A 3DVAR Land Data Assimilation Scheme: Part 1, Mathematical Design

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1 A 3DVAR Land Data Assimilatin Scheme: Part, Mathematical Design Lanjun Zu * a,,c Wei Ga a,d Tngwen Wu Xiafeng Xu Bingyu Du a,and James Slusser d a Sin-US Cperative Center fr Remte Sensing, Nanjing University f Infrmatin Science & Technlgy, Nanjing, PRC Laratry fr Climate Studies, China Meterlgical Administratin, Beijing, PRC c Jiangsu Key Laratry f Meterlgical Disaster, Nanjing University f Infrmatin Science & Technlgy, Nanjing, PRC d USDA UV-B Mnitring and Research Prgram, Natural Resurce Eclgy Laratry, Clrad State University, Frt Cllins, CO ABSTRACT Land surface states have significant cntrl t the water and energy exchanges etween land surface and the atmsphere. Thus land surface infrmatin is crucial t the glal and reginal weather and climate predictins. China has uilt aundant meterlgical statins that cllect land surface data with gd quality fr many years. But applicatins f these data in their numerical weather and climate predictin mdels are quite lw efficient. T take the advantages f land surface data in numerical weather and climate mdels, we have develped a three dimensin variatinal (3DVar) Land Data Assimilatin Scheme (LDAS). In Part f this paper, we present the mathematical design f the 3DVar LDAS. By assimilating a single pint servatinal datum int a ackgrund setup, the LDAS is tested t demnstrate its capaility and usage. In the ther part f this paper, we will demnstrate the results and errr analysis f assimilating China s air temperature servatinal data f the meterlgical statins int ECMWF s mdel ackgrund using the 3DVar Land Data Assimilatin Scheme. Key wrds: land surface, data assimilatin, 3DVAR. INTRODUCTION Land surface is an imprtant interface that exchanges energy with atmsphere as a main part f the earth. The mst key factrs f land surface that impact atmspheric circulatin are aled, sil misture and rughness f surface. Radiatin energy udget varies ver different lands due t different aleds, and sil temperature may directly affect the sensile heat in land-atmsphere exchange. Despite the thermal difference etween the land and the cean, the hetergeneity f the sil, vegetatin and terrain leads t the hetergeneity f sensile and latent heat flux, these thermal fluxes with turulence frm will impact the energy transfer f mes-scale and macrscale circulatin. Sil misture is nt nly related t aled, thermal capacity, radiatin and vegetatin grwth, ut als related t precipitatin, evapratin and runff, influencing the water and heat distriutin and redistriutin in land, thus it plays an imprtant rle t weather and climate [,]. It is clear that land surface is crucial t weather and climate predictin, hwever, uncertainties may exist in land data and * Address crrespndence t: wga@uv.nrel.clstate.edu, phne (97) , fax (97) 49-36,lanjunzu@gmail.cm; phne: Remte Sensing and Mdeling f Ecsystems fr Sustainaility III, edited y Wei Ga, Susan L. Ustin, Prc. f SPIE Vl. 698, 698E, (6) X/6/$5 di:.7/ Prc. f SPIE Vl E-

