoretical analysis of forecast first-guess error impact on retrievals The scatterplots of Fig. 10 reveal that temperatureretrieval algorithms not only respond directly to firstguess errors but may, in fact, systematically overcompensate for those first-guess errors as suggested by theresults at 850 mb. The authors believe that this behavioris a manifestation of implicit characteristics of the retrieval operator and algorithmic instability that may bestimulated by deep and systematic first-guess errors. Anattempt to explain this viewpoint is presented in thissection.a. A theory of the correlation between first-guess error and retrieval error Let us begin by formulating a theory of the relationship between first-guess errors and retrieval errors suchas that empirically illustrated in the scatterplots of Fig.10. The linearized perturbation radiative transfer equation is given by (1) and the general form of the inversesolution is given by (2), where the retrieval matrix Cis given by (3) or (4) or, perhaps, by some other regularized inverse solution. In these equations, the quantities, r, t, and i are perturbations from the given fore/~LAY 1994 THOMPSON AND TRIPPUTI 9136050.40504050~,0504030 60E NORTHERN HEMISPHERE M I DDLE LAT I TUDES . ~'-'/~'. ~____~,'- x5 ~ , ~ .~ ~ " ~:~,~'~--. ~ , ~ ~ ~-~ ~ ... ~ ,X'. ,, ~- ~.?~~.. ,, ~ . ~ ~ ~ ~...:..., v ~.. .~ ~ ~w-., ~ [;.~'~ ~ ~ ~ ~ ~ . % . . ~ ~J - , ~ ~ ~ ':'v.,'~ ~ d' ~ ~'~- ~ ~ ~'~L?~ ~_. ~._ .-~ ....... ~,.~.~ . ~. ~ ~ -~ - .~.~' ~* '~'-_X" ~~.~ .. ~ ....... . ', ~;,,~ ' ~' t:;-,' ~ ' ~' 'g I~(I ~ .... ~ ~~_ . ~ / .< ..... .. ~,.' ~ ~' ~~ ~ - ~~..~ ~g' .... ?~) ~:~,::,~';. ~.. / .... ~.~.?-~- ~ ~ - ~/F -~' ~. '~' -~ ~ "'-" ':' z- ,A~ '~- .' - ~, ~, ~ . .~.-~ ~, .~ ~~%:.-' o~ ~ . ............... i ....... ~=:~ ............ -~..w ~ ..... t.~,~ .....~._~. - ~. ~/~ ~ ....... i~ ~ ........ , ...... j~ ~ ~.. ,~. --- u ~ ~ ,"~- ~~. ~ .., ~,.,. ~ ~~ .. ~% ~-~ . ..... ~?~-~ ~,~ - ~. - . ~ , -~ ~ ~- ~. ' ,. ~ '~- ~! ,?~, ~. ~ ](. :~ .. " - "--~-(2 ;-..~-~% _:'~~;-~ ..-:, o -~ ~- . - ~ - . -~ / ' - '~ O.~...~ ~';~. "".~ ,. ~ . ..................~ ~ "~ .................... ~ '-'' I ,~-" ....... '~ '~.)~ ' ...... ~--'~' "~'~' '"e ................. ~ ~ t ~-~; ..... ~ . .. ~ c, ~2 ,-,~ ~ ,~ - ~ .~..'~'~g' 'L~:: ....... : ..... ~'"'?'"~"'['?&. ~2~' .............. X ........... ~ ...............~' ~'~'~ .... ~'..... .. '~, .~ ~. ~ ~ , ,~-~-.~ ., 1 ~0E ~ 50 120W 60W 60~OOZ06Z12Z18Z SOUTHERN HEM I SPHERE M I DDLE LAT I TUDES;-o~ - ~'./ .. ' ~"'--',~-7 .~ .... ~, . ~ ,-t ., ~/ ,.~ ~ / .... ,, . . . ....,o. ' ~'-"'-' ' .l~ ' ~.. u ..... .~ .............. ~ .................... '.~:...~ .......... ~ ....... '.*.~ .~ .... ) ~ - -~ ' ', V~ ' ~ - .~o ...... ~ ~ ...~ ~ ~.... ~~o ~ ~ . .40- ~ ~30 I120w 60W 0 66E 120E lfiOoozO6Z12Z18Z120WF~o. 8. Same as Fig, 7 except for TP-SVD retrievals.cast first-guess quantities. One may combine (1) and(2) into a single one that directly relates first-guesserrors to retrieval errors:T - T = B(T* - T) + Ce or AT = B(AT*) + C~,(s)where matrix B = I - CK' = I - A, and matrix A isthe Backus-Gilbert-Conrath "averaging kernel," A= CK', as given by Conrath (1972). The radiance errorvector e can be considered to consist of two distinguishable parts. First, actual random measurement errors by the satellite radiometer contribute to e by anamount that will be denoted as el. Second, (1) itself isonly an approximation to nature since a linearizing approximation to the radiative transfer equation has beeneffected. Thus, an additional contribution to e accruesdue to errors of linearization, which will be denoted ase2. This radiance discrepancy vector may be calculatedas the difference between "perfect" upwelling ambientradiance R and its linear approximation,e2 = R - [R* + K'(T - T*)], (6)where both R and R* may be simulated by full nonlinear radiance models using ambient and first-guess profiles T and T*; respectively. Thus, the theoreticalmodel of error correlations may be written asAT = B(AT*) + C~ + C~. (7)Equation (7), which is of the form of a theoreticalcorrelation equation, is not a linear relationship between first-guess errors AT* and retrieval errors AT,since matrices B and C each depend implicitly on thefirst-guess temperature profile T*. Before applying thisformalism to selected data of this study, we may makesome further theoretical observations about the expected behavior of (7).914 MONTHLY WEATHER REVIEW VOLUME 122 NORTHERN HEMI SPHERE M I DDLE LAT I TUDES~o ~,. ..~- . ~, ~..~.~ . ':~":~ ~. ~--_~o . ,~..-;.. ,~ ../~. :~-....~ .% .~ ~ ~.-.. ~ ~:~ ~ ~~.-:~,7.~~-~, ..'~ ~~~ ~)~.~ ~..~...... ..... .., ;, , 7~ % P~.~ ~.~ ~ - ~'.~~ ~, -~ ~ ~:' '~o. ~. _,-."~ - ~ ' ' A: ,~~'~"~ T~'~, v ?''' ~ ~ ~'~ ~ ' - ~.o ~ . ~- , ' ........ ' ........... ~-2 "O ~ - - ? ....... ? ..... -. ~2.~...~/.-~, .~L~.~ ...... ..~a '~'~.., ~:~..~ ~.~,.~ ....50- ~ ~ - .~~=[ - ~ ~ ~ ,~-~r~, ~ ~,~ ~. ~ ~,~.. ~,. ,..:::,. ~ .......... ~..z~ ....~.:_~~~ - 2,- 9' ':~ ~. ~ ~U. "__'Y '~.~~ ._~ ' ~e~/~' ~~%-~. ~ ~ ' ~ ~ ~ t~'.'~~- ' ~- -'~ "~---~-, .... /:~':-~----"~' -'~---~-*'-~~--~ ~-~2 ....... L~'~ : ,.:-,: ,,., ?.. ~ ~' ~ .~- ~ ~ ~,o- ........ ~,~, ........ ,m~j.~"7' .....,~ ................... ~ ,,.., ........ .,.:~ ..................... ~c~.~ ........ ~,.x "~ ";' ~'*' ~, % ' - / , ~~ . ~ ~OE ~20E ~o uow ~ow o00Z06Z12Z18Z60- SOUTHERN HEMI SPHERE M I DDLE LAT I TUDES~o,. ..... ~...~ ...... ...............~-' ~'c(il,.,.., .,.,. o % ~. - ooz5o ...... j- ~.. ~ .....~' ~'~y~.'. .... ' .... 06Z~o ............. ~q~ ........................... ~.. ............. ~ ..... . .' ............~_ ............. ~..~ ...... : ....................... . ............... ,.~ .............................