Analysis of Remotely Sensed Ocean Data by the Optimal Spectral Decomposition (OSD) Method

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1 Proceedings on IEEE/TS OCENS 9, Biloxi, ississippi, US, October 6-9, 9 nalysis of Reotely Sensed Ocean Data by the Optial Spectral Decoposition (OSD) ethod Peter C. Chu bstract new data analysis/assiilation schee, optial spectral decoposition (OSD), has been developed to reanalyze fields fro noisy and sparse data in a doain with open boundary conditions using two scalar representations for a three-diensional incopressible flow. The reanalysis procedure is divided into two steps: specification of basis functions in the spectral decoposition fro nowledge of boundary geoetry and velocity and deterination of coefficients in the spectral decoposition for the circulation solving linear or nonlinear regression equations. The basis functions are the eigenfunctions of the Laplacian operator with ixed boundary conditions. The optiization process is used to obtain unique and stable solutions on the base of an iteration procedure with special regularization (the filtration. The capability is deonstrated using various exaples. Key Words Optial spectral decoposition, rgo drifter, Lagrangian data, satellite data, rotation ethod. Introduction Sparse and noisy ocean data need to be reanalyzed before being assiilated into nuerical odels. ny field (teperature, salinity, or velocity) can be decoposed into generalized Fourier series using the Optial Spectral Decoposition (OSD) ethod. The three diensional field is then represented by linear cobination of the products of basis functions (or called odes) and corresponding Fourier coefficients. If a rectangular closed ocean basin is considered, the basis functions are sinusoidal functions. If a realistic ocean basin is considered, the basis functions are the eigenvalues of the three-diensional Laplace operator with real topography. The Fourier coefficients are deterined fro observational data through solving a set of linear algebraic equations. ajor benefit of using the OSD ethod is that the boundary conditions for the ocean variables (teperature, salinity, velocity) are always satisfied. Peter C. Chu is with the Naval Ocean nalysis and Prediction Laboratory, Departent of Oceanography, Naval Postgraduate School, onterey, C 93943, US (eail: pcchu@nps.edu). Generalized Fourier-Series Expansion Let (x, z) be horizontal and vertical coordinates and t be tie. physical variable c( x, z, t) at depth z is decoposed using the generalized Fourier series (Chu 999b; Ivanov and Chu 8; Chu et al. 3a, b, 5a, b) ( ) ( x ) c( x, z, t) = ( z, t) + z, t Ψ, z, x Rz ( ) = () where is the truncated ode nuber, Ψ ( x, z ) and ( z, t ) are the orthogonal basis functions (or called odes) and the spectral coefficients, respectively; R ( z ) is the area bounded by the lateral boundary Γ ( z ) at depth z. The basis functions { Ψ ( x, z )} are eigen-functions of the horizontal Laplace operator with the basin geoetry and certain physical boundary conditions. For teperature and salinity, the hoogeneous Neuann boundary condition is taen at the solid boundary Γ ( z) (i.e., no heat and salt fluxes), Ψ = λ Ψ ni Ψ =, =,,...,, h, h Γ () where / x + / y, and n is the unit vector h noral to Γ ( z). The basis functions { Ψ } are independent of the data and therefore available prior to the data analysis. The OSD ethod has two iportant procedures: optial ode truncation and deterination of spectral coefficients { }. fter the two procedures, the generalized Fourier spectru (3) is used to provide data at regular grids in space and tie. 3. Optial ode Truncation The optial ode truncation nuber ( opt ) is defined as the critical ode nuber with the set of spectral coefficients { } least sensitive to observational data sapling and noise. For saple size of P and ode truncation of, the spectral coefficients { } are