2 land surface variales are depicted carsely in GCMs, s it is nt enugh fr multi-mdel cnsideratin f climate system. Because f lack f servatin data with high quality and unsatisfied mdel results, hw t reduce the land data uncertainties and hw t use much mre data int the mdels are cnsidered, it is urgent t develp a gd land data assimilatin scheme fr data acquirement and mdel applicatins [3,4]. Land data assimilatin effectively cmines land surface data f different types with different spatial-tempral distriutins and different errr characteristics under the cmprehensive cnsideratins f servatin errr and mdel ackgrund errr. Cmparing t atmsphere and cean data assimilatin, land data assimilatin is a new field t take effrt in, hwever, the relatively mature cncepts and methds in ADAS (Atmsphere Data Assimilatin Scheme) and ODAS (Ocean Data Assimilatin Scheme) can e applied in LDAS. Hwever, LDAS is different with ADAS and ODAS ecause f the characteristics f land surface prcesses and land surface parameters. As this pint is cncerned, we develp a 3DVAR Land Data Assimilatin Scheme. Its design framewrk will e intrduced in this paper. A single pint servatin test and ECMWF ERA-4 air temperature data are used t demnstrate its validity and usaility. Iteratin steps and crrelatin length f ackgrund in assimilatin prcedure are determined depending n cst functin descending and crrelatin analysis f ackgrund [5,6,7,8,9].. SCHEME OF 3DVAR LAND DATA ASSIMILATION. 3DVAR land data assimilatin system scheme.. Cst functin The asic cncept f 3DVAR assimilatin is t find the minimum f a quadratic functinal (named as cst functin) that presents the difference f analysis t servatin and ackgrund. Cst functin in this land data assimilatin scheme is taken as T T Jx ( ) ( x x) B ( x x) [ yrx ( )] O [ y Rx ( )] () where x and x are analysis and ackgrund vectrs with length N respectively, while y is an servatin vectr with length m. N is the freedm degree f analysis field, m is the numer f servatins. B is an NN ackgrund errr cvariance matrix, while O is an mm servatin errr cvariance matrix. R is an servatin peratr, which maps analysis variales int servatin variales and servatin psitins. Under the cnditin that servatin variales cincide with mdel variales, R is a simple linear interplatin peratr, therwise R is a cmplex nnlinear peratr... Mdel f ackgrund and servatin errr Cst functin has tw parts, in essential, which is a weighted linear cminatin with analysis t ackgrund and servatin. And the inverses f B and O ( B and O ) are regarded as weightings f ackgrund and servatin cntriuted in analysis. Clearly, the result f assimilatin is up t ackgrund errr cvariance B and servatin errr cvariance O (Daley, 99). Vertical crrelatin f ackgrund errr is nt cnsidered in this assimilatin scheme, s we nly need t prcess hrizntal crrelatin and define ackgrund errr cvariance B as B ( i, j) NN,where i, j is errr cvariance etween the i-th and the j-th grid. Backgrund errr is regarded Gaussian, s the frm f ackgrund errr cvariance can e given as Prc. f SPIE Vl E-

3 B aexp( r / c cs ) () where a is ackgrund errr variance and c is hrizntal crrelatin length taken as Cnstant. r is the distance etween aritrary tw ackgrund grids and is the latitude f midpint etween the tw grids. Meanwhile, servatin errr is regarded as uncrrelated etween different servatin pints, which is reasnale ecause the measurements are independent fr cnventinal servatins, s the servatin errr cvariance matrix is diagnal and easy t inverse. This will greatly ring cnvenience and cut cmputatin cst in assimilatin prcedure...3 Principle f 3DVAR and minimizatin algrithm f cst functin T tain the ptimal analysis, the aim is t minimize the cst functin with respect t analysis variale x, i.e., t make the gradient f cst functin equal t zer. Thus, it makes the minimum prlem f functinal cnverted t the prlem f finding the slutin f J( x). Cncerning the tremendus dimensin f analysis variale x, increment analysis methd is adpted in this scheme. Als, B is decmpsed y Chlesky methd t reduce its cnditin numer. A variale W is set t meet W C ( x x ), where B CC T. Sustituting it int the cst functin, it fllws T [ ( )] T [ ( )] J W W yr x CW O yr x CW (3) T find the minimum f cst functin, it requires J / W, that is J T T W C L O [ yr( x CW)] g W R in which L is the tangent linear peratr f servatin peratr R, i.e. L. x x x The cnjugate-gradient methd was fund t represent a gd cmprmise in cnvergence rates and cmputer memry requirements etween simpler and mre cmplex methds f nnlinear ptimizatin (Fletcher and Reeves,964; Hestenes and Stieffel, 95; Plak and Riiere,969; Beale,97; Shann,97 and 978; Perry, 976, 977 and 978). Based n previus wrk f Hestenes and Steiffel(95), Fletcher and Reeves(964) prpsed a cnjugate-gradient methd applied t general nnlinear functins. Navn and Legler(987) assessed use f different availale cnjugate-gradient algrithms in large-scale typical minimizatin prlems in meterlgy cnsidering cmputatinal efficiency and accuracy as principal criteria. This cnjugate-gradient methd is used t calculate the gradient f J in this scheme. Tw preliminary ntes t e used in the iteratin algrithm are given as fllwings: i) If the cst functin J is defined as then T T J I C L O LC, where T T J( W) W W [ yr( x CW)] O [ yr( x CW)] (5) R L. x x x (4) Prc. f SPIE Vl E-3