~2ow 66w o 60E ~ 2bE ~ ~0 120W FIG. 9. Difference in absolute value of 1000-500-rob thic~css c~ors computed for S~ATREG and ~-S~5 retrievals. (~SVD5 thic~css errom arc subtracted ~om STATREG thic~css c~ors.) Results for No,hem and Southern Hemisphere midlatitudes (300-60- lat) at verification times of 0000, 0600, 1200, and 1800 UTC 28 January 1990. Co,tour i~tc~al is 5 m withthe zero contour omitted forb. First-guess error transformation--Signal contribution '\ The leading term~ in (7), B(AT*), will produce nocontribution to AT if matrix B = I - CKt is thenull matrix. This holds for the special case C= (KE-~Kt)-X(KE-~), which is the Penrose-Moorepseudoinverse solution. Unfortunately, the pseudo-inverse solution is mathematically unstable when formulated for satellite sounding radiometers. (TheSTATREG solution takes this value if the error covariance matrix S is replaced by the null matrix.) For thepresent solution algorithms, matrix 15 may be calculated using (3)' or (4): B(STATREG) = I - [I + (KE-'K'S)-']-~ (8) B(TP-SVD) = I - ~z~z ,. (9)Factors that produce what appears to be nonlinear behavior in the scatterplots of Fig. 10 include the temperature dependence of matrix 15, that is, the dependence of the radiative transfer kernel matrix K and singular vectors V on first-guess profile T*.c. Radiance error transformation--Noise and linearization contributions Equation (7) shows that retrieval errors also comeabout through a mappi~ng of radiance errors e~ and ~2into retrieval errors AT. In the context of the work ofBackus and Gilbert (1968) and Conrath (1972), itshould be appreciated that this error transfer is cruciallydependent on the mathematical stability or instabilityof the retrieval solution matrix C. If C is mathematically unstable, then even very small radiance discrepM^Y 1994 THOMPSON AND TR1PPUTI 915-12-1417.3 (100un) 33.9 , .. :',:' /t~...' .. ,.;? ,.:.: .. '5-39.1 9.71412 6 4 2 0 -8-10-14 11.4 (250~.) .40.1-32.2 16.2 -12-10 -8 -6 -414-2 2 4 6 8 10 12 -12-10-8-8-4-2 0 2 4 8 8 10 12 ~ i, T, i, i - i,i, ~ - ~, ~, ~ .'1, 12 -i4.1 (700Mill 31.7 % ~~ 10 - '~ ..~ ;.~ 0 ' '**e .p;'~~ 4 .. %a. - ~ ..~ ~'.'~ ._.'~ ~ o ..,...~ -4 /~~ ~g~ _6 - :2t~ .I 'q.7._ -:':~ ': ' ' .O w~ q~ :' . : -10 -~ 43.a 10.9 - -14 ' ~ ' ~ ~ ' ~ ' ~ ' ' ~ ' ~' ~ ~ ' ~ ' -12--10 --~ --6 -4 --2 0 2 4 ~ 8 IO 12 RETR! EVA I, ERRORS (K)141210 8 6' 4 2 0-2-4-6-8-10-12-1413.9 (850Mla) 35.4, "?:; .,~;. -- ... :. :~:r~.~-...; .... -- ';:,i! ~ El~:''..'5': ~:c.:.20.5 '- ~ . ~ - ~ - , ~ - i - , - , - ~ - ~ - , -16.4 (SOOMII) 27.8-...GL'i.'. ~ .~!;v '- .~.~ i ;ii-"",-ii-~. ~;"',?~ ~r-.' 'ffi. ~ .2/-,~-*. . : ;,.* ,,I..-42.6. 13.2. ~ . ~ , ~ ..~ , ~ , ~ ~ ~ ~ ~t2I06842-2-4-6-8I01214 -12-10-8 ~(I -'4 -2 2 4 6 8 10 12141210 4 2 0-2 -4 -8-12-1419.0 (lO00M!l) 33.319.2 28.5-12-10-8 -8 -4 -2 2 4 6 B 10 12 -12-10-8 ~6 -4 -2 0 2 4 6 8 10 12 B, ETI.! EVA l, I~RRORS (K) IUTRI EVA l, Eli. ROItS (K)FiG. 10. Scatter diagrams of first-guess error T* - T vs STATREG retrieval error ~' - T for the 1000-, 850-, 700-, 500-, 250-, and 100-rob levels in the 30--60-N midlatitude band at 0600 UTC 1 February 1990.ancies are manifest as very large profile retrieval errors.For example, the aforementioned Penrose-Moorepseudoinverse solution produces a null matrix B thatminimizes the signal contribution. But the pseudoinverse solution is unacceptable, for it incorporates theinverse of the highly ill-conditioned matrix KE-~Kt,and transforms even infinitesmally small radiance errors into unacceptably large retrieval errors. This willbe further examined in section 9. The formalism outlined here provides a pathway foranalysis of the bias and overcompensation aspects ofthe data shown in Fig. 10, including the role of a prioricovariance statistics in the distortions. The STATREGalgorithm is regularized, or stabilized, by the inclusionof a priori statistics in $-~ to "recondition" the illconditioned matrix operator KE-tW. The TP-SVD algorithm is regularized, or stabilized, by diagonalizingthe ill-conditioned operator Kle-~W and then "reconditioning'' it by an approximation in which or/ly a finiteand stable set of singular values and singular vectorsare retained. As covariance statistics in $ can affectonly the behavior of the STATREG solution, parallelcomparisons of the error response of both solutions willhelp distinguish between effects involving the physics(K and E) and effects involving statistics (0).9. Empirical analysis of first-guess and radiance error transfer into retrieval errora. The context of selected case studies Using the theoretical model of retrieval error contribution developed above, one may calculate the contribution terms from environmental data. To learn howthe error transformation relationships behave in detail,four particular retrieval cases at 0000 UTC 28 January1990 were carefully analyzed. Figure 11 shows a limited-area vertical cross section of forecast and retrievalerrors extending from 110- to 30-W, and averaged overthe latitude band 40--50-N. The deep "wavy" forecastfirst-guess error structure in the top panel of Fig. 11 istransformed by the STATREG algorithm into a retrieval error structure with foreshortened vertical scale..A somewhat similar shortening of vertical scale is evident in the TP-SVD retrievals although the phase isquite different. The horizontal structure of the STATREG retrieval error is clouded by evident biasing.916 MONTHLY WEATHER REVIEW VOLUME 122 rv.vg~ ~n~<o " ....... i Li.I IM Cr..a :~ ~ ~ ~<~, : .,>0 W I%) ~0 ~ [~ _ ~ .. ~~~ ~ ~ S~ ,~DO :'~0m~ ~. .....' -~ ~~~ ~: ...... 7. ~ ~ ~ i~'": ..".~ ~ ... ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ (qw) 3unss3ud ~: ~ %./ ~-. ......................... ~ g~ ~ OZ~ ~Z ~0 - >0 ~ ~~ ~0 : ~ ~ ~ ~ ~W~~ ~~0 ~0~o~ :~ ~o~ ~ ~ ~ ~ o~ ~ ~GO,I:,(qm) 3~lnSS3BdMAY 1994 THOMPSON AND TRIPPUTI 917Retrieval analysis experiments were performed at aolongitudes 40-, 60-, 85-, and 97-W corresponding 70roughly to the maxima and minima of the 850-mb first 10oguess error patterns. Following the theoretical ideasgiven in section 8, the retrieval error at the 850-rob ~olevel is expressed as an algebraic sum of three terms: ~ 2oo aAP(850) = B(850, p)AT*(p) + t3(850, vi)~(vi) '~e~o ral + 13(850, = +/xf2 g r,o r,o 400Each of these terms is a vector product of the 850-mb ~ ~0orow of matrix B or 13 with the column vector AT*, ~, chor ~2 as appropriate. ?00 8,q0 1000b. Signal contribution term The signal contribution term A;P~ for a selected retrieval solution method can be calculated for any pairof ambient and first-guess temperature profiles. Figure12 shows the signal contribution term for a single iteration of several retrieval solutions for the case at40-W. The vertical coordinate is an equally spaced gridof the 12 sounding levels in the NMC data. In additionto the STATREG algorithm, results are shown for theTP-SVD algorithm using 4, 5, 6, and 7 retained singularvectors. The matrix stability character of the TP-SVDalgorithm is directly controlled by the number of retained singular vectors, with TP-SVD4 quite stable andTP-SVD7 quite unstable. The four panels in Fig. 12show the following: (a) the row vector A(850, p) ofthe averaging kernel matrix A = CKt; (b) the row vector B(850, p) of matrix 15 = I - A, which maps theprofile of first-guess errors into the retrieval error at the850-mb level; (c) the profile of first-guess errorsAT * (p); (d) the profile of product elements in the vector product B(850, p)AT*(p). The signal term AiP~itself is the algebraic sum of the values plotted at the12 sounding levels in Fig. 12d, and the profiles showexplicitly how this signal contribution term is composed. It may be seen in Fig. 12d that the signal contribution to 850-mb retrieval errors is largely the algebraic sum of opposing contributions coming from the700- and 850-rob levels of the forecast error field. Fordeep layers of systematic first-guess error, such as thatshown in Fig. 12c, the 700- and 850-mb contributionsare opposite in sign and are attributable to the strongside-lobe characteristics of the averaging kernel. Evenweak side lobes of A yield strong sidelobes of B = I-A. The cumulative contribution from all levels ofthe first-guess error field is thus small in comparisonto each of the opposing contributions from 700 and850 mb. Table 3 shows values of A~'(850)/AT*(850), theratio of the 850-mb retrieval error to the 850-mb firstguess error; of A~(850)/AT*(850) = B(850, 850),the ratio of the 850-mb first-guess error contribution toA~'~ to the 850-mb first-guess error value itself; and ofAp~ (cum)/AT*(850), the ratio of the cumulative sig(a):..i l,.ii I I ~ i ~ , , 'rP B'VB6 m 1 t ! II!!l - r-,,'Sk4~till-0.8 -0.4 0.0 0.4- 0.8 -0,4- -0.2 0.0 0.2 0.4x(~5o.p) ,(o) il! ,;! d ': i 100 100 --8 --4 0 4 8 -4 --2 2 4 FG ERROR (K) RESPONSE (K) FiG. 12. (a) Profiles of row vector A(850, p) of the averagingkernel matr~ relevant to 850-rob temperature retrieval; (b) profilesof row vector B (850, p) of the e~or transfer averaging kernel matr~relevant to the 850-rob temperature re~ieval; (c) profile of forecastfirst-guess error, ~T *~ ) = T * - T; (d) profiles of elements of thevector product B(850, p)~T*~). Results are shorn for 40-W at0000 UTC 28 Janua~ 1990 for STATREG and ~-SVD retrievalalgorithms.nal contribution term to the 850-mb first guess error.Overcompensation of the 850-mb first guess error isindicated by a negative value of A~'(850)/AT*(850)and occurs at 40-, 60-, and 85-W while only the TPSVD4 algorithm produces overcompensation at 97-W.The signal contribution term at the 850-rob levelA~'~ (850) is itself always positively correlated with thefirst-guess error, A T * (850), since B (850, 850) alwayshas a positive value. The ratio AP~(850)/AT*(850),measuring signal transfer, decreases as the retrieval algorithm becomes less stable (that is, from TP-SVD4 toTP-SVDT). This is because as the retrieval algorithmbecomes less stable, the value of A(850, 850) gets918 MONTHLY WEATHER REVIEW VOLUME 122 TABLE 3, Values of the ratio of 850-mb retrieval error to 850-mbfirst-guess error, At(850)/AT*(850); ratios of the 850-mb value ofsignal response function to the 850-mb value of forecast first-guesserror, At~(850)/AT*(850); and ratios of the cumulative value ofsignal response function to the 850-rob value of forecast first-guesserror, At~(cum)/AT*(850) for four selected cases at 0000 UTC 28January 1990. Results are given for one iteration of STATREG andTP-SVD retrieval algorithms truncated to four, five, six, or sevenvectors,First-guess too First-guess toocold warm40-W 85-W 60-W 97-WA ~'(850)/A T *(850)STATREG -0.243 -0.33 -0.110 0.149TP-SVD4 -0.227 -0.199 -0.141 -0.186TP-SVD5 -0.102 -0.409 0.438 0.213TP-SVD6 -0.526 -0.541 -0.057 0.644TP-SVD7 -0.027 2.028 -3.579 0.059A ~'~(850)/ A T *(850)STATREG 0.271 0.353 0.307 0.284TP-SVD4 0.445 0.512 0.511 0.386TP-SVD5 0.213 0.275 0.260 0.207TP-SVD6 0.081 0.095 0.106 0.044TP-SVD7 0.039 0.045 0.045 0.037At,(cum)/zkT *(850)STATREG 0.037 0.079 0.023 -0.179TP-SVD4 -0.140 0.045 -0.378 -0.321TP-SVD5 0.187 0.188 0.408 -0.268TP-SVD6 0.066 0.107 0.023 0.152TP-SVD7 0.015 0.043 -0.040 0.152larger but the value of B(850, 850) gets smaller. However, the cumulative signal contribution term involvesa general cancellation of 850- and 700-mb effects leaving a small residual that may correlate negatively withfirst-guess error, as shown in Table 3 by negative values of the ratio A~'~(cum)/AT*(850). This decoupiing of A~'~ from its 850-rob value A~'~(850) owesdirectly to the sidelobe of the averaging kernel thatpicks up substantial contributions from the 700-mblevel of the first-guess field. As the instability of theretrieval algorithm increases, the 700-rob first-guess error becomes more influential to 850-mb retrievals thanis the 850-mb first-guess error itself! Comparing thevalues B(850, 700) and B(850, 850) in Fig. 12b, onlythe most stable TP-SVD4 retrieval algorithm has theproperty that the 850-mb first-guess error contributionto 850-mb retrieval error exceeds the 700-mb firstguess contribution. In these results, the STATREG algorithm shows signal transfer characteristics somewhere between those of TP-SVD4 and TP-SVD5. Inthis regard, according to Wahba (1985), the trace ofthe averaging kernel matrix A represents the "degreesof freedom of signal." For the TP-SVD