2 estiated by the least square difference between observed and calculated values J = J(,...,, P, ) ep P ( j) ( j) = c ( z, t) Ψ ( x, z) in P j = = where the sybol ~ represents the estiated value at (x, t). For hoogeneously sapled data with low noise and without systeatic error, the epirical cost function J ep should tend to onotonically as increases to infinity. The set of the spectral coefficients { } depends on the ode truncation. Optial estiation of { } is equivalent to the deterination of opt (Chu et al. 3 a, b). odified cost function (Vapni 98), J ep P ( ) J, (4) ln + ln( τ ) / P (3) is used to deterine the optial ode truncation opt. Here, τ is the probability of J J as increases. ep Usually, for a regional sea such as the Blac Sea, opt is 3 5 for the basin-scale (~3 ) variability and 5 for the esoscale (~ ) variability (Chu et al., 5a). For sparse and noisy data, it is difficult to get reliable and stable estiates of all the necessary spectral coefficients, but the first few spectral coefficients ( z, t ), ( z, t ),, ( z, t ) are reliable and stable. 4. Rotation atrix ethod for Regularization Deterination of the spectral coefficients is achieved by solving a set of linear algebraic equations of { ( z, t) } that are obtained fro the optiization procedure () and (4), â =QY. (5) where â is the estiated state vector (L-diensional) for the exact state vector a; is a P L coefficient atrix; Q is a P P square atrix (P > L); Y is a P- diensional observation vector, consisting of a signal Y and a noise Y, Y=Y+Y'. Due to the high level of noise contained in the observations, the set of algebraic equations is ill-posed and needs to be solved by a regularization ethod that requires: stability (robustness) even for data with high noise, and the ability to filter out errors with a-priori unnown statistics. The two nown atrices and Q are deterined by the physical process or field. Let be the Euclidean nor and QY ' L η =, η =, QY P ax(sigular values of ) η = 3 in(sigular values of ), (6) be the noise-to-signal ratio, diension ratio, and condition nuber of the atrix. For a particular syste, η is given. Usually, η and η 3 are large (called iperfect ), η, η >>, 3 which aes (5) difficult to solve. new rotation ethod for η < is developed to change (5) into a new syste with possibly iniu coefficient atrix and noise-to-signal ratio without a-priori nowledge of noise statistics. Nonsingular orthogonal transforation is conducted through ultiplication of (5) by a plane rotation atrix S fro the left, Sâ=SQY, (7) which changes the coefficient atrix and the source ter fro (, QY) to (S, SQY) and provides the opportunity to iniize the iperfection of the new syste (7), Where / η ( + η ) in, (8) 3 η SQY SQY, η SQY / a. (9) 3 iniization (8) leads to ( SQY * SQY ) J = - η + + η 3 SQY ax, () which is the procedure to obtain iniu values of η and η without a. Here, the sybol * indicates the 3 scalar product in the Euclidean space. The new transfored syste (7) can be solved by usual algebraic ethods such as the Gauss ethod. 5. rgo Data nalysis

3 Between Noveber 3 and January 5, over 56 float days (cuulative) of data were collected in the North tlantic ( o N-6 o N) in general at three paring depths:, 5 and. The floats paring at, depths shallower than, and unnown depths are excluded fro the analysis. Teperature at 95 and trajectories at and 5 are extracted fro all the existing rgo floats. The data fro and 5 were grouped together to represent the id-depth. The easureent cycle of an rgo profiling float includes four stages: ascending, surface drifting, diving and deep drifting. The rgo float can only get its position fixings while it ascends to the sea surface. The vector between two consecutive surface positions during the deep drifting divided by the tie interval is taen as the id-depth velocity vector. When the rgo float is diving, ascending and drifting below the sea surface, no data can be transitted to the ground stations in real tie. Velocity field after the first step analysis shows noisy circulation patterns (Figs. a, b) with large spatial gaps (fro 3 to 8 ). Uncertainty in the rgo float data causes errors in the velocity field. First, the data extracted fro the floats paring at two different levels: and 5 and grouped together to represent the id-depth ( ). This neglects the vertical shear. Second, the vertical shear causes increase or decrease of the distance between the points of ascending fro and diving to the paring depth. Third, the sequence of float trajectory segents only approxiates the real Lagrangian paths. Fourth, preliinary coputations (not included here to be published in a separate paper) show that high resolution eleents of circulation in the western North tlantic, such as the northern re-circulation gyre and the Deep Western Boundary Current (DWBC) are also revealed by the rgo floats. For exaple Figs. a, b clearly show the existence of DWBC. However, such a resolution is not available for the whole North tlantic. 