STATUS REPORT 

GRANT NAG8-059 

APPLICATION OF SATELLITE DATA IN VARIATIONAL ANALYSIS 

FOR GLOBAL CYCLONIC SYSTEMS 

G. L. Achtemeier 

Climate and Meteorology Section 

at 

University of I1 1 inoie 

Illinois State Water Survey 

2204 Griffith Drive 

Champaign, 1L 61820 

April 1987 

Prepared for 

NASA-George C. Marshall Space Flight Center 

Marshall Space Flight Center, Alabama 35812 

(NASA-CR- 180259) APPLlCAIICN CE SAIiELLITE Ne7-15852 

LATA 1N VARIATICbAk AEALYSIS FCE GLCBAL 

CYCLCNIC SYS'IEHS Status Be~crt (Illinois 

State Water Survey) 14 p CSCL OUB Unclas 

G3/47 43503 
? 

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1. Introduction 

Summaries of work underway during the first year of Grant NAG8-059 are 

presented in the following sections. Some of the research has progressed 

sufficiently for the preparation of manuscripts and the status of these is 

noted. Other ongoing research is not yet ready for publication. A 

bibliography of all papers and abstracts presented as part development 

of the variational objective analysis follows the summaries. 

of 

The goal of our research is a variational data assimilation method 

that incorporates as dynamical constraints, the primitive equations for a 

moist, convectively unstable atmosphere and the radiative transfer 

equation. Variables to be adjusted include the three-dimensional vector 

wind, height, temperature, and moisture from rawinsonde data, and 

cloud-wind vectors, moisture, and radiance from satellite data. This 

presents a formidable mathematical problem. In order to facilitate 

thorough analysis of each of the model components, we defined four 

variational models that divide the problem naturally according to 

increasing complexity. The first variational model (MODEL I) contains the 

two nonlinear horizontal momentum equations, the integrated continuity 

equation, and the hydrostatic equation. 

MODEL 11 contains MODEL I plus the thermodynamic equation for a dry 

adiabatic atmosphere. The introduction of this additional constraint 

violates the requirement that the number of subsidiary conditions (dynamic 

constraints) must be at least one less than the number of dependent 

variables (Courant, 1936). lnclusion of the same number of constraints as 

dependent variables overdetermines the problem and a solution is not 
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guaranteed. Therefore, we must develop a scheme to circumvent this problem 

or else the dynamically adjusted meteorological variables will not satisfy 

the closed set of primitive equations. 

MODEL 111 includes MODEL I1 plus radiance as a dependent variable and 

the radiative transfer equation as a constraint. MODEL 1V contains MODEL 

111 plus an additional moisture variable, a moisture conservation equation 

and a parameterization for moist adiabatic processes. 

In addition, the variational models will be made more responsive to 

the original observations. The direct methods for calculating derivatives 

require that the data be pregridded. As noted by Achtemeier (1975) and 

Williamson and Daley (19831, pregridding removes the dependence of the 

final analysis upon the original observations. This dependence can be 

reestablished by merging the variational models with the success<ve 

corrections interpolation method (SCM) through a cyclical procedure. 

Independently interpolated meteorological variables are merged 

variationally. The variational fields then serve as first guess fields for 

the next SCM analysis, and so on. 

The research currently is expanding into six areas. These are 1) 

conversion of MODEL 1 to the ISWS VAX 750 computer, 2) completion of MODEL 

11, 3) development of MODEL I11 (radiative transfer equation), 4) making 

the model more responsive to the observations, 5) extension of MODEL 1 

analysis periods from one to three, and 6) development of MODEL IV 

(moisture). Progress in each of these research areas and future research 

plans is briefly summarized in the following sections. 
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2. Conversion of MODEL I to the LSWS VAX 750 Computer. 

We have taken a module approach to the development of MODEL I. 

Mathematical equations, data gr idding , vertical interpolation, 

nondimensionalization and etc have been programmed separately. The 

advantages of this approach are that modules can be replaced by improved 

versions without disrupting the flow of other programs, revisions can be 

made within modules without extensive reprogramming, coding is easier and 

less error prone, and debugging is faster and less costly. The 

disadvantage is that the MODEL L requires a large number of data files. 

