DTIC ELEC.-TE S SEPi 71992

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1 AD-A Estimation of EOF Expansion Coeffiients from Inomplete Data DTIC ELEC.-TE S SEPi C J. D. Boyd Oeanography Division Oean Siene Diretorate E. P. Kennelly Neptune Sienes, In. Slidell, LA P. Pistek Oeanography Division Oean Siene Diretorate Approved for publi release: distribution is unlimited. Naval Oeanographi and Atmospheri Researh Laboratory, Stennis Spae Center, Mississippi NJl~lI i llllliilhil

2 These working papers were prepared for the timely dissemination of information; this doument does not represent the offiial position of NOARL.

3 Abstrat A least-squares minimization tehnique for estimating prinipal omponents (empirial orthogonal funtion expansion oeffiients) from inomplete oean profile data is developed, and its use for interpolating or extrapolating these profiles is desribed. A o fi ~,ý t o F o r M'IC QUALM'1 TITISM7,1D 3 a ~ F V -.*.4 t t y r, K\' :DiAt ipl.1a1

4 Aknowledgments This work was funded by the Offie of Naval Tehnology Code 230 under Program Elements N and N, Dr. R. Doolittle, ONT Program Manager, and Dr. E. Franhi, Naval Researh Laboratory Program Manager. ii

5 Contents 1. Introdution Theory Exam ple Coeffiient Estimation Software Conluding Remarks R eferenes... 7 Appendix A: FORTRAN Subprogram i1iie

6 Estimation of EOF Expansion Coeffiients from Inomplete Data 1. Introdution Among the earliest published reports on the use of empirial orthogonal funtions (EOFs) in oeanography are inluded suh artiles as Moore (1974), Kundu, et al. (1975) and Davis (1976). Sine then, the use of EOFs has beome inreasingly ommon for suh purposes as desribing and quantifying oeani variability (e.g., Halliwell and Mooers, 1979), in examining oean dynamis (e.g., Kundu, et al., 1975), in providing ompat haraterization of oeanographi data (e.g., Tolstoy, et al., 1991), and in examining interrelationships between surfae and subsurfae parameters (e.g., Carnes, et al., 1990). In this note we present a tehnique for using EOFs for interpolating or extrapolating missing data in oean profiles. This tehnique should prove partiularly useful for extrapolating shallow oean data profiles -- say, from shallow CTD (ondutivity-temperature-depth) asts or from expendable probes -- to deeper depths. The theory behind EOF omputations is straightforward. (See, for example, Lagerloef and Bernstein, 1988, for a partiularly ompat and luid desription.) As an aid to understanding our tehnique desribed in setion 2, we present a summary of the EOF omputation proedures here. Suppose we have N profiles with measurements at K disrete depths. Eah profile may be onsidered a vetor f., with omponents fk,. A set of perturbation profiles f'. is formed by subtrating the mean profile T from eah of the observed profiles. From these profiles the K x N matrix [F] is reated, having olumns whih are the N perturbation profiles f'.. Next the K x K ovariane matrix [R] is alulated from the [F'] matrix aording to [R)] =(1N) [F] [F] T, where [FIT is the transpose of [F']. Then the eigenvalues Ak and eigenvetors ek are found for the matrix equation [R] [E] = P[L [E]. [I is the matrix of eigenvalues 12

7 and [E] is the matrix of eigenvetors [E] = [e, e 2... e..,ekj where e. is a olumn vetor representing the mth eigenmode. eigenmodes may be alulated. A maximum of K The original data may be expanded in terms of these new basis funtions em aording to the relationship [F] = [El [C] (1) where the. elements of [C] are known as the expansion oeffiients, the modal amplitudes, or the prinipal omponents. The oeffiient matrix [C] is omputed from [C] = [E] T [F]. (2) The trae of the matrix [R] is the total variane of the data set. Sine Tr [R] = z x,, the relative value of eah xm is the fration of the variane of the data represented by the assoiated eigenvetor. If the eigenvalues are ordered by size (that is, by fration of the variane explained by the orresponding eigenvetor), it is usually found that only a limited number of eigenmodes are neessary to aount for a very large fration of the variane, say 3 to 5 (e.g., Kundu, et al., 1975; Servain and Legler, 1986; Fukumori and Wunsh, 1991). For many oeanographi appliations it is satisfatory to retain only those first M eigenmodes that aount for, say, 90 or 95% of the variane and subsequent profiles (the "syntheti profiles," as they were alled in Carnes, et al., 1990) are onstruted using only M eigenvetors. If a given profile f has data at eah of the K depths, estimation of the prinipal omponents is straightforward and is given by eq. (2) above. However, if the profile is inomplete, a modified proedure must be used. In this note we present one suh proedure based upon least squares minimization. 2. TIeory Aording to eq. (1) above, a "syntheti profile" p may be onstruted from M predetermined eigenvetors em aording to the equations Pk = "k + 1:m ei, where k is the depth index, k = 1,..., K. If all depths are present, eq. (2) shows the oeffiients, are alulated from the relationship 2

8 Cm -- Ek emk P'k" In pratie, however, some of the K depth values may be missing: the profile may not start lose enough to the surfae, or it may have terminated at too shallow a depth or there may have been noise ontamination within a ertain depth range. If only an oasional value is missing, it may be reonstruted by interpolation and the expansion oeffiients may then be omputed from a omplete data set. However, if interpolation is not satisfatory or feasible, a least-squares minimization approah may be used in whih we determine those priniple omponents m whih minimize the error e, the sum of the squares of the differenes between a perturbation profile P'k = Pk - "k and its "syntheti" reonstrution: C = Ek (P'k - E.m. ek.)2. This involves solving the system of equations ekl ekl e,.., ek e- ' P'k e., ek2 eki C2 P'k ek (3) ekm ekl... ek ep CM p'k C~ The resulting set of oeffiients {Cm} onstitute the desired estimates of the true oeffiients. 3. Example As an example of the appliation of this tehnique, onsider the 28 temperature profiles shown in Fig. la. Table 1 lists the measurement depths of the profiles. Fig. lb shows the mean temperature profile. The data represent a set of CJ-D temperature profiles taken in July during an experiment in the Sargasso Sea, but any other omparable data set would do as well for desribing our tehnique. Shown in Fig. 2 are the first five EOF modes (i.e., e 1 through e.), whih jointly explain 98% of the variane apparent in Fig. I. The first five prinipal omponents for eah perturbation profile in Fig. 1 as estimated from eq. (2) are given in Fig. 3a, and the root-mean-square (RMS) differene between the true profiles and the syntheti profiles generated with these five prinipal omponents and the five eigenfuntions in Fig. 2 is given in Fig. 3b. Using the five eigenfuntions, the maximum RMS deviation is about 0.25 OC near 50 nm, whih is loated somewhat below the base of 3

