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  • View in gallery

    AVHRR image (9 March 1985) of the western North Atlantic after spline interpolation and manual declouding. The color barshows the temperature in Celsius and the scales at the top and leftshow the longitude and latitude.

  • View in gallery

    Ensemble average sea surface temperature, Φ(x), derivedfrom the spline-interpolated ensemble.

  • View in gallery

    First four empirical eigenfunctions derived from the spline-interpolated ensemble. Projections onto these four eigenfunctions capture 99% of the variance on average.

  • View in gallery

    Eigenvalues derived from the spline-interpolated dataset.The 17th eigenvalue is indicated with an arrow.

  • View in gallery

    Modal coefficients, an(t), for the first four empirical eigenfunctions. The scale on the abscissa shows the day of the year 1985.The steady increase in a1 indicates the seasonal warming of the seasurface. The coefficients, a3 and a4, are a quarter period out of phase,which when combined with the spatial phase shift in eigenfunctionsψ3 and ψ4 produces a traveling wave.

  • View in gallery

    Temperature as a function of time averaged over 11.5 by11.5 km2 areas (5×5 pixels) centered on (34°N, 69°W) (solid);(37.5°N, 69°W) (dashed); (42°N, 69°W) (dotted); and (39°N, 74°W)(dash-dotted). Also indicated are the best fit, in a least square sense,straight lines. The letters next to the curves correspond to the pointslabeled in Fig. 4a.

  • View in gallery

    Coefficient of the first EOF after removing a linear trend(thin solid). Least square fit quadratic to the residual of the first EOF(dash–dot). Second EOF (dash). Least square fit quadratic to thesecond EOF (bold) and the negative of this fit (bold).

  • View in gallery

    Second EOF with regions of magnitude less that 0.025 masked (gray), 200-m isobath (white), continental outline (red), persistentfronts (crosshatched in black).

  • View in gallery

    (a) Time series of the first EOF (thin solid) and the sumof the first two EOFs (bold) at the entrance to the Chesapeake Bay(36.9°N, 75.8°W). (b) Time series of the first EOF (thin solid) andof the sum of the first two EOFs (bold) along the northern edge ofthe Gulf Stream (38.3°N, 70.6°W) and of the sum of the first twoEOFs (dash-dot) in the shelf/slope frontal band (39.3°N, 72.6°W).

  • View in gallery

    First four EOFs found in the regions associated with persistent fronts.

  • View in gallery

    Modal coefficients an(t) for EOFs in the regions associatedwith persistent fronts.

  • View in gallery

    The coefficient of the first empirical eigenfunction estimated from the clear regions of snapshots drawn from the years 1986–91. Though the data are irregularly spaced (and missing in the cloudyspring months) the annual cycle of seasonal warming is apparent.

  • View in gallery

    The coefficient of the first empirical eigenfunction estimated from weeklysnapshots in 1990.

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An Empirical Eigenfunction Analysis of Sea Surface Temperatures in the Western North Atlantic

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  • 1 Rockefeller University, New York, New York
  • | 2 Graduate School of Oceanography, University of Rhode Island, Narragansett, Rhode Island
  • | 3 Rockefeller University, New York, New York; Center for Fluid Mechanics, Brown University, Providence, Rhode Island
  • | 4 Center for Fluid Mechanics, Brown University, Providence, Rhode Island
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Abstract

The empirical orthogonal function decomposition is used to analyze time records of AVHRR sea surfacetemperature observations of the western North Atlantic from 32.9° to 43.6°N, 62.7° to 76.3°W. A manuallydeclouded dataset covering the spring of 1985 is analyzed. The majority (80%) of the variance about the meanis accounted for by an empirical eigenfunction, which is identified with seasonal warming. This eigenfunctionshows that the shelf water, excluding Georges Bank, warms the most rapidly; the surface water of the Gulf ofMaine warms a little less rapidly and the Gulf Stream and Sargasso Sea surface water warm the least rapidly.The SST of the Gulf Stream is also shown to behave more like that at 30°N than like Sargasso Sea waterimmediately to its south (∼35°N). The second EOF is found to be a small correction to the general warmingrate described by the first EOF. The third and fourth EOFs are determined primarily by meander propagation.Observations with partial cloud cover from the period 1985 to 1991 are also analyzed. Again, the dominanteffect is identified as seasonal warming.

Corresponding author address: Dr. Richard Everson, Department of Biomathematical Sciences, Mount Sinai Medical School, Box 1012, One Gustave L. Levy Place, New York, NY 10029-6574.

Email: rme@camelot.mssm.edu

Abstract

The empirical orthogonal function decomposition is used to analyze time records of AVHRR sea surfacetemperature observations of the western North Atlantic from 32.9° to 43.6°N, 62.7° to 76.3°W. A manuallydeclouded dataset covering the spring of 1985 is analyzed. The majority (80%) of the variance about the meanis accounted for by an empirical eigenfunction, which is identified with seasonal warming. This eigenfunctionshows that the shelf water, excluding Georges Bank, warms the most rapidly; the surface water of the Gulf ofMaine warms a little less rapidly and the Gulf Stream and Sargasso Sea surface water warm the least rapidly.The SST of the Gulf Stream is also shown to behave more like that at 30°N than like Sargasso Sea waterimmediately to its south (∼35°N). The second EOF is found to be a small correction to the general warmingrate described by the first EOF. The third and fourth EOFs are determined primarily by meander propagation.Observations with partial cloud cover from the period 1985 to 1991 are also analyzed. Again, the dominanteffect is identified as seasonal warming.

