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Methods for the Reconstruction of Vertical Profiles from Surface Data: Multivariate Analyses, Residual GEM, and Variable Temporal Signals in the North Pacific Ocean

Methods for the Reconstruction of Vertical Profiles from Surface Data: Multivariate Analyses, Residual GEM, and Variable Temporal Signals in the North Pacific Ocean
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  Methods for the Reconstruction of Vertical Profiles from Surface Data: MultivariateAnalyses, Residual GEM, and Variable Temporal Signals in the North Pacific Ocean B RUNO  B UONGIORNO  N ARDELLI AND  R OSALIA  S ANTOLERI  Istituto di Scienze dell’Atmosfera e del Clima, CNR, Rome, Italy (Manuscript received 17 June 2004, in final form 10 February 2005)ABSTRACTDifferent methods for the extrapolation of vertical profiles from sea surface measurements have beentested on 14 yr of conductivity–temperature–depth (CTD) data collected within the Hawaii Ocean Time-series (HOT) program at A Long-Term Oligotrophic Habitat Assessment (ALOHA) station in the NorthPacific Ocean. A new technique, called multivariate EOF reconstruction (mEOF-R), has been proposed.The mEOF-R technique is similar to the previously developed coupled pattern reconstruction (CPR)technique and relies on the availability of surface measurements and historical profiles of salinity, tem-perature, and steric heights. The method is based on the multivariate EOF analysis of the vertical profilesof the three parameters and on the assumption that only a few modes are needed to explain most of thevariance/covariance of the fields. The performances of CPR, single EOF reconstruction (sEOF-R), andmEOF-R have been compared with the results of residual GEM techniques and with ad hoc climatologies,stressing the potential of each method in relation to the length of the time series used to train the modelsand to the accuracy expected from planned satellite missions for the measurement of surface salinity, sealevel, and temperature. The mEOF-R method generally produces the most reliable estimates (in the worstcases comparable to the climatologies) and seems to be slightly less susceptible to errors in the surface input.Multivariate EOF analysis of HOT data also gave by itself interesting results, being able to discriminate thethree major signals driving the temporal variability in the area. 1. Introduction In recent years, different studies have investigatedthe coupling between surface measurements, integralquantities [such as geopotential thickness, i.e., stericheight (SH), or vertical acoustic travel time (   )] and thevertical structure of the ocean. The objective of thesestudies is the reconstruction of vertical profiles of hy-drological parameters from surface and integrated mea-surements that can be obtained from spaceborne re-mote sensing instruments and/or from moored instru-mentation, such as echo inverted sounders (IESs). Thisextrapolated information can be used to investigate theinternal ocean dynamics on a global scale with a highertemporal coverage than that obtainable with traditionaltechniques and to better initialize numerical predictivemodels adopted for operational uses [for a wide reviewon data assimilation, see De Mey (1997)].Passive infrared/microwave sensors provide mea-surements of surface parameters [e.g., sea surface tem-perature (SST)], while active microwave sensors (moreexactly radar altimeters) give us a sort of integratedmeasure of the density along the water column. In fact,the sea surface height (SSH) variations are given by acombination of volume and total mass variations at agiven location. Similarly, IESs provide a measure of theround-trip acoustic travel time, which is clearly relatedto the distribution of density along the water column (   is tightly related to the surface steric height, being de-fined as twice the integral along the water column of the specific volume divided by the product of gravita-tional acceleration and sound speed). Consequently, astrong effort has been made in the last years to propa-gate on the vertical the information from altimetricSSH data, from combined SST and SSH measurements,and from IES data, often starting from an analysis of the correlation between in situ temperature and stericheight measurements. Corresponding author address:  Bruno Buongiorno Nardelli, Is-tituto di Scienze dell’Atmosfera e del Clima, Sezione di Roma,Area di ricerca di Tor Vergata, CNR, Via del fosso del Cavaliere100, 00133, Rome, Italy.E-mail: bruno@gos.ifa.rm.cnr.it 1762  JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY V OLUME  22© 2005 American Meteorological Society  Dynamical (e.g., Hurlburt 1984), variational (e.g.,Thacker and Long 1988), statistical (e.g., Hurlburt et al.1990; Carnes et al. 1990, 1994; Gavart and De Mey 1997; Pascual and Gomis 2003; Buongiorno Nardelliand Santoleri 2004), and empirical analyses (Meinenand Watts 2000; Watts et al. 2001; Mitchell et al. 2004) were at the base of the different methods.Most of the statistical approaches begin with thesingle (univariate) empirical orthogonal function(sEOF) decomposition of observed hydrological verti-cal profiles, and then search a correlation/regressionbetween the amplitudes of the sEOF modes and somesurface measurements (e.g., Carnes et al. 1990, 1994;Pascual and Gomis 2003).Carnes et al. (1990, 1994), in particular, developedseveral different models based on the computation of the sEOF of temperature, salinity and steric height pro-files and on the least squares regression of the ampli-tudes of the most significant modes to both linear andnonlinear functions of the surface steric height and tem-perature. Including possible dependencies from nonlin-ear terms as powers and cross terms of surface SH andSST considerably improved the results with respect tothe simpler linear correlation models (Carnes et al.1994), even if a much higher number of degrees of free-dom (DOF) are absorbed by each nonlinear regression(8 DOFs per mode; see section 3 for details). Thishigher number of DOFs required by the method canlead to not trivial differences, in terms of significancelevels, when estimating the model parameters from alimited number of data, especially since the observa-tions could be significantly autocorrelated.In recent years, an increasing number of investiga-tions have been conducted on an empirical techniquecombining historical hydrography with integral mea-surements from IES. This method, called the gravestempirical mode (GEM) technique, consists of the pro- jection on surface geopotential height space (or, better,an approximation of the geopotential height space,such as two-way acoustic travel time or steric heightspace) of hydrographic data (Meinen and Watts 2000).A two-dimensional empirical mode (or GEM) is builtby applying to the hydrographic profiles as a function of     (or, equivalently, of the surface SH) a cubic smooth-ing spline. The smoothing is performed at one pressurelevel at a time and basically leads to the construction of a lookup table based on the integral measurementsavailable.Some refinements of the standard GEM techniquehave also been proposed, such as the residual GEM(Mitchell et al. 2004). Residual profiles are obtained byremoving a suitable climatology from each historicalhydrographic profile. The variability associated withthese new profiles is clearly expected to be less than thevariability observed in the srcinal dataset. Accord-ingly, the error associated to a GEM field defined fromthese data should also be reduced. In practice, the re-sidual GEM fields are the correctors of a climatologicalfield taken as predictor. A further parameterization in-cludes SST measurements to define bins on which dif-ferent GEM fields can be estimated (Mitchell et al.2004).The number of DOFs absorbed by the GEM tech-niques is 1 (given by the tension applied to the curve)plus the number of nodes used in the smoothing. Thelatter is generally chosen to be the maximum between afraction of the total number of profiles in the  learning dataset and a minimum of 3. As a consequence, theDOFs absorbed by a basic GEM are generally higherthan 4. If different bins are defined according to anyadditional surface parameter, the number of degrees of freedom needs to be multiplied by the total number of GEM fields computed. This clearly means that this fur-ther parameterization can be applied only if the GEMstructures can be adequately represented in parameterspace.Recently, we presented a new extrapolation method,called coupled pattern reconstruction (CPR), that al-lows extrapolation of the vertical profiles of tempera-ture and steric heights from corresponding surface data(Buongiorno Nardelli and Santoleri 2004). Unlikemany of the previous techniques, this method is notbased on sEOF but on a statistical technique for theidentification of the coupled modes of variability,known in literature as coupled pattern analysis (CPA;see Bretherton et al. 1992). CPA consists of the singularvalue decomposition (SVD) of the cross-covariancematrix constructed from temperature ( T  ) anomaly andsteric height (SH) anomaly profiles. The SVD identifiespairs of orthogonal vertical patterns, each one explain-ing as much as possible of the mean-squared covarianceof the two datasets. The amplitudes of the  T   and SHcoupled modes can be linearly related one to the other(verifying the effective coupling of the modes), and if we limit the expansions to the first two modes, verticalprofiles can be estimated from  T   and SH surface values,solving a simple linear system. Six DOF are absorbedby the CPR, two for each regression between  T   and SHamplitudes and two for the constraints imposed by thelinear system.The CPR was tested on the hydrological data col-lected by the Dynamique des Flux de Mati è re en Medi-terran é e (DYFAMED) program (1994 – 2002) at a fixedlocation in the Ligurian Sea (northwestern Mediterra-nean Sea). The analyses seem to indicate that CPRgenerally improves climatological estimates in the up- N OVEMBER  2005 BUONGIORNO NARDELLI AND SANTOLERI  1763  per layers and is less susceptible to errors associatedwith steric height estimation with respect to equivalentsEOF reconstruction (sEOF-R) methods, even if nocomparison has been done with GEM (BuongiornoNardelli and Santoleri 2004). Moreover, no specificstudies on the sensitivity of the methods to the length of the  learning/training  dataset were performed, and theconclusions of that analysis clearly cannot be general-ized to any region or dataset. The capability of any of these methods