9 the mixed layer at around 30 m. Otherwise the mean deviation is typially less than 0.15 oc. First onsider the impat on the oeffiient values of having missing data near the surfae. We estimated the prinipal omponents for six separate ases with inreasingly large amounts of missing data, as desribed in Table 2. The resulting oeffiients, normalized by their true values omputed from the omplete profiles (Fig. 3a), are shown in Fig. 4, and the RMS differenes between the true profiles and their orresponding syntheti profiles onstruted from five modes are given in Fig. 5. Not surprisingly, as the range of the missing data points beomes larger, in general the deviation of the estimated oeffiients from their true values beomes larger, as does the RMS differene aross the depth range of the missing data. However, the oeffiients of ertain modes are muh less sensitive than the oeffiients of other modes. The oeffiients of the first mode are least sensitive to data deletion, but by the time the first 100 m of data are removed (Case 5), the individual oeffiients begin to deviate notieably from their true values. Mode 4 oeffiients were the most sensitive, followed by mode 5, then mode 3, and mode 2. This orrelates well in a qualitative sense with the loation of extrema in the eigenfuntions of Fig. 2. However, even with the upper 100 (Case 5) to 150 (Case 6) m of data removed, the RMS deviation is less than 1.7 0C. Notie that the RMS differene below the depth of the missing data atually is less than it is when using the full profiles. This is beause when omplete profiles are used, a trade-off must our in the alulation of the prinipal omponents, where the various eigenfuntions must be fit as well as possible over the full depth range. When the highly strutured upper part of the profile is missing, the remaining portion may be better fit with the eigenfuntions. Next onsider what ours if inreasing amounts of data are missing at the bottom of the profiles. The six ases onsidered are desribed in Table 3, the normalized oeffiients are in Fig. 6, and the RMS differenes are in Fig. 7. Beause there is so muh less struture at the bottom of the profiles, the impat of missing data is less deleterious. In general, as the amount of missing data beomes larger, the deviation of the estimated oeffiients from their true values also beomes larger. However, the deviations are muh smaller in omparison with the estimated oeffiients onstruted with near-surfae data points removed. RMS differenes are also muh smaller; less than 0.3 "C in Case 12 when 1400 m of data were removed from the bottom of eah profile. Again the oeffiients of the first mode are by far the least sensitive to the deletion of data; they do not vary signifiantly from their true values even when a large number of nearbottom data points are missing from the profiles. Modes 2 and 5 are the most sensitive, but their deviations from their respetive true values are muh less than when the surfae values are removed. Deviations in mode 3 inrease quite slowly as the amount of missing data inreases. The RMS differene inreases as the profiles are progressively trunated. The most 4

10 sensitive part of the profiles lies between about 600 to 1200 M, where mode 2 in partiular has onsiderable struture. Deletion of data below this depth range has only a modest impat upon the resulting syntheti profiles. The RMS differene above the trunation depths again tends to be somewhat less than the RMS differene omputed using full depth profiles. Finally, we deleted inreasingly larger amounts of data in the depth ranges entered around 300 m (a minimum in the variability seen in Fig. 1) and around 600 m (a maximum in the variability), as desribed in Tables 4 (Cases 13, 14, and 15) and 5 (Cases 16, 17, and 18). The normalized oeffiients and RMS differenes are given in Figs. 8 and 9 and Figs. 10 and 11. Both of these sets of results show deletion of data inreases the RMS differenes between the syntheti profiles and the data over that omputed with omplete data; however, the inrease in the RMS is only in the viinity of the missing data and it is remarkably small. Case 18 was omputed with data between m deleted, but the RMS differene at 500 m inreased less than 0.1 C to about 0.2 a C. 4. Coeffiient Estimation Software To implement the theory desribed above, a FORTRAN subroutine ESTCOE was written and is presented in Appendix A. ESTCOE requires the input of two data files; one ontaining the profiles themselves and the other ontaining the mean profile and eigenvetors assoiated with the area of interest. For eah profile, the system of equations given in eq. (3) is solved for the set of unknown estimated oeffiients {m). The syntheti profile is then onstruted using the estimated oeffiients, the input eigenvetors, and the mean profile. The output oeffiients are stored in a two-dimensional variable, Coef(MaxProffles, Maxz), where the maximum number of profiles and maximum number of depths have been set (by a parameter statement) as MaxProfiles = 100 and Maxz = 50, respetively. The output syntheti profiles are also stored in a two-dimensional variable, pp(maxprofiles, Maxz). Other parameters are returned from ESTCOE. A full desription of these parameters as well as a more omplete desription of the algorithm, format of input files, output parameters, an example driver program, and the FORTRAN ode itself is found in Appendix A. 5. Conluding Remarks We have desribed a least-squares minimization tehnique for estimating prinipal omponents from inomplete oean profile data. An important appliation of this tehnique is to ompute from these estimated prinipal omponents a syntheti profile that interpolates or extrapolates the missing data values. This syntheti profile an then be used in plae of the original inomplete profile, or it an be merged with the original profile in some fashion to fill in the missing data. In partiular, the tehnique lends itself to the problem of 5

