MULTI-SOURCE DEM EVALUATION AND INTEGRATION AT THE ANTARCTICA TRANSANTARCTIC MOUNTAINS PROJECT

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1 MULTI-SOURCE DEM EVLUTION ND INTEGRTION T THE NTRCTIC TRNSNTRCTIC MOUNTINS PROJECT Yaron. Felus and Beata Csatho Byrd Polar Researh Center Sott Hall Room 8, 9 Carmak Road The Ohio State University Columbus, Ohio 43-, US felus.@osu.edu ; satho@ohglas.ms.ohio-state.edu Working Grou IV/ KEY WORDS: Data fusion, DTM/DEM/DSM, Geohysis, Mathematial models. BSTRCT Digital elevation models are essential tools in many glaiologial studies and eseially for mass balane studies, strutural geology modeling and advane remote sensing and geohysial roessing. However, due to the hostile limate and inaessible environment of the ntarti ontinent, there are insuffiient elevation databases and their quality is oor. In this aer, we analysis the satial distribution of error in the different DEMs that exists at the ntartia Transantarti Mountains. Based on this analysis, we investigate the various methods to ombine elevation models with different roerties (resolution, horizontal and vertial auray. There are five major data sets in the rojet area: The USGS :5 mas whih, overs the north west art of the rojet area and have 5 meter ontour line interval; USGS :5, taken from the ntarti Digital Database, whih, overs all our rojet area and have meter ontour line interval; satellite radar altimetry data derived from ERS- with 5 km resolution; airborne Radio- Eho Sounding rofile data at the north east art of the rojet olleted by Sott Polar Researh Institute and field surveying ontrol oints olleted by USGS. Our final goal was to omile all those elevation models into one uniform grid elevation model with the highest auray and resolution that an be obtained. Many tehniques and algorithm s exists for integrating database, some are based on interolation methods in the boundary zone, other tehniques erform simle data merging and aly various filtering funtions to make the transition smoother. We review those roedures and omare their roerties and aly some of them in our study. Last, we roose a method to ombine the different DEM into one set using universal Kriging onet. In this roess, we omute a ovariane matrix for every data set individually and a ross ovariane of the individual data set in the rediation omutation. INTRODUCTION. The Tamara Projet The Tamara rojet is an international researh aimed at integrating new aeromagneti data, aquired by a ooerative U.S.-German field amaign, with satellite imagery, geologial and strutural maing, and existing ground-based, airborne and marine geohysial data. With this omrehensive database we hoe to answer outstanding questions about the evolution of the Transantarti Mountains (TM - West ntarti rift- in southern Vitoria Land. The foundation of this database is a Digital elevation model (DEM whih is an essential tools in many glaiologial studies and eseially for magneti and gravity modeling. It is imortant to use a data set whih will have the best auray and with the highest resolution. However, due to the hostile limate and inaessibility environment of the ntarti ontinent, there are insuffiient elevation databases and their quality is oor. Consequently, we need to aly methods to ombine and integrate the different DEM's whih were aquired from different soures with different satial roerties.. Review of data fusion methods Many methods have been roosed for integrating multile data soures. For a omrehensive review on data fusion we refer the interested reader to bidy and Gonzales (99. Here we mention only a few methods that are imortant for understanding the roedures desribed in this aer. Ra (984 examines various tehniques; that an be used to ombine satellite gravity field information with terrestrial gravimetry. He is using sherial harmoni exansions (Fourier analysis to interolate the data and weighted least squares to solve the augmented observation equations and to omute the ombined interolation funtion oeffiients. Hahn and Samadzadegan (999 transform the data using International rhives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B. msterdam. 7