4 ii) The Euclidean nrm is defined as g ( g g g n ) L (6) n where g ( g, L, g ) R. n The minimizatin algrithm is given in the fllwings: ) Set W,, d,eps=.e-. (initial cnditin) ) D iter=,steps 3) Cmpute the gradient f J 4) Set 5) Stepsize is tained as J grad W C L O yr x CW (7) T T [ ( )] d grad d (descent directin) (8) T J grad T ( J) ( ) T d ( J) d d ( J) d (9) 6) Generate a new W y 7) Recmpute the gradient f J W W d () J grad W C L O yr x CW () T T [ ( )] 8) Update as 9) Endd grad () grad A cnvergence criterin fr stpping the iteratins is tested ( J gradi EPS r iteratin numer is larger than a given value, fr example, ), and, if it is satisfied, the iteratins must stp. We mdify the criteria f iteratin stp y cntrlling the nrm f cst functin and iteratin step numer in the prcedure. Practical applicatins in this scheme illustrate that this algrithm quickly cnverges in mst circumstances, which nly needs t deal with multiplicatin and additin f matrixes and vectrs. 3. SINGLE POINT OBSERVATION TEST T verify the validity and usaility f ur assimilatin system, ne typical single pint servatin test is carried ut. This test suppses the ackgrund value set as zer and the single pint servatin set as ne taking n the central grid Prc. f SPIE Vl E-4

5 f the ackgrund. China zne f lngitude as 75,5,35E and latitude as 5, 35, 55N is taken as ackgrund (all zeres) and single pint servatin (ne) is take in central grid (5E, 35N). Hrizntal crrelatin length is taken as 5km. Thus a list f results with different ackgrund errr cvariance times as ) are given in the fllwing tale(tale ): and servatin errr cvariance ( is.,.,,, Backgrund errr variance Oservatin errr variance Nrm f cst functin J after 3 iteratins e e-6...e e e E E-6...E E E E E-7.E E E-3 Tale. The nrm f cst functin descends with different ackgrund and servatin errr variances Fig. Single Pint Oservatin Test Over China This figure again verifies the validity f ur land data assimilatin scheme, ecause the central pint f assimilatin analysis is.5 and analysis elngs t Gaussian. In mst circumstances, the gradient f cst functin descends almst t zer in 3 steps, that is t say the Prc. f SPIE Vl E-5

6 cnjugate-gradient methd is quite effective. In rder t analyze the assimilatin result cncerning aut ackgrund errr variance and servatin errr variance, r is defined as the rati f and, i.e. r. When r (r r ), that is the effect f servatin (ackgrund) is larger than that f ackgrund (servatin), the assimilatin is clser t servatin (ackgrund) than ackgrund (servatin). When r, that is the effect f servatin equals t that f ackgrund, the assimilatin equals t half f ackgrund and servatin. In fact, errr variances reflect relative magnitudes f errrs, while their reciprcals reflect relative weights in the assimilatin. Assuming the ackgrund and servatin errr cvariance is equivalent, the single pint servatin test cmes t the fllwing result with Figure Cst Functin 5 Cst Functin Km Km Cst Functin Cst Functin 4 3 Km 4 Km Cst Functin 4 5 Km Prc. f SPIE Vl E-6