11 extrapolating shallow oean data profiles - say, from shallow CTD asts or from expendable probes suh as XBTs (expendable bathythermographs) or AXBTs (the air-launhed XBT) - - to a deeper preseleted depth. A ommonly used alternative is to use limatologial values below the maximum data depth, with some merging proedure to ombine the two. The tehnique presented here, however, makes use of the information ontained in the struture of the observed profile and the orrelations between that upper level struture and the deeper struture to produe what will, in many ases, be a better estimate of the deeper struture than an be made with limatology alone. To illustrate this, we omputed the RMS differene between our 28 temperature profiles in Fig. la and the appropriate GDEM (Teague, et al., 1990) and Levitus (Levitus, 1982) limatologial profiles (Fig. 12). We also simulated the use of T-7 XBTs by utting our 28 profiles off below 700 m, estimating the prinipal omponents as desribed above, and omputing the syntheti profiles from these prinipal omponents (this is Case 10 of Fig. 7). The RMS differene for Case 10 between the syntheti profiles and the data is also reprodued on Fig. 12. The RMS differene for the syntheti profiles is substantially lower than the RMS differenes for either of the limatologies, partiularly below 700 m. Hene, merging the syntheti profiles with the 700 m deep measured profiles would give a better depth-extended data set than using either limatology. These profiles happened to ome from the Sargasso Sea region, but we feel onfident our results would hold elsewhere as well. 6

12 6. Referenes Carnes, M. R., J. L Mithell and P. W. DeWitt (1990). Syntheti temperature profiles derived from Geosat altimetry: Comparison with air-dropped expendable bathythermograph profiles. J. Geophys. Res. 95: Davis, R. E. (1976). Preditability of sea surfae temperatures and sea level pressure anomalies over the North Paifi Oean. J Phys. Oeanog. 6: Fukumori, I., and C. Wunsh (1991). Effiient representation of the North Atlanti hydrographi and hemial distributions. Prog. Oeanogr. 27: Halliwell, G. R., Jr., and C. N. K. Mooers (1979). The spae-time struture and variability of the shelf water-slope water and Gulf Stream surfae temperature fronts and assoiated warm ore eddies. J. Geophys. Res.: 84, Kundu, P. K., J. S. Allen, and R. L. Smith (1975). Modal deomposition of the veloity field near the Oregon oast. J. Phys. Oeanog. 5: Lagerloef, G. S. E., and R. L Bernstein (1988). Empirial orthogonal funtion analysis of advaned very high resolution radiometer surfae temperature patterns in Santa Barbara Channel. J Geophys. Res. 93: Levitus, S. (1982). Climatologial atlas of the world oean. NOAA Teh. Pap., 3, 173 pp. Moore, D. (1974). Empirial orthogonal funtions - A non-statistial view. MODE Hot Line News, Ot , 1-6. Servain, J., and D. M. Legler (1986). Empirial orthogonal funtion analysis of tropial Atlanti sea surfae temperature and wind stress: J Geophys. Res. 91: Teague, W. J., M. J. Carron and P. J. Hogan (1990). A omparison between the generalized digital environmental model and Levitus limatologies. J. Geophys. Res. 95: Tolstoy, A., and 0. Diahok (1991). Aousti tomography via mathed field proessing. J. Aoust. So. Am. 89(3):

13 Table 1. Standard depths (m) used in the analysis Table 2. Depths at whih data values were eliminated for Cases 1-6. Case 1: 0m Case 2: 0, 10, 20 m Case 3: 0, 10, 20, 30 m Case 4: 0, 10, 20, 30, 50 m Case 5: 0, 10, 20, 30, 50, 75, 100 m Case 6: 0, 10, 20, 30, 50, 75, 100, 125, 150 m Table 3. Depths at whih data values were eliminated for Cases Case 7: 2000, 1750 m Case 8: 2000, 1750, 1500, 1400, 1300, 1200, 1100 m Case 9: 2000, 1750, 1500, 1400, 1300, 1200, 1100, 1000, 900 m Case 10: 2000, 1750, 1500, 1400, 1300, 1200, 1100, 1000, 900, 800 m Case 11: 2000, 1750, 1500, 1400, 1300, 1200, 1100, 1000, 900, 800, 700 m Case 12: 2000, 1750, 1500, 1400, 1300, 1200, 1100, 1000, 900, 800, 700, 600 m Table 4. Depths at whih data values were eliminated for Cases Case 13: 300 m Case 14: 250, 300 m Case 15: 200, 250, 300, 400 m Table 5. Depths at whih data values were eliminated for Cases Case 16: 600 m Case 17: 500, 600 m Case 18: 500, 600, 700 m 8

14 28 Temperature Profiles Mean Profile Perturbation Profiles E Temperature (oc) Temperature (0C) Temperature (00) Figure 1. The 28 temperature profiles used in the analysis. a: overplot of all profiles; b: mean profile; : overplot of perturbation profiles. 9

15 Mode 1 (78.6%) Mode 2 (11.6%) Mode 3 (4.5%) E (D 1000 ") Amplitude Amplitude Amplitude "E" a) 1000 C) Mode 4 (1.9%) Mode 5 (1.3%) 2000 ' Amplitude Amplitude Figure 2. The first five eigenvetors omputed from the data in Fig. 1. The number in parentheses is the perent of the total variane that is desribed by that partiular eigenvetor. 10

16 First 5 prinipal omponents o for 28 temperature profiles , E 600-0,- 800 E I! I n < Mode RMS Differene ( C) Figure 3. a: The expansion oeffiient values versus mode number for oeffiients alulated using full depth profiles; b: the RMS temperature differene over all 28 profiles between the real and the syntheti profiles omputed with five modes. 11

17 Case 1 Case "_0 15 -D < E 5 o:! < E 5 o i, I I 0-0 (1) N N CO -10 -Z -10 E ' O -15 z z Mode E Mode Case 3 Case D D Q- 0. E 5 E 5 ' I1 ' 0 I N* U CO -10 -O -10 E E E- -15 E Mode Mode Case 5 Case 6 20, F 1 20, 1, 01) 01) "_0 15 -o 15 DD S o.i * -~Qa E 5 - E 5,,,-oi ' _0 0 I I I I *0 I * D -.5 "- - 5 ; " - L4 -N C " E E " "-15 ~0 S-20 z~ Mode Mode Figure 4. The normalized estimated oeffiient amplitudes versus mode number for Cases 1-6 in whih data in the upper portion of the profiles were deleted. Table 2 desribes the data deleted for eah ase. 12