Table 1 summarizes the 84 modules of MODEL I. Even though the modules 

are small relative to the whole program several have large storage 

requirements that can be handled by the University of Illinois CYBER 175 

computer only during hours of minimal demand (late at night). The 

advantages of transferring MODEL I to the ISWS VAX 750 with its virtual 

memory are 1) the model can be run at any time, 2) useful peripheral 

equipment such as the ZETA plotter and VTlOO compatible terminals are 

linked directly to the VU, and 3) the VAX-VMS operating system is 

compatible with the operating system of the CRAY X-MP/24 supercomputer 

operated by the National Center for Supercomputing Applications. Plans are 

underway to move MODEL I to the supercomputer once compatibility is 

achieved. 

The transfer of MODEL I to the ISWS VAX was completed in February 

1987, Runs with the VAX version reveal significant discrepencies with the 

CYBER version. Investigations to reveal the source of the discrepencies 

are still ongoing. Computer truncation cannot be ruled out at this time. 
Table 1. Summary of program modules that make up MODEL I. 

~ 

Program Description 

Number of 

Programs 

TIROS-N and Rawinsonde data read and format 4 

Grid definition and map projection 1 

Data gridding 2 

Transformation into nonlinear sigma coordinate 1 

Nond imen s iona 1 iz e v ar iab 1 e s 1 

Initial variable fields 13 

Coefficients and forcing functions calculated 

from initial data only 12 

Variational ad justment model - programs in 

cyclical solution sequence 42 

Model statistics 6 

Tendency term recomposit ion 2 

.- -I.__- 
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The VAX has a 32-bit word in comparison with the 64-bit word of the CYBER 

and the CRAY. 

Once the conversion is complete, the following studies are to be done: 

1) The precision moduli will be varied over a large range of values to 

determine the model sensitivity to varying levels of input data accuracy. 

This study will also determine the ranges of the precision moduli for which 

the cyclical solution sequence will converge. 

2) The precision moduli will be varied horizontally. Although MODEL 1 was 

formulated for horizontally varying precision moduli, so far all analyses 

have been done with only vertically varying precision moduli. Horizontal 

derivatives of the precision moduli appear in the equations for the 

adjusted geopotential height and the adjustment velocity potential. These 

equations are general second order partial differential equations which are 

solved by traditional relaxation methods. This study will determine the 

impacts of horizontally varying precision moduli upon convergence to a 

solution. 

This effort has progressed to a limited extent through an extension of 

Barbara Chance's thesis. (See project bibliography for details, of her 

report.) Her work with mesoscale data showed the equation for the 

adjustment velocity potential to be particularly sensitive to large 

horizontal variations in the precision moduli. 

3) The adjustment of the velocity components tendencies will be further 

analyzed. To date, the results for these important hypersensitive terms 

have been very encouraging. However, the adjustment process has not been 
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understood as fully as the adjustments for the remaining variables. Some 

preliminary results obtained with the few model runs available point to 

inadequacy of the currently used tendency term formulation when the 

advective current is unusually strong. (The advective current is a 

smoothed version of the gridded 500 mb wind field.) An alternative 

formulation that corrects for this problem has been derived but is not yet 

coded. 

3. Completion of MODEL 11. 

The components of the variational adjustment equations that arise from 

the application of the Euler-Lagrange operator to the thermodynamic 

equation for dry, adiabatic flow were derived during mid 1986. These terms 

were, as a matter of course, expressed compatible with the staggerd grid 

used for MODEL I. These terms were coded into the appropriate modules and 

the programs are now in the final check-out stages. Test runs with MODEL 

II are expected within the next month. 

4. Development of MODEL 111 (radiative transfer eauation). 

In collaboration with Dr. Stanley Kidder, the variational equations 

for the radiative transfer equation for the four microwave TWS channels 

were derived for the nonlinear sigma coordinate system during spring of 

1986. was apparent that the solution of these equations with radiance 

and temperature as dependent variables was another temperature retrieval 

algorithm. The advantage of the variational formulation over other 

retrieval algorithms is that other meteorological variables such as winds 

and heights can be entered into the retrieval process. 