2 Yaron. Felus wavelet deomosition that yields a better loal interolation. (omared with sherial funtions. The merging roess of the two DTMs takes lae on the same wavelet funtion sale by least squares fit. Honikel (999 fuses digital elevation models derived from otial sensor (SPOT images and SR interferometry. She uses the orrelation oeffiient of eah data, i.e. the oeffiient generated by the automati DEM rogram or image mathing system for the otial sensor and the oeffiient whih was omuted from the interferometry hase unwraing roess. The mathematial model for this method is given in ( and reresent a simlified aroah to the least squares aroah. h ( x, y ρ f + hot ( x, y ρ ot + hsar ( x, y ρ sar ( hˆ( x, y = f ρ f + ρ ot + ρ sar Where: : resulting loal height estimate. hˆ( x, y h ( x, y : loal height of the filtered (f otial (ot and SR DEM. ρ : Correlation oeffiient derived for the aroriate data. Liu (999 uses a GIS aroah for identifiation and exhange of data in regions, where the height measurement of one of the ontributing sensors fails. We used a ombined aroah whih inororate Liu (999 roedure for merging overlaing DEM's but also uses a least squares aroah to fuse DEM's one into the other. Consequently we designed a two ste roess for the integration of data:. merging overlaing data sets to rodue the rimary DEM in our ase the :5k data from the DD and :5k data from USGS.. fusion of the base DEM with more aurate elevation data in seifi areas, to refine the rimary DEM and rodue a more aurate and with higher resolution DEM. In the subsequent setions, we desribe the data roessing roedures; setion desribe the different data soures and the evaluation methods used to estimate their auray; setion 3 desribes the merging tehnique; and setion 4 desribes a new statistial algorithm to blend two DEMs and imrove elevation models on a smooth area.. DIGITL ELEVTION MODELS NLYSIS Data soures The main soures of DEM S in the rojet area are (see also figure : N. The ntarti Digital Database, DD sheets ST 57-6 W E and ST53-56, whih were reated from the :5, USGS S ntartia Reonnaissane Series mas. These mas were mostly omiled from US Navy triamera aerial hotograhs taken in the 6 s. (DD 998. The digital ontour line ma has a m ontour interval, roviding a mean vertial auray of m (aording to the USGS ma standards the Tamara Data Sets auray equals half of the ontour interval. However, sine those mas are art of the ntartia Reonnaissane Series it may not omly with USGS standards and the auray of the data should be verified.. The USGS :5K digital data: whih overs the north west art of the rojet area, and has a 5 meter ontour line interval. gain we should examine the auray of those data sets. Figure : Tamara rojet, the different DEMs. 3. Ie ltimetry data - surfae elevations of the ntarti ie sheets derived from ERS- altimetry data, as roessed by the Oeans and Ie branh of the Laboratory for Hydrosheri Physis of NS/GSFC. The grid oints have a nominal saing of 5 km. The gridded elevations are derived from the data by a weighted fit of a bi-quadrati funtion (bi-linear where data distribution is oor to the elevation data that fall within a ertain radius of the grid loation. This data was reorted [Bamber 94] to have an error in the order of a few meters for elevations on sloes smaller then.65. This error omrises of random errors, originates from the satellite retraking error, and bias, from the geograhially orrelated orbit error and unertainties in the geoid. For regions with sloe greater then.65 the elevation estimates are not reliable i.e. have very large error, due to large footrint of the radar altimeter. 4. irborne Radio-Eho sounding rofiles data at the north east art of the rojet area olleted by Tehnial University of Denmark and Sott Polar Researh institute, University of Cambridge, U.K using an airborne Radio- Eho Sounding equiment oerating at 6 and 3 MHz [ Drewry 98 ]. The absolute auray of elevation is about 3m, However, the relative elevation auray is likely to be better than m; Data were olleted on subarallel lines 5km aart. Terrain learane and ie thikness data measured at interval of. km. TMR Boundary 5K Contours 5K ontours International rhives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B. msterdam. Radio Eho S. GPS oints 8 Miles