18 E C Case 1 Case 2 Case RMS Differene (OQ) RMS Differene (00) RMS Dif ferene (OC) E 600.z , 1000 " L Case 4 Case 5 Case 6 0 : -, 1 U j RMS Differene (CQ) RMS Differene (00) RMS Differene (OQ) Figure 5. The RMS differenes for Cases 1-6 between the real and the syntheti profiles omputed using the estimated oeffiients given in Fig. 4. The dotted urve is the RMS differene from Fig. 3b, shown for omparison. The shaded regions indiate the depth range over whih data was deleted. 13

19 Case 7 Case (I) 8 ' : 26 o - D3 6 n E 2 E 2 _ <0 0 <0 0 0) -2 G -2 N N "M 4-4 E -6 E -6 O z Zl Mode Mode Case 9 Case T 10 a) " " C 4 CL 4 E 2 E 2 <a 0 -. <0., *00 (D -2 0(D -2 N N "-4 3 CO - E -6 E z 10 z 101 0o Mode Mode Case 11 Case ) 8 a) 8 -o (D6 -u 6D CL 4 -_ 4. E 2 I E 2 < < -o 0-0 I: 0D N N " E -6 E-6 O z zl i Mode Mode Figure 6. The normalized estimated oeffiient amplitudes versus mode number for Cases 7-12 in whih data in the lower portion of the profiles were deleted. Table 3 desribes the data deleted for eah ase. 14

20 E 600 -C 800 g' QZ 1200 t, Case 7 Case 8 Case p RMS Differene (CC) RMS Differene (OC) RMS Differene (OC) Case 10 Case 11 Case E a) ti L 18E00L RMS Differene (0C) RMS Differene (OC) RMS Differene (0C) Figure 7. The RMS differenes for Cases 7-12 between the real and the syntheti profiles omputed using the estimated oeffiients given in Fig. 6. The dotted urve is the RMS differene from Fig. 3b, shown for omparison. The shaded regions indiate the depth range over whih data was deleted. 15

21 Case 13 Case S22 E 1 -< 0 0 0< 0 (D (1)._N -1_-N - SL 1 E -2 E -2 O 0 o Mode Mode 0) 3 Case 15 "2 " ' 0 _0 -l:= 1 ' jli E E -2 0 Z -3 ' o Mode Figure 8. The normalized estimated oeffiient amplitudes versus mode number for Cases in whih data around 300 m (a data variane minimum) were deleted. Table 4 desribes the data deleted for eah ase. 16

22 E 600 S 800 S Case 13 Case 14 Case I RMS Differene ( C) RMS Differene (OC) RMS Differene ( C). Figure 9. The RMS differenes for Cases between the real and the syntheti profiles omputed using the estimated oeffiients given in Fig. 8. The dotted urve is the RMS differene from Fig. 3b, shown for omparison. The shaded regions indiate the depth range over whih data was deleted. 17

23 Case 16 Case (D D " E E <0 0I - (D (D - -5 M -5 E o 0-10, 10, z Mode E Mode _0 C E -o ai) N Case 18 -O -5 E 0 z Mode Figure 10. The normalized estimated oeffiient amplitudes versus mode number for Cases in whih data around 600 m (a data variane maximum) were deleted. Table 5 desribes the data deleted for eah ase. 18

24 0 200 Case 16 Case 17 Case 18 E / S800 A S RMS Differene (0C) RMS Differene (0C) RMS Differene (0C) Figure 11. The RMS dbierenes for Cases between the real and the syntheti profiles omputed using the estimated oeffiients given in Fig. 10. The dotted urve is the RMS differene from Fig. 3b, shown for omparison. The shaded regions indiate the depth range over whih data was deleted. 19

25 Q)1000 E 600 fi: RMS Differene (0C) Figure 12. RMS differenes between the data in Fig. la and GDEM limatology (dotted line), Levitus limatology (dashed line) and syntheti profiles omputed from data trunated below 700 m to simulate T-7 XBTs (solid line). At all depths the syntheti profiles exhibit a smaller RMS differene and hene provide a better estimate of the profiles below 700 m than either of the limatologies. 20

26 Appendix A FORTRAN Subprogram 21

27 FORTRAN Subroutine ESTCOE I. Introdution Subroutine ESTCOE estimates EOF oeffiients for a set of input data profiles. It also alulates syntheti profiles using the estimated oeffiients. 11. Input ESTCOE requires the unit numbers of 2 input files: inunitl: The mean profile & eigenvetors. inunit2: The data profiles themselves. The mean profile & eigenvetors must be arranged in the first input file as follows: Depth values (of the mean profile) are stored in olumn 1, and the mean profile values at their respetive depths must be storeu, in olurnv 2. An "end of reord" flag is required at the end of the mean profile. This end of reord flag is a pair of real numbers, After the mean profile end r. reord flag, the eigenvetors are stored sequentially. For eah eigenvetor, the depth values should be stored in olumn 1, and the eigenveto, values at their respetive depths irus. appear in olumn 2. An end of reord flag is required at the end of eah ei,6envetor "reord." NOTE: The uumber oi depths in the mean profile/eigenvetor file has been set to 50 via a parameter statement: parameter(maxz = 50). This is also the maximum number of e','-envetors allowed in the mean profile/eigenvetor file. The a: a profiles must be arranged in the seond input file as follows: Depth values are stored in olumn 1, and the data values at their orresponding depths are stored in olumn 2. Depth values should be sequential; that is, a measurement taken at depth 10 m should be stored before a measurement taken at 20 m. An end of reord flag is required at the end of eah profile. NOTE: The number of depths in any single profile has been set to 6000 via a parameter statement: parameter(maxpts = 6000). III. Proessing First, the mean profile & eigenvetor input file is read. The user is asked to deide how many modes will be used to build the oeffiients (and subsequent syntheti profiles). NOTE: The maximum number of modes available is equal to the number of eigenvetors 22