in the reconstruction of the verticalstructure of the sea is in fact strongly dependent, on onehand, on the signals that characterize the area consid-ered and/or that can be resolved by available in situdata. On the other hand, it depends on the number of independent measurements needed to accurately trainthe model, with respect to the DOF absorbed by themodel itself.Consequently, the first objective of the present paperis the comparison of these techniques applied to a dif-ferent dataset. In particular, our work is focused on theanalysis and reconstruction of a long time series of CTD data collected in the framework of the HawaiiOcean Time-series (HOT) program in the North PacificOcean, using the in situ data as surface input, andevaluating the errors associated with the extrapolationwhen using training datasets of different lengths. Thesensitivity of the various techniques to the surface inputaccuracy has also been investigated. Data and valida-tion strategies are presented in section 2.The North Pacific area is characterized by three maincomponents modulating the temporal variability of thehydrological profiles: an intra-annual signal related tothe westward propagation of a baroclinic signal(Chiswell 1996); decadal/interannual oscillations linkedto the Pacific decadal oscillation (PDO), to the ElNi ñ o – Southern Oscillation (ENSO), and to the vari-ability in the associated winter rainfall (Lukas 2001);and a weaker component due to local seasonal forcings(Bingham and Lukas 1996). As we will see in section 3a,a strong salinity signal dominates the steric heightvariations at HOT, making it unfeasible to apply thestandard methods for the reconstruction of the profilesfrom surface temperature and sea level alone. We havetherefore hypothesized additional measurements of thesea surface salinity (SSS) to be available. From an op-erational point of view, these measures could comefrom drifting buoys, voluntary ships, and/or from theSoil Moisture and Ocean Salinity (SMOS) satellite mis-sions, scheduled for launch by the European SpaceAgency (ESA) in early 2006, and Aquarius/SAC-D,which has been recently approved for launch by theNational Aeronautics and Space Administration(NASA).Accordingly, a modified CPR (based on the CPA of salinity and steric height profiles), a modified sEOF-R(using SSS instead of SST to estimate the amplitudes),and proper residual GEM techniques have been ap-plied to the HOT time series in section 3b. A new tech-nique, called multivariate EOF reconstruction (mEOF-R), is described in section 4; mEOF modes for selected learning  periods are discussed in section 5; and a sum-mary of the results and conclusions are presented insection 6. 2. Data and validation methods The HOT program started in 1988 in the frameworkof the Joint Global Ocean Flux Study (JGOFS) andWorld Ocean Circulation Experiment (WOCE), aimedat the study of the interactions between physical,chemical, and biological processes that modulate theearth climate and can possibly modify the environment[for a complete description of the HOT program seeKarl and Lukas (1996) and all papers in a special issueof   Deep-Sea Res . (1996, vol. 43B, no. 2 – 3)]. The HOTexperiment is based on the regular sampling of varioushydrological and biochemical parameters at some sta-tions located near the center of the North Pacific sub-tropical gyre. In situ surveys have been conductednearly monthly since 1988.In this work we will concentrate on the CTD mea-surements collected at A Long-term OligotrophicHabitat Assessment (ALOHA) station, situated   100km north of Oahu, Hawaii (22.75 ° N, 158 ° W), between1988 and 2001 (Fig. 1). CTD data have been prelimi-narily processed directly by HOT personnel. They arequality controlled and binned at 2-db intervals. [Seealso Santiago-Mandujano et al. (2002) and the HOT F IG . 1. Location of the HOT CTD station ALOHA. 1764  JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY V OLUME  22  Data Reports, all available online at http://www.soest.hawaii.edu/HOT_WOCE/data_report.html. All mea-surements can be freely obtained at www.soest.hawai-i.edu/HOT_WOCE/ftp.html.] The CTD profiler used isan SBE911plus (Sea-Bird Electronics) that is calibratedregularly. We selected only the CTD deeper then 2000db and computed an average profile for each cruise inorder to filter out as much as possible of the higher-frequency variability (associated with internal wavesand baroclinic tides). In situ density was computed us-ing the standard equation recommended by the UnitedNations Educational, Scientific and Cultural Organiza-tion (UNESCO 1981). Steric heights were referencedto the 2000-db surface.CTD-derived temperature, salinity and geopotentialthickness data were first used both to train the variousmodels and to  “ simulate ”  the surface input used by themodels when testing their performances. However, asimple residual evaluation, that is, using the samedataset to train the model and to validate it ( hindcast validation ), does not give a reliable indication of howwell a model will do when it is asked to make predic-tions for data it has never seen. For this reason, wedecided to judge the methods ’  performance using animproved form of   holdout   cross validation, that is, asimple form of   k-fold  cross validation. The