3 5. GPS and traverse surveying oints. Colleted from many different soures and rojets but mainly from USGS field amaigns. The auray of those oints is a few entimeters, however due to the diffiult limate onditions the ntarti area have a auity of ontrol oints.. Data analysis It was imortant to evaluate the auray of our data and to get familiarity with the roblemati areas. One way to estimate DEM is to omare with another indeendent and more aurate data. In our researh, we use bilinear interolator to omute the height of a given oint from the DEM i.e. eah oint is omuted from the losest four grid oints. We did the following omarisons: USGS :5 DEM Vs. GPS oints : 63 oints, Mean differene: m, Standard Deviation of differene:.65 m; Maximum: 8.89; Minimum: Those oints were measured in the rough terrain area with rok outros of the Royal Soiety range and Mount Disovery in whih the aerial hotogrammetri methods are more aurate ( in ontrary to the flat uniformly white ntarti Plateaus, On the one hand, lak of texture on the ie make Photogrammetri interretation and measurement diffiult on the other hand in on a very rough and mountainous toograhy small error in the horizontal osition an ause big error in the vertial diretion i.e. elevation measurement. USGS :5 DEM Vs. GPS oints : 35 oints; Mean: 53.3; Standard Deviation: again in an area with rough Toograhy. Ie ltimetry data (ERS- data set Vs USGS :5 DEM; General statistis in all our rojet area; 359 oints Mean: 8. m and Standard Deviation: 38.7 m. This very high error is due to the very rough area (raid and big hanges of height e.g. from m to 368 m aross 35km However ;In a flat area we get for 463 oints Mean: 7.5 m and Standard Deviation: m; in an area with high sloes indeed the elevation estimate are not reliable e.g. 48 oints Mean: 37.8 m Standard Deviation: m. USGS :5K DEM Vs irborne Radio-Eho sounding rofiles : We have 674 measured oints; Mean:. m Standard Deviation: Maximum error is: 57. Minimum error is: -35. USGS :5 Vs USGS :5K; Statistial analyses of the differene grid gave us a Root Mean Square Error of 4m and exetation of 3m ( i.e. 3m datum shift so the 5k data has a higher datum Those result s were antiiated sine the interolation roess introdue interolation error and the ntartia Reonnaissane Series auray s are worse then the USGS normal maing standards. 3 MERGING THE 5K ND 5K USGS DT We used the original ontour line overage of the :5, and :5, USGS ma and rodue a grid from them. In the revious setion we estimated the height errors of our data but we also need to hek for irregular atterns in our data Consequently, we subtrated (overlay oeration the two data sets to get a differene grid. Close examination of this grid shows no signifiant trend or attern whih means that we an ombine the two sets without any datum transformation. ( see figure. fter liing the area of the 5k data from the 5k we hoose to merge the two dataset using r/info TOPOGRID ommand. The Toogrid ommand is based on Huthinson (996 interolation method. The interolation roedure has been designed to take advantage of the tyes of inut data (in our ase ontour lines. This method uses an iterative finite differene interolation tehnique. It is otimized to have the omutational effiieny of loal interolation methods suh as inverse distane weighted interolation, without losing the surfae ontinuity of global interolation methods suh as kriging and slines. It is essentially a disretised thin late sline tehnique, where the Figure : differene grid between the USGS ntarti DEM'S roughness enalty has been modified to allow the fitted DEM to follow abrut hanges in terrain, suh as streams and ridges. In order to imrove data ontinuity at sutures we ould either filter our data at the merge line or give a data free zone for the interolation. We hoose the seond otion and thus we ut km from the 5k ontours as a transition zone. With this transition zone the Toogrid interolation was able to merge the two dataset and rodue a seamless elevation grid at a resolution of m ( this grid resolution is a little high in the areas where we have only :5k data but we had to have a higher resolution grid for a SR retifiation roess. International rhives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B. msterdam. 9

Figure 3: :5k ma and :5K before the merge roess. The red line are SR images frames. Figure 3: :5k ma and :5K before the merge roess. The red line are SR images frames. Figure 3B: The two ma seamlessly ombined.

4 STTISTICL LGORITHEMS TO BLEND MULTI-SENSOR DEM'S It was imortant to further imrove our rimary DEM to inlude also loal DEM's that are more aurate i.e. irborne Radio-Eho sounding rofiles and the Ie ltimetry data in the flat areas.

Consequently, we roose to use a Least Squares Colloation ( whih is equivalent to simle Kriging with a trend to statistially interolate and blend the DEM's.

The measurement vetor y (elevations is equal to a random signal (on whih we have statistial rior information added to a linear trend vetor ξ and added to random vetor of noise e (error : y = X + ξ +

4 Figure 3: :5k ma and :5K before the merge roess. The red line are SR images frames. Figure 3: :5k ma and :5K before the merge roess. The red line are SR images frames. Figure 3B: The two ma seamlessly ombined. The red line are SR images frames. Figure 3B: The two ma seamlessly ombined. The red line are SR images frames. 4 STTISTICL LGORITHEMS TO BLEND MULTI-SENSOR DEM'S It was imortant to further imrove our rimary DEM to inlude also loal DEM's that are more aurate i.e. irborne Radio-Eho sounding rofiles and the Ie ltimetry data in the flat areas. However, sine the grid resolution of those DEM is small, we an not ut and relae the DEM as we did with the :5k DEM. Consequently, we roose to use a Least Squares Colloation ( whih is equivalent to simle Kriging with a trend to statistially interolate and blend the DEM's. We an use this statistial model sine our area varies slowly (the ntartia glaial lateaus. The basi mathematial model of our data is based on Least square olloation aording to Moritz 97. The measurement vetor y (elevations is equal to a random signal (on whih we have statistial rior information added to a linear trend vetor ξ and added to random vetor of noise e (error : y = X + ξ + e e (, σ X = e e Q Where: y is the observations at known oints; ξ is a trend that we want to detet, we used in our rojet d olynomial to desribe the trend and omute our design matrix. X a random signal that we want to detet with zero exetation and known variane. e noise is the weight matrix of the noise elements; Q is Cofator matrix of the random signal. and the basi solutions of olloation are the estimated trend oeffiients at a oint: ˆ ξ = ( where = ( T y + Q and the redited random signal at a required oint ~ T X P = Q P ( y ξ (4 where Q P is the ofator matrix omuted between the required oint and the other data oints. We need to ombine two data sets and thus we will slit the mathematial model of ( into two equations as follows: y y x = x x e = x e + e ξ + e e (, σ e e Q (, σ e Q Q Q (3 (5 International rhives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B. msterdam.