algorithm, thistrace is 4, 5, 6, and 7 for TP-SVD4 through TP-SVD7,respectively. For the STATREG algorithm, Tr(A)=5.013(40-W), 4.833(85-W), 4.997(60-W), and4.878(97-W), that is, about five degrees of freedom ofsignal for the 15 infrared and microwave soundingchannels used.c. Radiance measurement error contribution to retrieval error Let us turn to the radiance error contributions to retrieval error. Table 4 shows values of the radiance measurement error contribution to 850-rob retrieval errors,At2, and linearization error contribution, A~'3, eachgiven in kelvins. The radiance measurement errorswere drawn randomly from a Gaussian population withstandard deviations given by the diagonal elements ofnoise matrix i:, and the same set of channel randommeasurement errors were used for each of the fourcases in Table 4. The response to measurement errorstends to be larger, in magnitude, the more unstable theretrieval algorithm. The values shown suggest that adverse measurement error transfer may contribute onlya relatively small contribution to temperature retrievalerror for algorithms with reasonable stability characteristics. However, the contribution to retrieval errorowing to linearization assumptions is a different matter.This error transfer also tends to be more problematicthe more unstable the algorithm. Values are substantially larger than the measurement errors and may contribute several degrees to the retrieval error.d. Correction of radiative transfer model linearization errors Errors due tO linearization should presumably be"correctable" by iteration of the retrieval algorithm.The TP-SVD algorithm unambiguously submits to aniteration process since it involves only the radiativetransfer kernels that are updated as successive estimatesof the temperature profile are calculated. The STATREG algorithm, on the other hand, would require a reconsideration of, and recalculation of, the $ covariance TABLE 4. Values of measurement error contribution A~'2 (K) andlinearization error contribution A~3 (K) for four selected cases at0000 UTC 28 January 1990. Results are given for one iteration ofSTATREG and TP-SVD retrieval algorithms truncated four, five, six,or seven vectors.40-W 85-W 60-W 97-WSTATREG 0.05 -0.15 -0.17 0.15TP-SVD4 -0.09 -0.16 -0.09 ' -0.14TP-SVD5 0.08 -0.09 -0.02 0.27TP-SVD6 0.28 -0.29 -0.51 0.48TP-SVD7 -1.59 -3.38 -4.21 -0.43STATREG 2.84 4.20 -0.23 0.58TP-SVD4 0.99 2.56 0.80 0.44TP-SVD5 2.92 5.96 0.11 0.81TP-SVD6 5.84 6.66 -2.72 0.62TP-SVD7 2.02 -16.14 -6.37 0.23M~Y 1994 THOMPSON AND TRIPPUTI 919matrix after the first iteration since an a priori hydrodynamic forecast error covariance matrix is certainlynot relevant to second- and higher-order iterations. Forthe four cases analyzed in this section, additional iterations were forced of all algorithms (arbitrarily holding$ fixed for STATREG) so that we might examine thebehavior of the response factors upon such an iterativeadjustment. Table 5 shows values of the retrieval matrix condition number F defined after Golub and VanLoan (1983) as the ratio of the largest to smallest eigenvalue of the retrieval matrix; the 850-mb first-guesserror AT*(850); the retrieval error A~r(850); and parameters A~'~, A~'2, and A~'3 for three iterations of eachalgorithm at the four test longitudes. The role of a prioristatistics in the STATREG algorithm has the effect ofmaintaining a rather consistent condition number overthe iterations of these four very different cases. Thecondition numbers for the statistics-free TP~SVD algorithms show more variability between iterations andbetween cases. It may be seen that the STATREG algorithm profited by a second iteration in three of thefour cases (40-W, 85-W, 97-W). The STATREG retrieval at 60-W was the only case where an optimumresult occurred in a single iteration. For all retrievalalgorithms applied to the four cases (except the unstable TP-SVD7 algorithm that diverged for three of fourcases), it may be seen that the error contribution dueto linearization was substantially reduced from the firstto second iteration.10. The influence of retrieval iteration on characteristics of error transfer It would appear from the selected cases in Table 5that contributions to retrieval error attributable to thelinearization of the radiative transfer equation may besubstantial. Using the error analysis theory of section8, the STATREG retrieval errors illustrated in the scatterplots of Fig. 10 can be partitioned into signal, noise,and linearization components. Figure 13 shows such apartitioning of the 850-mb STATREG errors of Fig.10. For this figure, and similar ones to follow, thereader should also consult Table 6, which shows ensemble mean bias errors and rms errors for these experiments. The leftmost panel of Fig. 13 showing totalretrieval error at the first iteration is identical to thecorresponding 850-rob plot in Fig. 10. The partitioningof these errors into their signal, noise, and linearizationcontributions is shown in the subsequent three panelsextending to the right. The signal contribution termshows a weak but detectable correlation with forecastfirst-guess error, but this term does not explain all ofthe interesting characteristics of the error scatterplot.While about 66% of the retrievals were too warm, leading to the mean bias of 0.617 K, only about 53% of thesignal contributions are too warm. The measurementnoise contribution is generally small with no apparentcorrelation with first-guess errors, and insignificantbias. The linearization error contribution is a very interesting panel in Fig. 13. This term is highly biasedtoward producing overly warm retrievals, with slightlygreater than 70% of the cases showing positive linearization contributions to 850-mb retrieval error. Theeffect is similar no matter the sign of the first-guesserror. Retrieval experiments were repeated for the independent Northern Hemisphere midlatitude data of 0600UTC i February 1990 (the data of Fig. 10) such thatthree iterations were performed for