7 Figure Cst functin vs. iteratin steps with different crrelatin length 4. DETERMINATION OF ITERATION STEPS AND CORRELATION LENGTH FROM ECMWF ERA-4 AIR TEMPERATURE DATA ECMWF ERA-4 has gd and the mst time-cnsistent prduct quality fr the gle as a whle, and it is a cmmn-used data set fr climate research, ut it als has uncertainties, we chse air temperature frm ERA-4 t assimilate Chinese statin data as a test t determine the iteratin steps and crrelatin length in assimilatin prcedure. ECMWF ERA-4 has a hrizntal reslutin f.5.5 degree. 4. Determinatin f iteratin steps It must take assimilatin accuracy and efficiency int cnsideratin at the chice f iteratin steps. Iteratin steps depend n the iteratin algrithm, servatin lngitude and latitude, ackgrund grid distance and selected assimilatin regin. Assimilatin system always iterates steps with ackgrund hrizntal crrelatin distance taken as,, 3, 4 and 5 km respectively, and then the descent velcity f cst functin is examined t find ut the iteratin step satisfying accuracy and lw-cst time requirement (See Fig. ). The value f cst functin rapidly descends with iteratin. Cst functin is very small at the tenth step cmparing t that at the first step when ackgrund hrizntal crrelatin length is taken as,, 3, 4 and 5 km respectively, s the step numer f this assimilatin scheme is taken as. 4. Determinatin f ackgrund hrizntal crrelatin length Backgrund errr cvariance matrix B, cntrlling the way that infrmatin spreads ut at servatin psitin, presenting unifrm analysis increment at adjacent mdel grids and vertical levels, and ensuring mdel variales dynamically cincide with each ther, is crucial t analysis results. Suppse that the distriutin f ackgrund errr is istrpic and unifrm, then ackgrund errr cvariance can e mdeled y Gaussian functin, i.e., B a r c exp( / cs ). Crrelatin length must e derived frm statistics f numerus ackgrund data. Crrelatin Cefficient (elw Km) Average Crrelatin Cefficient. Crrelatin Cefficient Distance (Km) Fig. 3 Crrelatin cefficients vs. distance Black dts: crrelatin cefficient; Red line: Average crrelatin cefficient Prc. f SPIE Vl E-7

8 After statistics aut crrelatin cefficients f aritrary tw f all the grids and average crrelatin cefficients at every distance, the distance with crrelatin cefficients ave.9 is taken as crrelatin length, that is 5 km (See Fig. 3). 5. SUMMARY AND CONCLUDING REMARKS Cncerning aut the lw efficient applicatin f Chinese statin climatic data int the mdel, a 3DVAR Land Data Assimilatin Scheme is develped and its single pint servatin test is carried ut in the first part f this paper t verify the validity and usaility f this 3DVAR LDAS. Iteratin steps and crrelatin length f ackgrund are cautiusly determined as and 5 km in assimilatin prcedure y using ECMWF ERA-4 air temperature data as ackgrund and China statin data as servatin. This 3DVAR LDAS can e used t assimilate statin servatin efficiently and will greatly help the research f climate. Acknwledgements ECMWF ERA-4 data used in this study have een prvided y ECMWF.,USDA UV-B Mnitring and Research Prgram under a grant frm USDA CSREES( ), Natinal Natural Science Fundatin f China (44797). REFERENCES. Beale, E. M., 97: A derivatin f cnjugate-gradients. Numerical Methds fr Nn-linear Optimizatin. F. A. Ltsma, Ed., Academic Press, Fletcher R. and C. M. Reeves, 964: Functin minimizatin y cnjugate-gradients. The Cmputer Jurnal, 7, Hestenes M. R. and E. Stieffel, 95: Methds f cnjugate-gradients fr slving linear systems. J. Res. Natl. Bur. Stand., 48, Navn I. M. and D. M. Legler, 987: Cnjugate-gradient methds fr large-scale minimizatin in meterlgy. Mnthly Weather Review, 5, Perry, A., 976: A mdified cnjugate-gradient algrithm. Discussin Pap. N. 9, Center fr Mathematical Studies in Ecnmics and Management Sciences, Nrthwestern University. 6. Plak, E. and G. Riiere, 969: Nte sur la cnvergence de methds de directins cnjuguées. Rev. Franc. Infrmat. Rech. Operatinnelle, 6, R. Deley, 99, Atmspheric Data Analysis, Camridge University Press. 8. Shann, D. F., 97: Cnditining f quasi-newtn methds fr functin minimizatin. Math. Cmp., 4, X. L. Zu, F. Vandenerghe, M. Pndeca and Y. H. Ku,997, Intrductin t adjint techniques and the MM5 adjint mdeling system, NCAR Technical Nte, NCAR/TN435STR. Prc. f SPIE Vl E-8