Corresponding author address: Dr. Richard Everson, Department of Biomathematical Sciences, Mount Sinai Medical School, Box 1012, One Gustave L. Levy Place, New York, NY 10029-6574.

Email: rme@camelot.mssm.edu

1. Introduction

This paper presents a study of Advanced Very HighResolution Radiometer (AVHRR) sea surface temperature observations in the western North Atlantic usingempirical orthogonal functions. Assimilation and interpretation of the wealth of satellite observations is a major task facing the scientist. The EOF expansion givesan optimal modal expansion of the data and permits theidentification of particular modes with relevant physicalprocesses.

This study examines a 1200×1200 km2 region of thewestern North Atlantic from the eastern seaboard ofNorth America to the Grand Banks and from the SargassoSea to Nova Scotia. The predominant contribution to seasurface temperature variability is found to be the overallseasonal warming and cooling of the ocean. A modedescribing this effect is found from both a manually declouded dataset covering the spring of 1985 and also fromdata covering 1986–1991. The spatial form of the modeshows distinct warming rates in the Sargasso Sea, overthe Grand Banks and in the shelf and slope waters. Twomodes describing the meandering of the Gulf Stream areused to obtain the average phase velocity and wavelengthof meanders during spring 1985.

The EOF procedure is closely related to factor analysis (Harman 1960), principal components analysis (Hotelling 1933), and singular value decomposition (Goluband Loan 1983; Moler and Morrison 1983). The technique, which has a long history (Sirovich and Everson1992; Stewart 1993) in the guise of the Karhunen–Loèveexpansion (Karhunen 1947; Loève 1963), was introduced to geophysical studies by Lorenz (1956) and inrecent times has found wide application in turbulence(Sirovich 1991; Berkooz et al. 1993). The snapshotmethod (Sirovich and Everson 1992; Sirovich 1987a–c) enables the application of the technique to datasetswith a great many pixels. The technique has been applied to the analysis of sea surface temperature (SST)fields obtained from both in situ observations (primarilyexpendable bathythermographs) and satellite-derivedfields. The source of the data has in turn determined thescales, both temporal and spatial, of the analysis. Giventhe sparse coverage of in situ observations, EOF analyses of SST fields derived from these data have focusedon large spatial and long temporal scales. Tanimoto etal. (1993) analyzed 37 years of SST anomalies in theNorth Pacific; Naiyou et al. (1992) considered 35 yearsof data for the western North Pacific; Panitz and Speth(1986) performed an EOF analysis on data for 1979 inthe equatorial Atlantic; and Thompson et al. (1988) examined 30 years of engine intake temperatures in theshelf–slope region of the northwest Atlantic. The primary focus of such studies has been large-scale atmospheric forcing and the response of the SST field as ameasure of the response of the upper ocean in general.

EOF analyses of satellite-derived SST fields havetended to focus on much smaller spatial scales (∼100km) and temporal scales (∼100 days): Kelly investigated the relationship between sea surface temperatures,winds, and topography over the north California slopewith EOFs (Kelly 1985); Lagerloef and Bernstein haveused EOFs to investigate currents in the coastal watersoff California (Lagerloef 1992; Lagerloef and Bernstein1988); Paden et al. have used the technique to investigate tidal and atmospheric forcing in the Gulf of California (Paden et al. 1991); Gallaudet and Simpson examined large-scale and mesoscale processes in the openocean on the opposite side of the Baja Peninsula fromthe region considered by Paden et al. (Gallaudet andSimpson 1994); Fang and Hsieh analyzed the variabilityof the summer SST field off Vancouver Island with 8years of AVHRR data (Fang and Hsieh 1993); Garzoliet al. compared EOFs derived from satellite SST fieldswith those derived from numerical model runs for boththe Brazil/Malvinas region and the Kuroshio/Oyashioregion (Garzoli et al. 1992); and Chiswell investigatedthe waters off New Zealand (Chiswell 1994). It is interesting to note that except for the work of Garzoli etal. (1992) and Chiswell (1994), EOF analyses ofAVHRR data have been for the eastern North Pacific.To date there have been no published EOF analyses ofAVHRR-derived fields for the North Atlantic despiteLagerloef and Bernstein (1988) suggestion that “. . .EOF analyses of remote sensing data could be directedto such strong frontal features as the Gulf Streamfront. . . .” The work presented herein attempts to redress this omission.

In the next section we describe the data used in thisanalysis and preliminary processing of these data. Thisis followed by a brief outline of the EOF procedure. A120-day spline interpolated set of SST fields obtainedfrom the AVHRR sensor is then analyzed. Data spanningseveral years, but with partial cloud cover, are examinedin the final section of the paper.