It 
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Reprogramming of the microwave channel weights for compatibility with 

the MODEL I vertical coordinate was begun. Dr. Kidder left the project 

for a one-plus year leave of absence in August 1986. 

5. Making the Model more Responsive to the Observations. 

The direct methods for calculating derivatives for the various terms 

of the variational model require that the data be pregridded. Past 

approaches (Achtemeier, 1975) have treated the gridding and adjustment 

processes separately and sequentially and therefore the dependence of the 

final analysis upon the original observations is removed. This dependence 

can be reestablished by merging the variational models with the successive 

corrections interpolation method (SCM) through a cyclical procedure. 

Independen tly interpolated meteorological variables are merged 

variationally. The variational fields then serve as first guess fields for 

the next SCM analysis, and so on. 

Before the SCM is merged with the variational model, we must take into 

consideration 1) how the data is to be gridded and 2) the best SCM method 

to grid the data. As regards how the data is to be gridded, we have 

investigated the calculation of the geopotential gradients by triangulation 

and then gridding the gradients directly. The current approach calculates 

the geopotential gradients from the gridded heights. The advantages of the 

triangulation method are as follows: a) The locations of the data points 

are at triangle centroids not at the observation locations. Therefore the 

interpolation error caused by irregular distribution of the geopotential 

gradient will be independent from the error of interpolation for the 

geopotential. These errors therefore can be more easily removed by the 
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variational adjustment. b) The gradients are not calculated from the same 

grid as that used for the variational adjustment. This better satisfies 

the theoretical requirement for the variational analysis namely that the 

observations be mutually independent. 

The disadvantages of the triangulation method are currently not known. 

As regards the best method to grid the data, much effort has been 

expended to determining the best SCM method. We have undertaken a thorough 

analysis of the Barnes (1964, 1973) objective analysis method. Some of the 

results appear in the papers (Achtemeier 1986, 1987) that accompany this 

interim report. Still further work is being done on the Barnes methods and 

will be the subject of a third article to be published in the near future. 

We have found that replacing the traditional 2-pass Barnes method with a 

three-pass version a) better restores short but resolvable wavelengths 

without introducing undesirable shorter wavelengths into the analyzed 

fields, b) does not introduce high frequency noise caused by irregular 

station spacing, and c> gives smoother, more meteorologically realistic 

derivatives. The later improvement is crucial to the variational analysis 

which requires the calculation of numerous first and second derivatives 

from analyzed fields of data. 

It is anticipated that these new methods will lead to improved 

analyses of the initial fields along with the incorporation of the SCM into 

the variational analysis. We plan to continue this research effort. 

6. Extension -- of MODEL 1, Analysis Periods from One to Three. 
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During the past year, we performed MODEL I SAT and NOSAT analyses for 

the 00 GMT and 12 GMT 11 April 1979 periods. As with the analyses for 12 

GMT 10 April (Achtemeier, etal.; 1986) the purpose the new analyses is for 

model error analysis and sensitivity studies. These analyses are evaluated 

according to three criteria that measure a) the extent to which the 

assimilated fields satisfy the dynamical constraints, b) the extent to 

which the assimilated fields depart from the observations, and c) the 

extent to which the assimilated fields are realistic as determined by 

pattern recognition. The last criterion requires that the signs, 

magnitudes, and patterns of the hypersensitive vertical velocity and local 

tendencies of the horizontal velocity components be physically consistent 

with respect to the larger scale weather systems. 

The percent reduction of the initial RMS error is used to determine 

the extent to which the SAT and NOSAT blended data sets converge to the 

solution of the four dynamical constraints. These results show same levels 

of convergence for the two new analyses as for 1200 GMT 10 April 1979; 

90-95 percent error reduction for the two horizontal momentum equations, 

and 90-100 percent for the integrated continuity and hydrostatic equations. 

Regarding the analysis of hypersensitive variables, the vertical 

velocity was analyzed well at 00 GMT but not for 12 GMT 11 April. This 

problem has been traced to the inadequacy of the currently used tendency 

term formulation when the advective current is unusually strong. (The 

advective current is a smoothed version of the gridded 500 mb wind field.) 