holdoutmethod requires that the srcinal dataset is separatedinto two sets, called the  learning / training  dataset (usedto train the model) and the  testing  dataset (used toindependently evaluate the model performance). Themethod adopted here consists of repeating the holdoutvalidation  k  times each time the learned dataset isshifted by a few months. The advantage of this methodis that it is more robust than both the residual and thesimple holdout methods, even if it introduces only fewcomplications from the computational point of view.Different training period lengths (2, 5, 8, and 11 yr)have been chosen out of the srcinal 14-yr series to testthe techniques. CTD profiles corresponding to thetraining periods were also used to compute ad hocmonthly climatologies, built up by simply averaging thecasts for each month. The climatological profiles wereused as reference for the evaluation of the performanceof the different techniques considered and for the esti-mation of residual GEM fields.All the methodologies presented aim at estimatingthe vertical structure of the sea directly from SST andeventually SSS data measured from space, or by drift-ers/buoys and/or ships of opportunity measurements,coupled with altimeter estimates of SSH or with     mea-sured by IES. As a consequence, it is important to em-phasize that a basic validation based on virtually  “ error-free ”  input data (surface and integrated parametersmeasured/computed directly from CTD data) seems in-adequate, as fully independent data could be contami-nated by errors of various kind. In particular, SST ac-curacy from space is of the order of 0.5 ° C (Kearns et al.2000), while satellite SSS measurements (still onlyplanned with SMOS and Aquarius/SAC-D) plan toreach an accuracy of 0.1 psu with a spatial resolution of 200 km every 10 days (Kerr et al. 2001), even if thereare still many uncertainties about the real capability of these sensors in achieving these requirements. On theother hand, the accuracy in estimating steric heightsfrom altimeter data is affected both by instrumentalerrors and by the method adopted to adjust altimeterSSH to a steric level (which is quite a complicated prob-lem, clearly beyond the objective of the present paper).Hence, we present here a simple analysis of the sensi-tivity of the methods to the surface SH, SST, and SSSerrors. This has been done analogously to our previouswork on CPR (Buongiorno Nardelli and Santoleri2004), that is, adding a random noise to the observedsurface values used as input for the vertical reconstruc-tion. The white noise errors were generated throughthe Interactive Data Language (IDL; version 5.3, Re-search Systems, Inc.) RANDOMN function, which usesthe Box-Muller method for generating normally distrib-uted (Gaussian) random numbers. Two different noiselevels have been added to the independent testingdatasets, one corresponding to the accuracy expectedfor future satellite data and a second one considering amore realistic estimation of present capabilities. Therms errors added to surface SH, SST, and SSS in thetwo cases are 2 cm (3 cm), 0.4 ° C (0.5 ° C), 0.1 psu (0.2psu). 3. CPR, sEOF-R, and residual GEM applied toALOHA data a. Reconstruction of vertical profiles knowing surface SH and SST  The first method that has been applied to ALOHAtime series is the CPR of temperature and stericheights. The CPA of   T   and SH profiles (limited to thefirst 500 db) showed that most of the covariability (onaverage more than 93%) between the two parameters isexplained by the first two modes, with the first modegenerally explaining more than 85% of the covariance,whatever the length of the learning dataset considered.These percentages clearly satisfy the first hypothesisrequired by the CPR (most of the covariability ex-plained by the first two modes). However, the CPR didnot perform quite acceptably in the reconstruction of   T  profiles, with rms errors often comparable to or above N OVEMBER  2005 BUONGIORNO NARDELLI AND SANTOLERI  1765  those associated with the climatology, always up tomore than 1 ° C around 80 db, and also worse (over1.2 ° C) in the layers below 300 db with 2 yr of training.Better results were found in the layer between 200 and300 db (rms   0.5  0.7 ° C) and in the first 30 – 40 db.Effectively, the method is forced to reproduce valuesimposed at the surface, so it is not surprising that theclimatological error was soon approached, adding anoise to the input surface SH and temperature values(Figs. 2a,d).The second method tested here is the sEOF-R of temperature profiles from SST and surface SH as de-scribed by Carnes et al. (1994), reported in their paperas model 3 (or simplified model 6). The EOF of   T  ( z ,  t  )identified three main modes, generally explaining  90% of the variance in any training period. The re- F IG . 2. Rms error of CPR temperature profiles extrapolated from SST and surface SH and associated with climatology, computedfrom (a) 2-, (b) 5-, (c) 8-, and (d) 11-yr learning datasets; rms of CPR salinity profiles extrapolated from SST and surface SH andassociated with climatology, computed from (e) 2-, (f) 5-, (g) 8-, and (h) 11-yr learning datasets. 1766  JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY V OLUME  22
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