6 N x 6 N 3 The high-energetic esoscale eddies and narrow boundary currents are classified as noise and reoved fro the analysis. Fifth, there are large spatial gaps in rgo float trajectories. The teperature observation has higher quality than the velocity observation since the forer has higher resolution and less spatial gaps. Fig. shows typical onthly coverage of observations. Interested readers can find the detailed discussion on the navigation errors and easureent errors caused by teperature sensors in Three quality control steps are perfored to identify and reove teperature profiles corrupted by large easureent errors. () Explicitly absurd profiles were reoved after a visual inspection. () portion of teperature profiles, which were outside of the prescribed accuracy of the cliatic data (Levitus et al.998), were also excluded fro the further analysis. (3) Teperature snapshots coputed by the Optial Spectral Decoposition (OSD) ethod (Chu et al. 3a, 4) should not show explicit outbreas in teperature structure. ny teperature profile contributing to such outbreas was a subject to reoving. paring depth for each rgo float was extracted fro a eta file as the variable PRKING_PRESSURE ( To control the paring depth the variable PRES (a pressure easured along the float trajectory) fro a file containing float trajectory data, was used. ost floats launched in the area of interest have easured this variable. 6 N 5 N 4 N 3 N N 6 N 5 N 4 N 39 N N N 5 N x 5 N N 4 N Γ 4 N Teperature ( C) 3 N N N 4 c/s Γ Γ 3 3 N N N 4 c/s Figure. Circulation velocities (tiny arrows) estiated fro the original RGO float tracs at for Dec 3 ar 4 and ug 4 Nov 4. The figure scale is given for tiny arrows. Red arrows are circulation velocities obtained by averaged over appropriate bins (fro Chu et al., 8). Figure. onthly teperature at 95 for Feb 4 and Dec 4. Circles are points where teperature was easured (fro Chu et al., 8). Identification of long Rossby wave of tropical North tlantic at id-depth (~ ) fro the rgo trac data is taen as an exaple to deonstrate the usefulness of the OSD ethod to reanalyze sparse and noisy ocean data (Chu et al., 7). rgo float data (subsurface tracs and teperature profiles collected fro arch, 4 through ay, 5) are used to detect signatures of long Rossby waves in velocity of the 3

4 currents at depth and teperature, between the ocean surface and 95, in the zonal band of 4 o N -4 o N in the Tropical North tlantic. Different types of long Rossby waves (with the characteristic scales between and 5 ) are identified in the western [west of the id-tlantic Ridge (R)] and eastern [east of the R] sub-basins. Current velocities coputed along the original (nonsoothed) rgo tracs in Noveber-Deceber, 4 are shown in Fig. 3 as red arrows. strong contribution fro intensive eddies, such as that shown in inset B, narrow jets and easureent errors is clearly identified here. Visually, the velocity pattern corresponding to the original data loos quite chaotic, and there are 5-6 spatial gaps in observation coverage. To understand the reconstruction sill for such data we applied three criteria: () the foral ean square error (the reconstruction error) coputed by the lainar enseble technique, () statistics of angle (α ) between the reconstructed and observed velocity at float locations, and (3) stability degree of the reconstructed snapshot on observation sapling. Rossby wave propagation is also identified fro reanalyzed teperature field at 55 depth (Fig. 4). 3 N N 3 N N N N c/s 5 c/s Figure 3. Sensitivity of the reconstructed circulation patterns to filtration of the original data: OSD is applied to the original data (Noveber-Deceber, 4); the original data (Noveber-Deceber, 4) filtered with a - onth window; (c) the original data (October-Deceber, 4) filtered with a 3-onth window; (d) the original data (October-Deceber, 4) filtered by 4 o 4 o bin averaging; Blue and red arrows correspond to the reconstructed circulation and original/filtered data. For α -histogras (inserts ) the x-axes is the angleα and the y-axes is the nuber of coparisons (%) (after Chu et al., 7). (d) 3 N c/s 5 c/s N N B N N N 3 N c/s 5 c/s 3 N N N N N N N 3 N c/s 5 c/s (c) Teperatute anoaly ( C) Figure 4. Spatio-teporal structure of sei-annual teperature anoaly at 55 depth: ay 5 th and June 9 th 4. Here, the gray arrow shows the direction of the positive teperature anoaly (after Chu et al. 7). 