28 in this input file. Then, one at a time, the profiles are read, oeffiients are alulated, and syntheti profiles are onstruted using the following approah: a. When a profile is loaded, it is heked against the eigenvetor depths for missing data points (relative to the eigenvetor depths). If a profile ontains data at depths that are not in the eigenvetor fie, these "extra depths" are ignored in subsequent oeffiient estimation subroutines. If a profile ontains "missing data" relative to the eigenvetor file, this is noted in a "flag" array. b. For a single profile, the system of equations given in eq. (3) is solved using standard Gauss-Jordan elimination routines 1.. The eigenvetor matrix [ee 2... e,] is reated using only the eigenvetor depths where input data values are present. To solve for the oeffiient vetor represented by the set of values {m}, the inverse of the eigenvetor matrix is found. The determinant of the resulting matrix is displayed. If the determinant is too small (relative to a set tolerane of 1.OD-14), the user is warned that there may be large errors in the resulting estimated oeffiients. The user is enouraged to rerun the program using a smaller number of modes with whih to build the oeffiients. d. The right-hand side vetor in eq. (3) is reated using the input profile data, the mean profile, and the eigenvetors. The entire system of equations is then solved. e. The syntheti profile is built using the estimated oeffiients. The oeffiients and the syntheti profile are stored, and the program "loops" through again using the next profile's data. IV. Output Output from ESTCOE onsists of the variables listed below. Parameter values are set for the maximum number of depths allowed in the mean profile/eigenvetor input file (Maxz = 50) and the maximum number of input profiles (MaxProfiles = 100). Num.probes: e_ndep: Num eigen: e depth: Number of profiles in the input data file Number of depths in eah eigenvetor (as well as the mean profile) Number of eigenvetors (or modes) used in alulating the estimated oeffiients Array of eigenvetor depths; e depth(maxdepths) 'Press, William H. et al. (1986). Numerial Reipes -- The Art of Sientifi Computing. New York (NY): Cambridge University Press, pp

29 Coef: 2-D array that stores the estimated oeffiients; Coef(MaxProfiles, Maxz) pp: 2-D array that stores the syntheti profiles; pp(maxprofiles, Maxz) 24

30 V. Example of input files and driver program (1) Input file 1: Mean temperature profile and eigenvetors (depth 0 m, mean profile value at 0 m) (depth 2000 mi, mean profile value at 2000 m) (end of reord) (depth 0 m, eigenvetor 1) (depth 2000 m, eigenvetor 1) (end of reord) (depth 0 m, eigenvetor N) (depth 2000 m, eigenvetor N) (end of reord) (2) Input file 2: Temperature profiles (depth 0 m, temperature value at 0 m, profile 1) (end of reord) (depth 0 m, temperature value at 0 m, profile N) (end of reord) 25

31 (3) Example of a driver program that alls the subroutine ESTCOE. $1arge program driver Purpose: To test the subroutine ESTCOE Compiler: Mirosoft FORTRAN v 4.01 To ompile: fl - driver.for To link: link driver + ESTCOE; To run: driver Parameter values set: Maximum number of depths in any input file is set at Maxz = 50 Maximum number of input profiles is set at MaxProfiles = 100 parameter(maxz = 50, MaxProfiles = 100) double preision Coef(MaxProfiles, Maxz), pp(maxprofiles, Maxz) integer inunl, inun2, outunl, outun2 integer Num eigen, Num_probes, e_n dep, mode, probe real edepth(maxz) Set unit numbers for input/output files inunl = 10 inun2 = 11 outunl = 12 outun2 = 13 Give input filenames open(inunl, file = 'eve.dat') open(inun2, file = 'profles.dat') Give the output filenames open(outunl, file = 'oeff.dat') open(outun2, file = 'synthet.dat') Call the estimate oeffiient subroutine all est oef(inunl, inun2, Num_probes, e_n_dep, Numeigen, * e-depth, Coef, pp) Write the oeffiients to output filel 4 format(ix, i4, 1x, f10.4) do 10 probe = 1, Num.probes do 20 mode = 1, Numeigen write(outunl, 4) mode, Coef(probe, mode) 26

32 20 ontinue 10 ontinue Write the syntheti profiles to output file2 5 format(lx, flo.2, Ix, flo.4) do 30 probe = 1, Num.probes do 40 k = 1, e_n_dep write(outun2, 5) edepth(k), pp(probe, k) 40 ontinue 30 ontinue Close all open files lose(inunl) lose(inun2) lose(outunl) lose(outun2) End of program stop end 27

33 $large C ****S****************************************************.************** subroutine ESTCOE(inunl, inun2, Num_probes, e_n_dep, * Numeigen, edepth, Coef, pp) Purpose: Input: To find EOF oeffiients for input profiles whih have missing depth values and to reate syntheti profiles. inunh: Unit number of input profiles inun2: Unit number of Mean profile & Eigenvetors Output: Numprobes: Number of profiles in the input data file e_n_dep: Number of depths in eah eigenvetor (as well as the mean profile) Numeigen: Number of eigenvetors (or modes) used in alulating the estimated oeffiients edepth: Array of eigenvetor depths; edepth(maxz) Coef: 2-D array whih stores the estimated pp: oeffiients; Coef(MaxProfiles, Maxz) 2-D array whih stores the syntheti profiles; pp(maxprofiles, Maxz) C Limitations: Maximum number of input profiles is set at MaxProfiles = 100. Maximum number of depths in any single input profile is set at Maxpts = Maximum number of depths in the mean/eigenvetor file is set at Maxz = 50. Matrix Singularity riteria set at det tol = L.OD-14. Preision: Double Preision Compiler: Mirosoft FORTRAN v 4.01 To ompile: fl - ESTCOE.for To link: link driver + ESTCOE; Code uses routines from NUMERICAL RECIPIES ROUTINES: L U Deomposition (LUDCMP) Bak Substitution (LUBKSB) NOTE: Modifiations were made to the original ode for these two subroutines; real variables were hanged to double preision; unneessary output variables were deleted from alling and subroutine delaration statements. 28