TP-SVD retrievalalgorithms retaining 4, 5, 6, and 7 singular vectors. Inthe iteration procedure, spectral transmittances, kernelmatrix, and SVD decomposition are recomputed ateach iteration and at each grid location. For each iteration of each case, the signal, noise, and linearizationcontributions to retrieval error were computed as wellas the total 850-mb retrieval error A~'(850). Figures14 and 15 show decomposed scatterplots for three iterations of the most stable TP-SVD4 and least stableTP-SVD7 algorithms, respectively. Readers shouldnote that "first-guess errors" relate to NMC forecasterrors only at the first iteration; thereafter, guess fieldsare the previous iteration of the retrieval algorithm. Thetendency for the error correlation to converge towarda straight line is an indication of algorithm convergencesince the (n + 1)th iteration should be identical to then th in the limit of a converging sequence of linear problems. Table 6 shows that iteration reduced the meanbias error at 850 mb for each of these experiments.Even the least stable TP-SVD7 algorithm moves in thedirection of convergence upon iteration even thoughthe rms retrieval error after three iterations still exceedsthe first-guess error. The data of Table 6 strongly suggests that the mean bias error of the STATREG retrievals may be due to an undesirable degree of matrix instability. It is seen in Fig. 14 that the signal contributionfor TP-SVD4 also tends to converge in the sense described above, while that for the unstable TP-SVD7does not (Fig. 15). For the more unstable TP-SVD7algorithm, the retrieval errors become more related toradiance noise amplification and linearization errors. Atthe first iteration, the correlation between the signalcontribution term and forecast error is strongest for(TP-SVD4) and weakens, or disappears, as the algorithm becomes unstable. This may be appreciated fromthe results in Fig. 12, where it is seen that the moreunstable the algorithm, the smaller the value of B(850,850). In the limit, as the number of retained singularvectors approaches 12 (the maximum), the error transfer kernel 15 becomes the null matrix so that there is nosignal contribution at all. In Figs. 14 and 15, and others not shown for TPSVD5 and TP-SVD6, the contribution to the 850-mbretrieval error due to radiance measurement error is farless problematic for stable algorithms and becomes increasingly important as the algorithm destabilizes. This920 MONTHLY WEATHER REVIEW VOLOMB 122MAt 1994 THOMPSON AND TRIPPUTI 921is exactly as expected based on the Backus-GilbertConrath concepts. The contribution to the 850-rob retrieval error due tolinearization assumptions of (1) involves the same retrieval operator matrices but significantly, different radiance errors. While measurement errors are introducedas random errors, errors due to linearization assumptions should be systematically related to the first-guessfields since first-guess errors are a direct measure ofhow good or bad the linearization might be. Onceagain, one may see in Figs. 14 and 15 that for stableretrieval algorithms (e.g., TP-SVD4), such error transfer into temperature retrieval errors is not such a seriouseffect. As the algorithm becomes more unstable, thisbecomes a significant contribution to retrieval error andmay produce rather unexpected effects, such as thesomewhat peculiar shapes of the point distribution forTP-SVD7. Given what has been said above about converging sequences of linear problems, the iteration ofunstable retrieval algorithms may be toward a conditionwhere linearization errors are "frozen" into the solution rather than removed from the solution.11, Iteration of the truncated, physical singular value decomposition solution The experiments with iteration of the TP-SVD algorithm in the previous section shows the tendency foriteration to improve both the bias and rms errors provided it is not too unstable. Ordinarily, one stops aniteration process when some radiance convergence criterion is met. This decision is made on the grounds thatone has no rational reason to iterate further within the"noise level." The present study provides such rationalgrounds for further iteration. Based on the analysishere, the authors argue that even at such a "converged" step, the radiative transfer kernels that wereused in that step are known to be in error since theywere evaluated for a profile condition (the previousiteration) known to be imperfect since it did not produce a convergent condition. Thus, the radiative trans TABLE 6. Mean bias and root-mean-square errors at 850 mb forNMC T80 spectral model forecast and retrievals using the STATREGalgorithm and iterated TP-SVD algorithms with four, five, six, orseven retained singular vectors. Statistics are for 0600 UTC 1February 1990 in northern midlatitudes.Mean bias error 850 mb rms error 850 mbForecast -0.096 2.37STATREG 0.617 1.76TP-SVD4Iteration 1 0.070 1.53Iteration 2 -0.015 1.48Iteration 3 -0.031 1.48TP-SVD5Iteration 1 0.089 1.48Iteration 2 -0.002 1.42Iteration 3 -0.025 1.42TP-SVD6Iteration 1 0.845 2.34Iteration 2 0.141 1.60Iteration 3 0.045 1.51TP-SVD7Iteration 1 1.799 5.55Iteration 2 0.640 3.63Iteration 3 0.163 2.86TP-SVD5-plus -0,016 1.42fer kernels should be updated and the algorithm repeated one additional time. To test this idea, the authorsdeveloped a simple iterative version of TP-SVD5 andapplied it to the independent data shown previously inthis paper. The iteration scheme is straightforward: theTP-SVD5 algorithm was iterated until it achieved radiance convergence in the sense that root-mean-squarebrightness temperature errors, over the 15 HIRS-MSUchannels, fell below the rms noise values composingmatrix ~=. Upon such convergence, the algorithm wasiterated one additional time. This iteration procedure isdesignated TP-SVD5-plus and its performance may' RIgTRIE'VAL SIGNAL NOISg LINRARIgATION~o I i 'l 'l r I ' [,. [.'l 'l'l'i'l'l 'l't'lrl'*'. .'l'l'l'I'l' I 'l'l'l'l'~ .' I'l '~'l '1' 'l' I ' ['1 ' [ 'om [cr13.9:"'""'"" , 35.4_" ~13.1 ,.. 