6. Global Surface Current Vector Data N N N Near real tie ocean surface current ( 3 depth) analyses real tie (OSCR) has been produced by NO on 5-day interval fro sea surface height (TOPEX/Poseidon), surface vector wind (SS/I) and sea surface teperature (VHRR in situ easureents) with a diagnostic odel using quasi-linear and steady physics and the absolute velocity using the ean dynaic height inferred fro the World Ocean tlas (WO). This satellite-derived global ocean surface 4

5 current vector data, representing cobined geostrophic, Ean, and Stoel shear dynaics, has been undergone thorough quality control procedures such as error analysis using drifter, ship, and oored observations. Detailed inforation can be found at the website: During constructing the OSCR data, the lateral boundary condition is never considered. This leads to the absence of ajor western boundary currents such as Gulf Strea, Kuroshio, Soali Current, California Current, Brazil Current, etc. (Fig. 5b). fter reanalyzing the OSCR data with the OSD ethod, the surface ocean current field is shown in Fig. 5a (June 5, 7). Coparison between Figs.5a and 3b shows the powerfulness of the OSD ethod. 5 saple the current velocity (speed and direction), teperature, and salinity every 5-in to -hour (ostly 3 in). The drifting buoys record the location at various ties with inhoogeneous area coverage. The data were interpolated into a unifor teporal grid with tie step of 3 hours. eteorological data are fro 7 buoys (represented by squares in Fig. 6) fro the National Data Buoy Center (C-N) in the LTEX- area. Wang et al. (998) have estiated the zero-crossing spatial scales of winds (~35 ) and other eteorological paraeters over the TLCS shelf after analyzing extensively the eteorological data collected during the LTEX- Progra observation period. Since the scale of the atospheric systes is larger than the scale of the continental shelf, the horizontal ean wind speed and direction fro 7 buoys are used for analysis. Figure 6. Geography and topography of Texas-Louisiana continental shelf, LTEX- current eter stations (represented by the sybol ), and eteorological buoys (represented by the sybol ) (fro Chu et al., 5a). Figure 5. Coparison of global surface ocean currents on June 5, 7 fro OSCR data with OSD and without OSD. 7. Lagrangian Drifter Data The OSD ethod was also to process the current velocity data collected fro the Texas-Louisiana Shelf Physical Oceanography Progra (LTEX-) and first segent of the Surface Current and Lagrangian Drift Progra (SCULP-) into teporally-spatially gridded data. With this data set, the TLCS current structure and reversal ay be identified. In conjunction with the local surface wind data, the wind effect on the TLCS synoptic current reversal ay also be identified. Data for the TLCS fall/winter current reconstruction were collected fro 3 near-surface current eter oorings during LTEX- at -5 depths (Cho et al. 998) fro pril 99 to Deceber 994 (Fig. 6) and fro drifting buoys deployed during SCULP- fro October 993 to July 994. The current eter oorings Trajectories of 77 SCULP-I drifting buoys show current reversals during January -7, 994 (Fig. 7). The OSD reconstructed horizontal velocity vector field shows dynaical characteristics of the synoptic current. The flow pattern on 3 Deceber 993 is characterized as a cyclonic gyre with a strong westward TLCS flow as the northern flan (Fig. 8a). This current copletely reverses to the east on 3 January 994 (total reversal, Fig. 8b), and the cyclone re-occurs (6 January 994) with the westward current at the north-central shelf and the eastward currents at the western shelf and shelf brea (partial reversal, Fig. 8c). The total reversal has a width up to - isobath (depth for shelf brea), and the partial reversal has a width saller than - isobath. During that period (3 Deceber 993 to 6 January 994), southwesterly winds prevail, and no evident offshore eddy appears at the shelf edge. Recurrence of the TLCS current reversals is estiated using the OSD reconstructed velocity fields (3 days of duration). period T (fro 5 to 4 days) is selected as the tie window. The nuber of total (or partial) reversals is counted in each T day window. Let (n, n, n, ) be the nubers of -current reversal, - current reversal, -current reversals, and be the all 5