5 in model (5 we assume no orrelation between our noise element of different data sets C(e,e = the signal elements in ontrast have orrelation whih should be omuted. The derivation of this ombined data set least square olletion solution is done by lugging the aroriate matries into equation (3 and (4 and using matrix arithmeti's eseially the formula for the inverse of a * matrix, we will ski this derivation and resent the solution: (a similar derivation an be found at Helmut & Hans (978 ˆ ˆ T ξ = ξ ( ˆ + y y ξ Where: P = T = = + P T ordingly in this sequential or stewise formula we first omute the trend ξ due to the first data set and then we add the new data set effet. The sequential form for the random arameter is : ~ X P ~ T = X + ( P ( y y ˆ ξ P s mentioned before, X ~ ; ˆ ξ P are the solution of the interolation roess done on the first data set, with the introdution of the seond data set we uses formulas (6 and (7 to udate our interolation. Obviously, we an use this sequential roess with as many data set as we need as long as we omute the aroriate ovariane matrix for eah data set and the ovariane with all the others. We omute our ovariane funtion from the data in an emirial way and using: C{ s, } = E{[ s E( s][ E( ]} = ( n h C s = CX ( h [ si + h si ] ˆ β ( n h i= (8 n Where ˆ β = s i n i= C is the ovariane, β is the exetation, s and denotes the satial osition, h=s- is the dislaement vetor or lag. This is under the onditions of ergodiity and Isotroy, Cressie (993 roves that variogram estimation is to be referred over ovariogram estimation. The main reasons for that are: When our roess is only a seond order stationary (not intrinsially stationary then both the variogram estimator and ovariogram estimator are biased. However, the variogram bias is of smaller order. If our data has trend ontamination then it has disastrous effet on attemts to estimate the ovariogram while on the variogram it has a small uward shift. Sine we assume seond order stationarity we an write the relationshi γ ( h = C( O C( h (9 Note that C( =σ, the variane of the random funtion and we an estimate the semivariogram by: ˆ (6 (7 E{[ s s + h] } γ ( h = = ( n h [ s s + h] Using (8 and ( we omuted the semivariogram in figure 3a. 3b. 3. i= i n i ( V(x 5, 4, 3,,,,, 3, distane between airs 4, V(x 3, 5,, 5,, 5, -5,, 4, 6, distane between airs 8, V(x x Comutation of the two data set mutual ovariogram s suggested by Cressie 93 our semivariograms and ovariagrams inludes Figure only 3a: half Emirial of the maximum Variogram ossible ( greenlags Figure and 3b: only Emirial those lags Variogram ossessing ( more Blak - going then u 3 and airs. Covariogram Note that for (red the - mutual and data Covariogram set ovariogram/ (Blue Semivariograms of the Radio going funtion down we of the used USGS a smaller DEM. samle of Eho the data Sounding so we need Data. to fit a funtion over a smaller range of values ( shorter y,x-xis Distane between airs x 4 Figure 3: Emirial mutual Variogram ( blue and mutual Covariogram (green of the Radio Eho Sounding Data and USGS DEM International rhives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B. msterdam.