36.3_- -~5.7 .. 23.6': -215.6 ,, 33.7_-~ .". .... : i..,..::!.~ .: .:~.,.. C .~ ; ,f.:.. .. ~ _z :~':3;'..'" ' [' :,ffi ~?..5:,. 5 ~ "~"(! i '~. ~ .- ;2 c~;:~ ,-. i i. a ~ i i .~,,.- . V . ~:~ ' . :":" ..... .~o.5 ,'? , :?,.f., ,~'~ ',l,l,l,~,~,] ,~, ,~,~ ,~'~' -i0 0 10 -10 0 10 -10 0 t0 -10 0 I0 CONTRIBUTION CONTRIBUTION C0NTRIBU~ON ' ~0NTRIBUTION F~G. 13. Scatter diagrams of partitioned contributions to retrieval errors for the STATREG retrieval algorithm for 0600 UTC 1 February 1990 between 30- and 60-N. From left to right: total retrieval error at 850mb ~'(850) - T(850) versus 850-mb forecast first guess error T*(850) - T(850); signal contribution term/x ~ versus first-guess error; measurement noise contribution term A f2 versus first-guess error; linearizationerror term Af3 versus first-guess error.922 MONTHLY WEATHER REVIEW VOLUME122 ITERATION 1RETRIEVAL SIGNAL NOISE LINEARIZATION~0 to ~i"~'"". '"'"'"'"~.?.6-~ -~'"'. "'""""~.~..02 ':~'"'. '"~'"'~4.5~ ~""9.? ..... "'~'"""'""~9.6~ ....~ .~ , : -~' .:: ' . __ : .. ~ .... 7'.. ::gt-~V.4::?~:'/.:'lB.3~ h8.;:"~~~';:::"i3.7 25.3 25.4 ~8.4 :~ ~g.g~~ '~.~.~.~.~..~.~.~.~.~.' -.~.~.~.~.~..~.~.~.~.~. .,.~.t.~.~..~.~.~.~.~ ~.~.~.~.~..~.1.~.~.~. -10 0 10 -10 0 10 -10 0 10 -10 0 10 ~ON ~ '"'"' .... '"'"'"" ~'d6'"' ..... "'"~" ~2'~'"' ""~'~' ~'~" .... '"'""~~ 5 2.0 48.05 - 45.82 - - 5 - 28.9 /--____ ./. .... _- _- -~ ~ --~l~ ~,' ~ ~,'~ ~' ~,'~ ?~' ~.9:' ~,o.a~:. . . .~.~..~.l.~.. . .... ].~..~.~.~ .~.~..~.~ .... - ~]~ ~.~.t.~.~.~-10 0 10 -10 0 10 -10 0 10 -10 10 1TERA~ON 3o~ lCm0~tv~ -10 0 10 -10 0 10 --10 0 10 -10 0 10 CONTRIBUTION CONTRIBUTION CONTRIBUTION CONTRIBUTION FiG. 14. Scatter diagrams of partitioned contributions to retrieval errors for three iterations of the TPSVD4 retrieval algorithm for 0600 UTC 1 February 1990 between 30- and 60-N. Panel quantities arc thesame as in Fig. 13.now be appreciated in Fig. 1 and in Tables 1 and 6.Figure 16 shows the 1000-500-rob thickness errors fornorthern and southern midlatitudes using TP-SVD5plus and may be compared against Figs. 6, 7, and 8.Substantial improvement occurs with iteration. Figure17 shows the difference in absolute errors betweenSTATREG and TP-SVD5-plus, comparable to Fig. 9.This iterated solution appears to produce very good results in the lower troposphere, in comparison to STATREG, and quite competitive results in the upper le~,elsas well.12. Conclusions In this paper, the performance of two linearized inverse algorithms for satellite temperature profile retrieval were examined in an interactive mode wherebyeach algorithm was initialized by a numerical forecasttemperature profile produced by the NMC T80 spectralmodel and compared against "ground truth" consisting of NMC verification analyses. The two algorithmsmay be distinguished by the manner of regularizing theill-posed nature of the associated inverse problem. TheSTATREG technique regularizes the problem by introducing a priori forecast model error covariance statistics into the matrix operator in a manner such as toreplace small eigenvalues of the radiative transfer operator with larger, statistically derived numbers so thata stabilized inverse may be calculated. The TP-SVDtechnique regularizes the problem by truncating thespectrum of eigenmodes of the radiative transfer operator in such a way as to retain a well-conditionedsubset but allowing no influence of statistics to modifythose eigenvalues retained. The TP-SVD approach maytherefore be argued to be more "physical" than STATREG by definition of the regularization. Each, however, is initialized by a grid-located first-guess profileobtained from the numerical weather prediction model. The results of intercomparisons show that both algorithms, when initialized with a hydrodynamicweather forecast model, produced temperature profileretrieval errors that were reduced in magnitude but thatM~- 1994THOMPSON AND TRIPPUTI ITERATION 1RETRIEVAL ' SIGNAL NOISE L1NEARIZATION""'"'"" '"'~e'~" "'"'~"'" '"'"'"' " '~'"'"' .......... '" "'"'"'" "'"8'~"~0.4 .... _-'-~3. . 25.7: 2 .0 ..;..25.4_-.-..~0.5 . .. 2. ii~.~.';'! i:.~ "~ ~- '.' .-~i,- ~ :.,.; ' :!~- ,' ~.,,...?..:,' - ~ . :.,.. ~-'.-.; ::~' : : ~:i' .~i ~' .c. ~.~; -. ~ :' - .,~ .... ~,..,... .'~.... ., ".~,~ ! g': ~ - ,. ~-~ ~ ..... - ~ - ,~ '~',~ .....- . ~?,:~--:;~_ ,.v-~ ,~ - ...~-- .~.... . ........., -.~.., ~.,...,...... ,. -, .... ~ ~'..'. - - ".~x'~ "~ ..': .','.~,~ ~ ~.~ ~ ~' . ..,..~ ~,' : : ...~ ,:~l~ ~;:'.~ L' ~: - ' .'-~ ,~" - - - ' .'re--, ..... *a . . ~1. ~ - - ' -. :; I. - - ' - ' ' . - .'...'. -. ' ' - . - - _.. _ "' " '"~ ';'~'~.24.2~ i12.1 ' :" "::[15.4 g5.3~ '~1.3 .~ 19.4~ ~6.5 - ." 38.6'.~.~.~.~.~,i,1,1,~,1,' '.~.~.~.~.~ ~,t.~.~.~,- -.~.~.~.~.,. ,~,~.~.~.~.- -.~,~.~.~.~ ~.~.~.~,~.~ '"'"'" '""?;"'- ." '" ' " ' "i" " "" "- -'"'"' ..... '~'"' ...... '"'""'" ..~m- lo 15.5 ;.~72 ~7.3~ 26.9-- .~6.4 ~ (' 37.8:35.7 .:~5~= .~ '~,~. : : ,. ~ '.~ .~... - ~ . .,;' .~',:' ~ 'X~ : - ;y~ -~. - : . ..~ ' .;~Z' .,T':': : ~ - .:,i .... '.:,- ....:?~ :. ~ [' ~...:~ .:~, . ~ -;~ ~ :..:~.;;:- . ~ : ;~ ~; - ~:' ~s~ ..... '~; : .~.~ .j - : .'.., ':t '; .:: +m~ '~f:~' ~' ~ ~ i : S I .~ ,~::I ' I : I ~;~ : . "':X' '~ - ~ : :,,?, ~'~::~-' ~.o~ ~.~ ~ ,o.~ ~.~':.'::;F.~. 9.a ~6,.~:?: 9a L, I, I, ~. ~ Ii [ iIi [i [ll~ '~1 ,i.l~l ,L. [,~ ",1, ~,1.1.~.[,1 *1 . [, I*1 I I, .I ,;.I., ,I,1,~,~,1, -10 0 10 -i0 -0 10 -10 0 10 -10' 0 10 ~ON 3. L'. - . :... .':. !m ........ . .....~.?. - .,.~ ~o.~ ':. ..:,~ i ,~ ~ '?;.:-a "i'? - '. ~ ~,' :4..: ., -.- ..:_, - "":"-" ...:i.i, .-::': - : - -. ,., '~ :-.: .... ~.".:-.-:- ,' . . -.:::'.:::: ? , i_ ? ~_ ...... ...:.:,- . ::. :_:4,0,!.a, ~,! ~,: ~!~ ,.,' 1, o ~,~ . 5 i~3.2 . . 31.8' 13.62~ ', , , ,I,~ ~.~,~ [ . .... , rlllr,i i, i~,1 ,i i' , , , , l, I. I* [, [. I ,I i,I, [, -10 0 10 -10 0 10 -10 0 10 -10 0 10 CONTRIBUTION CONTRIBUTION CONTRIBUTION CONTRIBUTIONFIG. 