6 realizations for a given T day window. The probabilities for -, -, and -current reversals are calculated by ( ) n P T =, ( ) n PT =, ( ) n P T =, () where P (T), P (T), and P (T) depend on the period T (Fig. 5). With T larger than days, the probability for zero current reversal is less than.. With T of 5 days, the probability for one reversal reaches.5. The probability P (T) is fitted to the Poisson distribution (Fig. 9) P( T) = ( μt) exp( μt), =,, ; ()! where μ =.8 day - is the ean rate of current reversal. 6 Figure 7. TLCS current reversals detected fro SCULP- buoy trajectories during January -7, 994. The blac dots show the starting positions of buoys (fro Chu et al., 5a). Figure 9. Epirically (dashed curve) and theoretically (Poisson distribution, solid curve) estiated recurrence probabilities: P (T), P (T), and (c) P (T) as functions of duration T (fro Chu et al., 5a). 8. Conclusions Figure 8. OSD reconstructed circulation: cyclonic gyre on Deceber 3, 993, total reversal on January 3, 994, and (c) partial reversal on January 6, 994 (fro Chu et al., 5a). Without nowing the bacground field and decorrelation scale, the OSD ethod can process sparse and noisy data. This ethod has three coponents: () deterination of the basis functions, () optial ode truncation, and (3) deterination of the Fourier coefficients. Deterination of basis functions is to solve the eigen-value proble. Chu et al. (3a, b) also developed a theory to obtain the basis functions with open boundaries. The basis functions are only dependent on the geoetry of the ocean basin, not dependent on the oceanic variables. This is to say, no atter which variable (teperature, salinity, or velocity) is concerned, the basis functions are the sae, and can be predeterined before the data analysis. For data without error, the ore the odes, the ore the accuracy of the processed field. For data with error, this rule of the thub is no longer true. Inclusion of high-order odes leads to increasing error. The Vapni variational principal is used to deterine the optial ode truncation. fter the ode truncation, optial field estiation is to solve a set of a linear algebraic equation of the Fourier coefficients. This algebraic equation is usually ill-posed. The rotation ethod is eployed to 6

7 change the atrix of the algebraic equation fro illposed to well-posed such that a realistic set of the Fourier coefficients are obtained. The OSD ethod is a power tool to process teperature, salinity, and velocity data fro Lagrangian and Eulerian, reotely sensed and in situ observed ocean data. 7 cnowledgent. This wor was sponsored by NO/NODC. References Chu, P.C., L.. Ivanov, T.. argolina, T.P. Korzhova, and O.V. elnicheno, nalysis of sparse and noisy ocean current data using flow decoposition. Part. Theory. Journal of tospheric and Oceanic Technology,, , 3a. Chu, P.C., L.. Ivanov, T.. argolina, T.P. Korzhova, and O.V. elnicheno, nalysis of sparse and noisy ocean current data using flow decoposition. Part : pplication to Eulerian and Lagrangian data. Journal of tospheric and Oceanic Technology,, 49-5, 3b. Chu, P.C., L.. Ivanov, and T.. argolina, Rotation ethod for reconstructing process and fields fro iperfect data. International Journal of Bifurcation and Chaos, 4 (8), , 4. Chu, P.C., L.. Ivanov, and O.. elnicheno, 5a: Fall-winter current reversals on the Texas-Louisiana continental shelf. Journal of Physical Oceanography, 35, 9-9. Chu, P.C., L.. Ivanov, and T.. argolina, 5b: Seasonal variability of the Blac Sea Chlorophyll-a concentration. Journal of arine Systes, 56, Chu, P.C., L.. Ivanov, O.V. elnicheno, and N.C. Wells, 7: On long baroclinic Rossby waves in the tropical North tlantic observed fro profiling floats. Journal of Geophysical Research,, C53, doi:.9/6jc3698. Chu, P. C., L.. Ivanov, O. V. elnicheno, and R.-F. Li, 8: rgo floats revealing biodality of large-scale id-depth circulation in the North tlantic. cta Oceanologica Sinica, 7 (), - Ivanov, L.., and P.C. Chu, 8: On stochastic stability of regional ocean odels to finite-aplitude perturbations of initial conditions. Dynaics of tosphere and Oceans, in press Vanpni, V.N., 98: Estiation of Dependencies Based on Epirical Data. Springer-Verlag, New Yor, 379 pp. 7