An alternative formulation that corrects for this problem has been derived 

but is not yet coded. 
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The results for the local tendencies continue to be better than 

expected. For reasons not yet understood, the patterns of 3-h velocity 

component tendencies generated by MODEL I better correspond to observed 3-h 

forward tendencies than to 3-h backward tendencies or to the 6-hr average 

centered at the analysis time. 

7. Develoment of MODEL IV {Moisture) 

Our work on developing methods to meld and grid moisture data and to 

build the moisture conservation equation and mois turelheat 

parameterizations into the themodynamic equation is at the conceptual 

stage. 

8. References 

Achtemeier, G. L., 1975: On the Initialization problem: A variational 

adjustment method. Mon. Wea. Rev., 103, 1090-1103. 

Achtemeier, G. L., 1986: On the impact of data boundaries upon the objective 

analysis of limited-area data sets. Mon. Wea. Rev.. 114. 40-49. 

Achtemeier, G. L., 1987: On the Concept of Varying Influence Radii for a 

Successive Corrections Objective Analysis. Mon. Wea. Rev., 115, (in 

press) 

Achtemeier, G. L., H. T. Ochs Ill, S. Q. Kidder, R. W. Scott, J. Chen, 

D. Isard, and B. Chance, 1986: A Variational Assimilation Method for 

Satellite and Conventional Data: Development of Basic Model for Diagnosis 

of Cyclone Systems. Nasa Contractor Report 3981, 223 pp. 

Barnes, S. L., 1964: A technique for maximizing details in numerical weather 

map analysis. J. ADD^. Meteor., 3%-409. 

Barnes, S. L., 1973: Mesoscale objective analysis using weighted time-series 

observations. NOAA Tech. Memo. .ERL NSSL-62, National Severe Storms 

Laboratory, Norman, OK 73069, 60 pp. (NTIS COM-73-10781). 

Courant, R., 1936: Differential and Integral Calculus. Vol. 2, (E. J. 

McShane, translator 1, Wiley - Interscience, pl98. 

Williamson, D. L., and R. Daley, 1983: An iterative analysis initialization 

technique. Mon. Wea. Rev., 111. 1517-1536. 
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9. Proiect Biblionrauhy 

The following is the bibliography of all publications involving the 

effort to develop a diagnostic variational model for data assimilation. 

a) Published Papers 

Achtemeier, G. L., S.Q. Kidder, H. T. Ochs 111, 1985: Variational Objective 

Analyses for Cyclone Studies. NASA/MSFC FY-85 Atmospheric Processes 

Research Review, NASA Conf. Pub. 2402, Columbia MD, 8-12 July 1985, pp. 

26-2 8. 

Achtemeier, G. L., 1986: On the impact of data boundaries upon the objective 

analysis of limited-area data sets. Mon. Wea. Rev.. 114. 40-49. 

Achtemeier, G. L., H. T. Ochs 111, S. Q. Kidder, and R. W. Scott, 1986: 

Evaluation of a multivariate variational assimilation of conventional and 

satellite data for the diagnosis of cyclone systems. Variational Methods 

- in Geosciences. Develoments Geomathematics & Y. Sasaki, ed. 

Elsevier, Amsterdam. 

Achtemeier, G. L., H. T. Ochs 111, S. Q. Kidder, R. W. Scott, J. Chen, 

D. Isard, and B. Chance, 1986: A Variational Assimilation Method for 

Satellite and Conventional Data: Development of Basic Model for Diagnosis 

of Cyclone Systems. Nasa Contractor Report 3981, 223 pp. 

Achtemeier, G.L., 1986: Modification of a Variational Objective Analysis Model 

for New Equations for Pressure Gradient and Vertical Velocity in the Lower 

Troposphere and for Spatial Resolution and Accuracy of Satellite Data. 

Final Report to USRA, Contract NAS 8-36474, 28 pp. 