6 Following the generation of exerimental semivariogram in the above figures; we used an interative method to find the initial funtion and then used weighted least squares to fit the exat orresonding semivariogram funtions. We should mention that to omute the samle semivariogram, we hoose area in out data that have no visible trend ( the seletion was made using the ontour line ma. We were able to fit a Gaussian funtion to all the data sets that will also give us the required sill for equation (9 -C(. The USGS fitted Gaussian funtion has the following arameters: Nugget=g=366; Range=r= 533; Sill=s= ; The Radio Eho Sounding Data fitted Gaussian Variane funtion has the arameters of: Nugget=g=3; Range=r= 695; Sill=s= 356 and the mutual data sets fitted Gaussian funtion has a Nugget=g= 5; Range=r= 8843; Sill=s=. With those variogram models we followed the roess of equation (6 and (7 to grid our data at meter resolution and rodue a grid with a better auray. 5 CONCLUSIONS ND FURTHER RESERCH. We have designed a omlete shema to integrate DEM's aquired from different soures, and we demonstrated the exeution of this roess in an atual roblemati ase study area - The tranantarti mountains in ntartia. We divide the integration roess into two searate lasses of algorithms, namely; merging of overlaing data sets and fusion of one data set in the other. For the first tehnique, we suggest an interolation zone of km, this means, that we smooth and derease the effet of a 4 m variane over a distane KM ( more then times the variane is a good rule of thumb. For the fusion roess, we roose to use a geostatistial interolation - Least square olloation - it is not ommon to see statistial interolations when using elevation models, those methods are being used extensively for otential fields, for geologial analysis and satial environmental examination. We deided to use those methods in our area sine we assume a smooth behavior of the DEM over the ie toograhy of the ntarti lateaus. More over, our interolation algorithm is designed to work with small loal subset or suort whih make it more suitable to deal with moderately varying data suh as elevations model. The main advantage of the geostatistial mathematial sheme is that it fits a unique ovariane/semivariogram model whih enomass the measurement errors and the intrinsi data relationshi in it, based on this analysis the algorithm interolates the data. This is in ontrast with other methods, whih assume a ertain data behavior in advane. Moreover, using geostatistis, we an get an estimate for our interolation disersion, The mathematial develoment of the sequential disersion equation is long but follows the same line of arguments as the sequential least squares olloations. Further researh is needed to evaluate the results of this model and omare it's erformane with reset to other models. CKNOWLEDGMENTS The authors would like to thank Dr. Terry Wilson and Dr. Burkhard Shaffrin for their useful disussions. This work was artially suorted by an NSF grant (OPP REFERNENCES bidy M.. and R. C. Gonzales 99. Data fusion in Robotis and Mahine intelligene. ademi Press, In., San Diego, C. ntarti Digital Database (DD Version. [998], Maing and Geograhi Information Center British ntarti Survey on behalf of the SCR Working Grou on Geodesy and Geograhi Information at: htt:// Bamber L. Jonathan 994. digital elevation model of the ntarti ie sheet derived from ERS- altimeter data and omarison with terrestrial measurements. nnals of Glaiology, vol Proeedings of the fifth international Symosium on ntarti Glaiology (VISG, Cambridge, U.K. Cressie N. (993. Statistis for Satial Data. Wiley & Sons, New York. Drewry J. David 98. Ie Flow, Bedrok and Geothermal Studies from Radio-Eho Sounding Inland of MMurdo Sound, ntartia. Craddok C(ed, Hahn M. and Samadzadegen F Integration of DTM s Using Wavelets. International rhives of hotogrammetry and remote sensing, Vol 3 art W6, Valladolid,Sain, 3-4 June 999. Helmut Moritz 97. generalized least squares model. Studia Geohysia Et Geodaetia, Vol 4. P International rhives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B. msterdam.

7 Helmut Moritz and Hans Sunkel 978. roximation methods in geodesy, Letures delivered at the seond international. Summer shool. Herbert Wihmann Verag Karlsruhe. Hongxing Liu 999. Generation and refinement of a ontinental sale digital elevation model by integrating artograhi and remotely sensed data: GIS aroah. Ph.D dissertation. The Ohio-State University. Honikel M Strategies and methods for the fusion of digital elevation models from otial and SR data. International rhives of hotogrammetry and remote sensing, Vol 3 art W6, Valladolid, Sain, 3-4 June 999. Huthinson, M. F loally adative aroah to the interolation of digital elevation models. Proeedings, Third International Conferene/Worksho on Integrating GIS and Environmental Modeling, Santa Fe, NM, Santa Barbara, C: See: htt:// Ra H. Rihard 984, The determination of High degree otential oeffiient exansions from the ombination of satellite and terrestrial gravity information. Reorts of the deartment of geodeti Siene and surveying, No 36 Preared for NS. Shaffrin B.99. Biased Kriging on the shere?. Geostatistis Troia 9, editor Soares milar,kluwer ademi Publishers, -3. International rhives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B. msterdam. 3

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p[4] p[3] p[2] p[1] p[0] CMSC 425 : Sring 208 Dave Mount and Roger Eastman Homework Due: Wed, Marh 28, :00m. Submit through ELMS as a df file. It an either be distilled from a tyeset doument or handwritten, sanned, and enhaned