15. Same as Fig. 14 except for the TP-SVD7 retrieval algorithm.923were undesirably correlated with the forecast firstguess errors. This correlation was not only apparent ina statistical sense but was also manifest in systematic,synoptically correlated patterns of retrieval error similar to patterns of forecast error. The error transfer wassuch as to maintain systematic distortions to hydrostaticand baroclinic stability fields that existed in the forecastfield. From the point of view of dynamical weather prediction, this is precisely what one does not want to happen when incorporating independent observational datafrom weather satellites into the forecast problem. Initializing the satellite profile retrieval problem with numerical weather predictions of the ambient atmosphericstate would seem to guarantee that the actual independent satellite radiance data is transformed into a sounding consistent with the forecaster's model of what thedata ought to be. The dependence of satellite profile retrieval errors onforecast first-guess errors has been carefully examined.Were first-guess errors random in time and space, theimpact of such errors on retrievals--and subsequentforecast cycles in a fully interactive system--might notbe a serious problem. However, in the database examined here, the NMC T80 forecast model producederrors that were disturbingly systematic in the spacetime domain, often manifest as moving, evidently synoptically correlated error centers. Owing to the substantial depth and breadth of such first-guess error centers, the resulting retrievals show similar features. Suchbehavior raises concern that interactive retrieval methodologies might introduce false, "satellite-retrieved"hydrostatic and baroclinic features to forecast modelsthat are actually unrelated to the satellite measurementsthemselves.~ 1An anonymous reviewer takes exception to our interpretationshere, insisting on inclusion of the following rebuttal statements:"[The reviewer] strongly disagrees with the authors' interpretationof their own results. The authors express concern that retrieval errorsare influenced by first guess errors, and feel that use of a forecast firstguess for retrievals might result in an amplification of errors anddegrade future forecastS. A comparison of Fig. 8 with Fig. 6 showsthat while retrieved thickness errors are in phase with first-guess errors, they are greatly reduced .... I interpret this as a positive, nota negative, result. If the system were fully interactive, the next forecast error would be much smaller than if no retrievals were assimi924 MONTHLY WEATHER REVIEW VOLUME12260,504050 J405040~oI3O ~0- NORTHERN HEMISPHERE M I DOLE LAT I TUDES ,::., ;,.~ - ~:-~%,- ..~ ~ ~,.--~j ~.,..,, ~ - ~~ .'.. 't~:'...'.f... ~ ~ ...... ' : .' .... ~ .... ~ ~ ~ ~ /~ .?;~ .... ' ~ .. /~.. ~ ~ ~.~ C:' ~ ~t:.~ ~,........ ,.;. ~.~',~ ~... ~. ~.~: .- ...... ~.:~ :~.~ ~ .~"' ~'~';L ,~ _ ~_ ..~ ..... ~ .... ~;~ ..."~ ~. ~' ~ . ...~ ~ ',~> ::~ -~ ~ -~~'::. ' ' - '~- ......... , ..... '~ . ........ ~ ............~ ~ ~C~ ..... t~.~_ .... ~ . -: ..... ~,.~,.~, ~:.'~ .~ .... .~ ,: .... . ~.~~ ~. - ' -~- ' _ ~-~; -~- ~_ "~ ,,-~ ~ ~ ~ '~ ~z?~ ' ' ............. ~ ........ ~ .. 5/...: ..... ~ ......... X~ ...... ,~..'N? .... ~.r> .... ~..~,.'~i'~ ...... ~ ~ , , ~ ~.-.~,J~ ~ ~ ..., ~ '., ~P" - - ~ .~ ~2~%~1......... ~ - .~ ~ ..... ~., . .... ..~~, ,:' ~ ~ '. -~, ~'~~ / .-. '~- ?, ~ g' .... ~" ~ '" ';~;~. ~ -~-.~- :'~ / ....... (" ....... , ...................... ~ .......... ~ ........... ~' ' ' '~' ' *;~ ........ ,~ ......... ~:; ..... / ...2 .... m .... ;;~ ~ "..' ...................... i.:.~:';: ..... ~-~.C 5..' ..... : ..... ~m~,~..~t '~ ~' - ~ L-' ~ '~4 ~ ~ ~~ ~'~ - n., ,-*' _, X , '-~ , / 120E 180 120W 60W 0 60EOOZ06Z12Z18Z SOUTHERN HEM I SPHERE M I DOLE LAT I TUDES40 ......... ~.~ . ...... - ; .......... ; ............... ~ .... w ............ .... , ..........50 ..... '~ - ' d" '40 ...... ~ .,~ ~ e ................. .............. /...~ ..................... .W .......................... '....~~ ...... ~ .................~o- ~../ , ~ . .~ ~ .~o ......... ~-~. .... . .......... . ~ -,:~ . ............. ............... ..................... ............... ............. ...... ................. /~ ~3 - ~? -~o ................. ~ ...... j ..........~ .............. i ........ . ............................... "' .................'3~2ow 66w d 66E 120E 11~0OOZ06Z12Z18Z120W I~G. 16. TP-SVD5-plus retrieval 1000-500-mb thickness forecast error for. Northern and Southern Hemisphere midlatitudes(300-60- lat) at verification times of 0000, 0600, 1200, and 1800 UTC 28 January 1990. Contours are drawn at 1-dam intervalswith the zero contour omitted for clarity. The results of this intefq0mparison show that thetruncated, physical SVD approach using HIRS-MSUsounding channels can produce improved retrievals inthe lower portion of the atmosphere from which thereis a greater abundance of information in the radiancemeasurements. The TPLSVD algorithm exhibits poorerperformance in the high levels where radiance inf6rlated and the system may well converge rapidly. I require this alternative interpretation be addressed in the paper and statements aboutuse of an interactive system be more balanced .... [The authorsmust] express my point of view that a lessening of first-guess errormay not be bad in an interactive system if it leads to a reduction insubsequent forecast errors. This can be expressed as an alternativepossibility to the authors'-claim that this may (not will) cause forecasts to get worse." The authors would agree that additional researchis necessary to fully sort out the evident advantages and disadvantages of fully interactive retrieval systems.marion content is weaker and where good a priori.thermal information may therefore be positively influential. The results further reveal that a retrieval algorithmmay overcorrect, or otherwise adversely respond to,first-guess errors when those errors are systematic overa deep atmospheric layer. An analysis of this behaviorhas been given where we have traced several, somewhat independent influences that contribute. Whenforecast first-guess errors occur in coherent three-dimensional spatial patterns--associated with orography, synoptic weather systems, or other systematic influences-then "side lobes" in the retrieval averagingkernel can produce contributions to retrieval error, attributable to first-guess errors at adjacentlevels, whichare highly coupled, thereby systematically degradingthe retrieval. This feature, along with adverse radianceerror transfer, is dependent on the degree of matheMAY 1994 THOMPSON AND TRIPPUTI 925 NORTHERN HEM I SPHERE M I DDLE LAT I TUDES~o ~ -~.:.,~ ...;, - '~./s ',~ '~, ~ ~ ~'-'~,.