34 C Program written by: Eileen P. Kennelly Neptune Sienes, In Slidell, LA Variables: A: "A" matrix in A *x =B; Used in LUDCMP B: "B" vetor in A *x =B; x = (inv)a * B; The,effiients for a partiular profile are temporarily stored in this vetor; Used in LUBKSB Coef: 2-D array whih stores alulated oeffiients; Coef(MaxProfiles, Maxz) determin: Determinant of the LUDCMP matrix; heks for singularity det tol: Tolerane hek of the determinant of the LUDCMP output matrix; set at parameter (dettol = 1.0D-14) E: Eigenvetors are stored in this 2-D matrix; E(Maxz, Maxz) eof: End of input file flag ejdepth: Array of eigenvetor depths; edepth(maxz) en dep: Number of depths in eah eigenvetor f: Shallow water data (input profiles) is stored here one profile at a time; a I-D array, f(maxpts) fbar: Mean profile assoiated with the eigenvetors; a l-d array fbar(maxz) flag: Logial array; Set to TRUE if there exists an eigenvetor value at a profile depth f.depth: Array of profile depths; fdepth(maxpts) f.n.dep: Number of depths in a partiular profile fprime: At depth z, f' = Profile value - Mean; f'(z) = f(z) - fbar(z); a ID array f.prime(maxpts) indx: Array used by LUDCMP and LUBKSB inunl: Unit number of input Mean profile and eigenvetors inun2: Unit number of input profiles MaxProfiles: Maximum number of input profiles; parameter (MaxProfiles = 100) Maxz: Maximum number of depths in either input file; parameter (Maxz = 50) N: Number of eigenvetors in the eigenvetor file Numeigen: Number of eigenvetors whih are atually used in the onstrution of estimated oeffiients Num~probes: Number of profiles in input file p: 1-D array whih stores a single syntheti profile; p(maxz) pp: 2-D array whih stores all syntheti profiles; pp(maxprofiles, Max) probe: Profile ounter 29

35 double preision dettol parameter (Maxz = 50, MaxProfiles = 100, Maxpts = 6000) parameter (dettol = L.OD-14) double preision A(Maxz, Maxz), B(Maxz), f(maxpts), determin double preision E(Maxz, Maxz), fbar(maxz), f.prime(maxpts) double preision p(maxz), Coef(MaxProfiles, Maxz) double preision pp(maxprofiles, Maxz) integer inunl, inun2, indx(maxz) integer Num.eigen, probe, endep, f.n_dep, N, Num_probes logial eof, flag(maxz) real e.depth(maxz), f_depth(maxpts) Get the Eigenvetors and the Mean vetor from first input file all EigMea(inunl, N, E, en_dep, edepth, fbar) all LdEig(N, Num-eigen) do 10 probe = 1, MaxProfiles Let user know something is happening... write(*,*) 'Calulating oeffiients for profile ', probe Load an input profile, f all Ld Pro(inun2, f, fldepth, fn-dep, eof) if (eof) goto 89 Find missing depths and set the flag array to indiate this all Miss(f depth, fndep, e depth, endep, flag) Load the matrix A (in A * x = B) with the Eigenvetors all AMatx(endep, E, Numeigen, flag, A) Call the Numerial Reipe LUDCMP to find the inverse of the A matrix in A * x =B all LUDCMP(Num eigen, A, indx) Look at the determinant of the inverse A matrix. Chek it for singularity all Det(Numeigen, A, determin) if (dabs(determin).it. det tol) then all Messageo endif Calulate f.prime; f'(z) = f(z) - tbar(z) al Ld fpr(f, f.depth, fn.dep, e.depth, e..n..dep, 30

36 * fbar, f prime) Load vetor B (in A * x = B) with f * E all LA_B(E, edepth, e.n_dep, fdepth, fn dep, * f prime, Numieigen, B) Solve for x: A*x-=B (resulting oeffiients stored in variable B) all LUBKSB(A, Num eigen, indx, B) Create the syntheti profile, all Syn(B, E, fbar, e_ndep, Numeigen, p) Store oeffiients in a 2-D array all MkCoe(probe, Num eigen, B, Coef) Store syntheti profile in a 2-D array all MkSyn(probe, e-n-dep, p, pp) 10 ontinue 89 Numprobes = probe- 1 return end subroutine EigMea(lun, N, E, en_dep, e_depth, fbar) Purpose: To load the Mean vetor, and the Eigenvetors parameter (Maxz = 50) double preision E(Maxz, Maxz), fbar(maxz) integer lun, N, endep real ej.epth(maxz) Load the Mean vetor, fbar End of mean profile flag set at do 10 k = 1, Maxz readoun, *) ejdepth(k), fbar(k) if (e depth(k).gt and. tbar(k).gt ) goto ontinue 20 endep = k- I Load the eigenvetors, E(eigenmode, depth) End of eah eigenvetor flag set at

37 do 30j = 1, Maxz do40k = 1, Maxz read(lun, *, end = 50) edepth(k), E(j, k) if (edepth(k).gt and. E(j, k).gt ) goto ontinue 31 ontinue 30 ontinue Note that there are (j - 1) eigenvetors available 50 N = j - 1 return end subroutine LdEig(N, num-eigen) Purpose: The user deides how many modes will be used to build the oeffiients (and eventually the syntheti profiles). parameter (Maxz = 50) integer N, num_,;g,.n 10 format(lx, A, 13, lx, A, A) 20 write(*, 10) 'There are', N, 'eigenvalues in the', * 'input eigenvetor file. You an use' write(*, 10) 'a maximum of', N, 'eigenvalues in the', * 'onstrution of the syntheti profile.' write(*,*) 'How many do you want to use? read(*,*) numeigen return end subroutine Ld Prolun, f, f.depth, fndep, eof) Purpose: To load a single profile parameter (Maxz = 50, Maxpts = 6000) double preision f(maxpts), vals(2, Maxpts) integer lun, fn_dep logial eof real fldepth(maxpts) Load a profile, f End of profile flag set at