Achtemeier, G. L., H. T. Ochs 111, S. Q. Kidder, and R. W. Scott, 1986: 

Evaluation of a multivariate variational assimilation of conventional and 

satellite data for the diagnosis of cyclone systems. PREPRINTS, 2nd Conf. 

on Sat. Meteor./Rem. Sensing and Appl. Williamsburg, VA, 13-16 May 1986, 

Amer. Meteor. SOC. 

Chance, Barbara A., 1986: Application of satellite data to the variational 

analysis of the three dimensional wind field. Master's Thesis. NASA 

Contractor Report 4022, 86pp. 

Chance, B. A., and G. L. Achtemeier, 1986: Application of satellite data to 

the variational analysis of the three dimensional wind field. PREPRINTS, 

2nd Conf. on Sat. Meteor./Rem. Sensing and Appl. Williamsburg, VA, 

13-16 May 1986, Amer. Meteor. SOC. 

Kidder, S. Q., and G. L. Achtemeier, 1986: Preparation of satellite 

soundings for insertion in a variational objective analysis model. 

PREPRINTS, 2nd Conf. on Sat. Meteor./Rem. Sensing and Appl. 

Williamsburg, VA, 13-16 May 1986, Amer. Meteor. SOC. 

Kidder, S.Q., and G. L. Achtemeier, 1986: Day-night variation in 
f 

operationally-retrieved 

1175-117 8. 

Achtemeier, G. L., 1987: On 

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TWS temperature biases. Moa. Wea. Rev.. 114. 

the Concept of Varying influence Radii for a 

Sucessive Corrections Objective Analysis. Mon. Wea. Rev.. 115, (in Press) 

b) Abstracts of Forthcoming Papers 

Achtemier, G. L., 1987: A 3-Pass Near Optimum Barnes Method for the Univariate 

Objective Analysis of Mesoscale Phenomena. (To be presented at the Third 

Conference on Mesoscale Processes, August 21-26, 1987, Vancouver, B.C., 

Canada) 

ABSTRACT: The Barnes objective analysis method is widely used by 

mesoscale meteorologists for gridding irregularly distributed weather data. 

Results from a thorough reanalysis of the Barnes method revealed that the 

filter fidelity optimizes (filter fidelity measured by how well the method 

restores desired wavelengths and removes undesired wavelengths) if the 

filter parameters that determine the influence of observations are 

unchanged for both the initial and correction passes (analogous to a 

constant radius of influence). These results were obtained from families 

of response curves that satisfied the requirement that the final responses 

were equal at an arbitrary reference wavelength. 

Subject to the above requirement, the Barnes method with filter 

parameters unchanged (fixed method) was compared with a more traditional 

approach (reduction parameter equal to 0.2) through theoretical and 

analytical tests. The analyses with analytical monochromatic sine and 

cosine waves were done with uniform and highly nonuniform data 

die tributions. 

The theoretical studies and analytical analyses with uniform data 

distributions confirmed the superiority of the fixed method for wavelengths 

ranging from twice to twelve times the average observation separation (SI. 

The greatest improvements, found for the 3s to 6s wavelengths, ranged upto 

seven percent of the amplitudes of the original wavelengths. These 

improvements were locally large and were gained without the method exciting 

undesirable very short wavelengths. However the traditional approach gave 

the best results for the longer wavelengths when the data were nonuniformly 

distributed. This was because the smoother first pass of the traditional 

method gave better representation of the area near the inflection points of 

these waves. 

The information about how different filter parameters modify the 

performance of the standard 2-pass Barnes method was used to develop a 

3-pass hybrid method that incorporates the advantages of both the fixed and 

traditional methods. Comparisons between analyses of derivatives of the 

height and wind fields surrounding intense jet streaks on 00 GMT 10 April 

1979 by the hybrid method and the traditional method are presented in this 

paper. The comparisons show that (1) the hybrid method restores more of 

the mesoscale features inherent in the data, (2) the hybrid method better 

preserves linear features such as shear vorticity zones and divergence 
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zones, (3) the hybrid method better preserves the magnitude and shape of 

patterns of the laplacian of the height field, and (4) the hybrid method 

does not as severely contaminate the analysis with undesirable small scale 

centers that arise from analysis of nonuniformly distributed data.