~: ,- ::~ '_~' ~ 2~ '='.,~, ~ .- ~ - .... ' ' ~ ~'" g~- ~ ~ .~" ~ ~,.~ ~' ~- ~ ~.~ ~, ~, ,, ~.o ~~,. ~~'(2~.?~ ~...:, o. ~ 2~~~i'~ ~i ~ '~ ~~':~'J ~ ~. ' __ ~ ~ ~o~' ,, r,-,% ~,~,,,~ ~/~' -' ~ M ~ , ~ ~ '~" ~> ..... a % - n be'-~ - ~' ' ~ *.' ~ ~ ' . :~' "~ ~ ~ . . ,:~ .v~ .~.. ~.~ ' ~ ,. ~% ' ~ ~.0~ ~, -~.%~, ~.~ - ~,. ,,.-'~. - ~~ -~ ~~ - ~ ~C~ % -- ': ~ e /~ ~ ~~ ~~~ ' ~ ' ~~ '~-' ~ - ~~~~~~'~~~~~~~ .~'~';:~' ~ ~-~'.a~',o,o I ~ ..... ~~~-~~~~~~~~~~~~"~ ~ ~::~'50~ ~"' ~"-- "~~ "~ ~~~)~ "~~~~~~~.~ ~ ~"' ...... '~"p-~ - ~~~~ ~~~~~~~~: ~.& '~/~ ~ ,.~,o ~ ..... ""~~~~~ '~~ ~'~ ' '~~~~ -" '~ ' "~' ~ .X ~ ,~~,:~ -. - . ,~?.. / ~ ~'~.~-~ ~60Eooz06Z12Z18Z120E 180 120W 6~W 60E SOUTHERN HEM I SPHERE M I DDLE LAT I TUDES5o ........... n '~'~+;~'" .......... ' ............... ~ ......................... ~'~ ' '"~ .............. 12Z~ .....120W 60W 0 60E 120E 180 120W FI~. ~7. Difference in absolute value of l~0-500-mb thic~ess e~o~s computed for STA~EG aad ~-SVDS-plus [et~ievals.(TP-S~5-Plus thic~ess e~ors ~e sub,acted from STA~G thic~ess e~ors.) Results for No,hem and Southern Hemispheremid]ati~des (300-60- ]at) at verification times of 0000, 0600, 1200, and 1800 UTO 28 January 1990. Contour inte~a] is 5 mwith the zero contour omitted for clarity.matical instability of the retrieval algorithm. Furthermore, retrieval algorithmic instability is itself a systematic function of the forecast first-guess field. These influences may lead to the production of satelliteretrievals with systematic error patterns that are notvery simply related to model performance or to satelliteradiance observations. Theory has been given that suggests that such complex behavior is related to propertiesof the signal response of a retrieval algorithm, to adverse transfer of radiance measurement errors due toalgorithmic instability, and to adverse response to intrinsic errors attributable to the linearizing assumptionswithin the retrieval theory. The singular value decomposition approach has shown that this behavior is intimately linked to the degree of mathematical instabilityof a retrieval algorithm and further suggests that aninteractive version of the so-called minimum variancemethod may involve undesirable degrees of instabilityfor tropospheric sounding. Examples have been given showing that some of thesystematic error transfer effect may be reduced by iteration of the TP-SVD retrieval algorithm. Such iteration is unambiguous in the TP-SVD approach since noredefinition of a covariance matrix at each iterative stepis required. Indeed, we propose that the truncated,physical singular value decomposition retrieval algorithm can be iterated to advantage at least one additional time after it has converged below noise level ina radiance sense. Parallel testing of this procedureagainst the STATREG retrieval procedure producedsuperior tropospheric definition of the thermal field inthe cases analyzed here. The singular value decomposition method for deriving retrieval algorithms reopens an avenue for further926MONTHLY WEATHER REVIEWVOLUME 122research. The. SVD approach provides an orth0gonalrestructuring of the source functions in the radiativetransfer equation in such a way that one may freelyincorporate many overlapping weighting functions ofmodern sounding instruments while controlling the adverse effect of redundancy through Strategies of truncatioh, or of regularizing only those structures that mapradiance errors adversely into retrieval errors. Not onlyis this approach a competitive candidate for processingsatellite data, it has proved quite useful in studying thebehavior of the'well-known statistical regularization, orminimum variance approach. The singular value decomposition retrieval strategy would appear to be ideally suited .for the advanced, plenti-channel soundinginstruments Of the Earth Observing System era since itwould not require exotic a priori statistics to regularizethe highly overlapping suite of potential soundingchannels. Restructuring the radiative transfer problemusing the SVD method would als0 appear to be a usefulapproach to investigate for the purpose of extractingclimate and global change variables from satellite-observed radiance data without such a heavy requirementto introduce apriori statistical or model-generated estimates of what thai change might be. Acknowledgments. This research was sponsored inpart by the National Aeronautics and' Space Administration through Grant NSG5-730, and by the NationalOceanic and Atmospheric Administration, NESDIS,through Grant NA27WCO275-01. The authors wish tothank J. Susskind (NASA/GSFC) for use of his HIRSMSU transmittance computation code; and W. Baker(NOAA/NMC) and B. Katz (General Sciences Corp.)for providing NMC spectral model forecast and analysis data used in this study. REFERENCESBackus, G. E., and J. F~ Gilbert, ~968: The resolving power of grossearth data. Get'phys. J. Roy. Astron. Soc., 16, 169-205.Chedin, A., N.. Scott, C..Wahiche, and P. M0ulinier, i985: The im proved initialization inversion method: A high resolutio~i phys ical method for temperature retrievals from satellites of the TI ROS-N series. J. ClimateAppl. Meteor., 24, 128-143.Conrath, B~; 1972: Vertical resolution of temperature profiles ob tained from remote sensing measurements. J. Atmos. Sci., 29, 1261-1272:Crosby, D. S., and M. P. 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