38 do 10 k = 1, Maxpts read (lun,*, end = 40) vals(1, k), vals(2, k) if (vals(1, k).gt and. vals(2, k).gt ) goto 20 fdepth(k) = vals(1, k) f(k) = vals(2, k) 10 ontinue 20 f.n.dep = k- I goto eof =.true. 30 ontinue return end subroutine Miss(fLdepth, fn_dep, e_depth, e n_dep, flag) Purpose: To find the missing points in a profile parameter (Maxz = 50, Maxpts = 6000) integer endep, fn.dep logial flag(maxz) real e.depth(maxz), fdepth(maxpts) do 10 k = 1, endep flag(k) =.false. do 20 nn = 1, f n_dep if (f.depth(nn).eq. ecdepth(k)) then flag(k) =.true. goto 10 endif 20 ontinue 10 ontinue C return end subroutine AMatx(e_n.dep, E, Numeigen, flag, A) Purpose: To load the "A" Matrix in A * x = B parameter (Maxz = 50) double preision E(Maxz, Maxz), A(Maxz, Maxz) integer e._n_dep, Num-eigen 33

39 logial flag(maxz) real sum Create the A matrix using only the eigenvetors needed. do 10 i = 1, Numeigen do 20 j = 1, Num eigen sum = 0.0 do 30 k = 1, e_n_dep if (flag(k)) then sum = sum + E(i, k) * E(j, k) endif 30 ontinue A(i, j) = sum 20 ontinue 10 ontinue return end C * subroutine Det(N, A, prod) Purpose: Chek the determinant of the (inv)a matrix for possible singularity parameter (Maxz = 50) double preision A(Maxz, Maxz), prod integer N prod = 1.0 do 10i = 1, N prod = prod * A(i, i) 10 ontinue 4 format(ix, A, IX, E8.2) write(*,4) 'Determinant of the Eigenvetor LUDCMP matrix =', prod return end C ********************************************************************** subroutine Ld fpr(f, fldepth, f_n_dep, edepth, e_n_dep, * fbar, f prime) Purpose: At eah depth z, alulate f'(z) = f(z) - fbar(z) parameter (Maxz = 50, Maxpts = 6000) 34

40 double preision fbar(maxz), f prime(maxpts), f(maxpts) integer f_n_dep, en_dep real ecdepth(maxz), fdepth(maxpts) do 10 k = 1, e_n_dep do 20 nn = 1, f_n_dep If (fldepth(nn).eq. edepth(k)) then f prime(nn) = f(nn) - fbar(k) endif 20 ontinue 10 ontinue return end C ********************************************************************** subroutine LdB(E, e_depth, e-n dep, f_depth, f_n_dep, * f.prime, Num.eigen, B) Purpose: Load the vetor "B" in A * x = B parameter (Maxz = 50, Maxpts = 6000) double preision B(Maxz), f.prime(maxpts), E(Maxz, Maxz), sum integer Num_eigen, f_n_dep, endep real e.depth(maxz), fdepth(maxpts) do 10 i - 1, Numeigen sum = 0.0 do 20 k = 1, e_n_dep do 30 nn = 1, f_n.dep if (fldepth(nn).eq. edepth(k)) then sum = sum + fjprime(nn) * E(i, k) endif 30 ontinue 20 ontinue B(i) = sum 10 ontinue return end S ******************************************* *********i** subroutine Syn(C, E, fbar, endep, Num eigen, p) Purpose: At eah depth z, alulate the syntheti profile, p p(z) = Mean(z) + sum (Coeffiient * Eve) Note: This is for a single profile 35

41 parameter (Maxz = 50) double preision C(Maxz), fbar(maxz), E(Maxz, Maxz), p(maxz), sum integer Num_eigen, e_n_dep Syntheti(z) = Mean(z) + sum (Coeff * Eigenvetor) do 20 k = 1, en.dep sum = 0.0 do 10 j = 1, Num-eigen sum = sum + C(j) * E(j, k) 10 ontinue p(k) = fbar(k) + sum 20 ontinue C return end subroutine MkCoe(probe, Numeigen, B, Coef) Purpose: To store the alulated oeffiients in a 2-D matrix parameter (Maxz = 50, MaxProfiles = 100) double preision B(Maxz), Coef(MaxProfiles, Maxz) integer probe, Numeigen do 10 k = 1, num-eigen Coef(probe, k) = B(k) 10 ontinue return end subroutine Mk Syn(probe, ejndep, p, pp) Purpose: To store the alulated syntheti profiles in a 2-D matrix parameter (Maxz = 50, MaxProfiles = 100) double preision p(maxz), pp(maxprofiles, Maxz) integer probe, endep do 10 k = 1, en.dep pp(probe, k) = p(k) 10 ontinue return end 36

42 subroutine LUDCMP(N, A, indx) Purpose: To solve a Matrix A into L U Deompostion form... from NUMERICAL RECIPES parameter (Maxz = 50) parameter (nmax = 100, TINY = 1.0E-20) double preision A(Maxz, Maxz), VV(nmax), aamax, sum, dum integer N, indx(maxz), D, imax do 26 i = 1, N indx(i) = 0 26 ontinue D=I do 12 i=l,n aamax =0. do 11 j=i,n if (dabs(a(i, j)).gt. aamax) aamax = dabs(a(i, j)) 11 ontinue if (aamax.eq. 0.) pause 'Singular matrix.' VV(i) = 1. / aamax 12 ontinue do 19j = 1, N if(j.gt. 1) then do 14i = 1, j- 1 sum = A(i, j) if (i.gt. 1) then do 13 k = 1, i - 1 sum = sum - A(i, k) * A(k, j) 13 ontinue A(i, j) = sum endif 14 ontinue endif aamax = 0. do 16i = j,n sum = A(i, j) if (j.gt. 1) then do 15k= 1,j-1 sum = sum - A(i, k) * A(k, j) 15 ontinue A(i, j) = sum endif dum = VV(i) * dabs(sum) if (dum.ge. aamax) then 37

43 imax = i aamax = dum endif 16 ontinue if (j.ne. imax) then do 17k = 1, N dum = A(imax, k) A(imax, k) = A(j, k) A(j, k) = dum 17 ontinue D=-D W(imax) = W(j) endif indx(j) = imax if j.ne. N) then if (A(j, j).eq. 0.) A(j, j) = TINY dum = 1. / AO, j) do 18 i =j + 1, N A(i, j) = A(i, j) * dum 18 ontinue endif 19 ontinue if (A(N, N).eq. 0.) A(N, N) = TINY C 8 return end subroutine LUBKSB(A, N, indx, B) Purpose: To solve the system A * x = B for x by bak substitution... from NUMERICAL RECIPES parameter (Maxz = 50) double preision A(Maxz, Maxz), B(Maxz), sum integer indx(maxz), 11, N ii=0 do 12 i = 1, N 11 = indx(i) sum i B(l) B(nl) = B(i) if (ii.ne. 0) then do l1j = ii, i - I sum = sum - A(i, j) * B(j) 11 ontinue elseif (sum.ne. 0.) then 38

44 11=i endif 3(i) = sum 12 ontinue do 14 i = N, 1, -1 sum = B(i) if (i.it. N) then do 13j = i + 1, N sum = sum - A(i, j) * B(j) 13 ontinue endif B(i) = sum / A(i, i) 14 ontinue C 8 return end subroutine MessageO Purpose: To flash a warning message if the determinant of the LUDCMP matrix is too small write(*,*) ' ' write(*,*) 'WARNING. In trying to alulate oeffiients, it', 'was found that the' write(*,*) 'eigenvetor matrix was singular; the oeffiients', 'may be erroneous.' write(*,*) 'It is suggested that you re-run this program and', 'hoose a smaller number' write(*,*) 'of eigenvalues to onstrut the oeffiients.' write(*,*) 'Proessing ontinues...' write(*,*) return end 39

45 Distribution List Commander Dr. Jim Merer Code 64H Applied Physis Laboratory RDT & E Division U of Washington NCCOSC 1013 NE 40th Street San Diego, CA Seattle, WA Dr. Rihard Doolittle Dr. Peter Worester Offie of Naval Tehnology, Code 230 Sripps Institution of Oeanography, 800 N. Quiny Street A-013 Arlington, VA University of California at San Diego La Jolla, CA Dr. Eri Hartwig Code 7000 NRL Code 125L (10) Naval Researh Laboratory Code 125P Washington, DC Code 110 Code 200 Robert Hearn Code 253 M. Fagot Code 7104 Code 300 RTD & E Division Code 322 M. Carnes NCCOSC Code 330 A.W. Green San Diego, CA Code 331 R. Hollman Code 331 J. Boyd (20) Dr. Rihard Heitmeyer Code 331 W. J. Teague Code 5120 Code 331 D. Johnson Naval Researh Laboratory Washington, DC NAVOCEANO Code OPTR (attn: T. Bennett) Dr. Brue Howe Applied Physis Laboratory Commanding Offier U of Washington Attn: Code TD library 1013 NE 40th Street Naval Oeanographi Offie Seattle, WA SSC, MS Dr. William Hodgkiss Commander Marine Physial Laboratory Naval Oeanography Command Sripps Institution of Oeanography SSC, MS San Diego, CA Commanding Offier Dr. Robert Knox Fleet Numerial Oeanography Center Sripps Institution of Oeanography, Monterey, CA A-013 University of California at San Diego Superintendent La Jolla, CA Naval Postgraduate Shool Monterey, CA 93943

46 Diretor Woods Hole Oeanographi Institution PO Box 32 Woods Hole, MA University of California Sripps Institute of Oeanography PO Box 6049 San Diego, CA Johns Hopkins University Applied Physis Laboratory Johns Hopkins Road Laurel, MD University of Washington Applied Physis Laboratory 1013 NE 40th St. Seattle, WA Naval Researh Laboratory Attn: Library Washington, DC

47 DCUMNTATON AGEForm Approved REPOT DOUMETATIN PAE [ OBM No. 0704o0188 Pubi ww"ni burden tor this oualon of imfornitn is ashurnatd to average 1 hour per fsoapr. inluding the trime for reviiiiivng inatrutiofis. "arhing existing data soures. gathering and rnsoraitafilthedaa needed. and owirlting and rwngethelliin Worrntesaion. Send imennents negasdingthisburd~en ofany other mwe ofthe olletion ofi mlornison. rluiing suggestion$ for reduingthis burdrto Weshwon HeIadquaruus serviles. Direlttorae for inomtion Operations and Reports, 1215 Jeffesmon Davis Hightiay. Suite Ariiogtoni. VA : and to the Offi de Menagement andt Budget. Paperworki Redution Protet ( ). Washington, DC Ageny Use Only (Loam, blank). 2. Report Date. 13. Report Type and Date* Covered. IAugust 1992 Final 4. Title and Subtitle. S. Funding Numbers.0635N Estimation of EOF Expansion Coeffiients from Inomplete Data Program Element Nlb N Projet No ,3503, 6. Author(s). Task No. JOD J.D. Boyd, E.P. Kennelly% and P. Pistek Aession No. DN Work Lrt Ab 13312A 7.Performing Organization Name(s) and Address(es). 8. Performing Organization Naval Oeanographi and Atmospheri Researh Laboratory Report Number. Oean Siene Diretorate Stennis Spae Center. MS NOARL Tehnial Note Sponsoring/Monitoring Ageny Name(s) and Address(es). 10. Sponsoring/Monitoring Ageny Offie of Naval Tehnology Report Number. 800 N. Quiny Street Arlington, VA NOARL Tehnial Note Supplementary Notes. *Neptune Sienes, In. Slidell, LA a. Distribution/Avalability Statement 1 2b. Distribution Code. Approved for publi release; distribution is unlimited. 13. Abstrat (DMaxh,,um 200 wods). A least-squares minimization tehnique for estimating prinipal omponents (empirial orthogonal funtion expansion oeffiients) from inomplete oean profile data is developed, and its use for interpolating or extrapolating those profiles is desribed. 14. Subjet Taines. 15. Number of Pages. Aousti Arrays, Oeanographi Data, Underwater Sound Signals Prie Code. 17. Seurity Classifiation 16. Seurity Classifiation 19. Seurity Classifiation 20. U.mltation of Abstrat. of Report, of This Page. of Abstrst Unlassified Unlassified Unlassified SAR NS S001-8-S Standard Form 296 (Rev. 2-89) Piewribed by ANSI Sid. Z &-02