2. Data preparation

a. Preprocessing

The measurements that we analyze consist of AVHRRimages of the North Atlantic collected by the NOAApolar orbiting satellites. These images stretch from theeastern seaboard of North America to the Grand Banks(62.7° to 76.3°W) and from the Sargasso Sea in the southto Nova Scotia in the north (32.9° to 43.6°N). This region covers the development of the Gulf Stream and itsmeandering into the mid-Atlantic. Each image comprises 512 × 512 pixels on a rectangular grid with apixel spacing of approximately 2.3 km. The sea surfacetemperature is derived from multichannel infrared measurements. See Cornillon et al. (1987a) for details ofthe processing.

Cloud cover and missing data are major hindrancesto the analysis of sea surface temperatures. Data maybe missing from a particular image because the area ofinterest does not lie completely in the swath of the observing satellite. For NOAA polar orbiting satellites,this occurs approximately 12% of the time for the areaof interest. Clouds are a more serious problem. A roughestimate of the cloud cover over the study area for themonths March–June 1985 suggests that on average thesurface is obscured 28% of the time in late spring andearly summer. Taken together, these two effects resultin missing data values for approximately 35% of eachsatellite pass. There are two ways in which these datamay be analyzed with the EOF procedure: 1) the datagaps may be filled by interpolation or 2) the proceduremay be modified to deal with gappy data. The first approach is undertaken here using a declouded and spline-interpolated dataset. This dataset was obtained by firstmanually flagging all pixels in each image thought tobe contaminated by cloud cover. The remaining “clear”pixels define an irregularly spaced time series at eachpixel location in the study area. For each of these locations (approximately 170 000 excluding land), a continuous function of sea surface temperature in time wasestimated by performing a least squares approximationto the temperature time series using cubic splines withfixed knots (de Boor and Rice 1968). A total of 17 cubicsegments were used to fit the data at each location andthe knots connecting these segments were fixed at thesame times for all locations. A new set of images wasgenerated by sampling each of the splines every 12hours and then combining all interpolated values corresponding to the same times. Finally, the images weremedian filtered with a 7 × 7 pixel box. This last steptended to replace extraneous sea surface temperaturevalues resulting from splines fit made with insufficientdata. The dataset covers the spring of 1985 from 1March to 30 June and consists of 244 images; a typicalimage is shown in Fig. 1. The cold continental shelfand Gulf of Maine waters show up as shades of blue inthe upper half of the image. The Gulf Stream, extendingfrom the lower left-hand corner of the image across itsmiddle, is defined by the warmest (reddest) water, witha very sharp sea surface temperature gradient on itsnorthern edge. To the south the slightly cooler (reddishyellow) water defines the Sargasso Sea. Between theGulf Stream and the shelf water is the slope water(greenish yellow in the figure). The warmer approximately circular features in the shelf water are warm corerings. Two cold core rings are evident to the south ofthe stream. One, immediately to the east of the streamin the western part of the image, is defined by the semicircular band of warm water drawn off of the streamby the ring. The other is south of the stream on theeastern edge of the image and shows up as a relativelycooler near-circular region surrounded by a slightlywarmer band.

b. Empirical orthogonal function analysis

We denote by ϕ(x, t) the fluctuation, at location xand time t, of the sea surface temperature about theensemble mean temperature, Φ(x) = 〈Φ(x, t)〉. The EOFprocedure expresses the dataset of images {Φ(x, tn)} asa modal expansion:
i1520-0485-27-3-468-e1
The modal coefficients, an(t), and the empirical orthogonal functions, ψn(x), are determined by demanding that,for any N, the approximation error averaged over theentire ensemble 〈∫dx|ϕ(x, t) − ΣNn=1 an(t)ψn(x)|2〉 is aminimum. The solution of this minimization problemleads to an eigenfunction equation for the EOFs (Sirovich 1987a–c; Sirovich and Everson 1992).

The empirical orthogonal functions that minimize (1)form the best basis in which to represent the ensemble.The basis is intrinsic to the particular dataset becauseit is wholly determined by the dataset itself, hence thedesignation empirical.

The modal coefficients, too, are orthogonal:
i1520-0485-27-3-468-e2
The eigenvalue, λn, thus measures the mean squaredprojection of the data onto the nth empirical eigenfunction. The empirical eigenfunctions are ordered indescending order of the corresponding eigenvalues. Bychoosing to represent the dataset in terms of the first Nempirical eigenfunctions one therefore captures, on average, a fraction ΣNn=1λn;shΣn=1 λn of the variance or energy, the maximum that can be captured using any Nbasis functions.

3. Results: Spline interpolated data

We have computed the empirical eigenfunctions forthe 244 manually declouded and spline interpolated seasurface temperature images. The spectrum of eigenvalues is shown in Fig. 2. The mean temperature distribution, Φ(x), about which the eigenfunctions are fluctuations, is shown in Fig. 3, and the first four eigenfunctions are depicted in Fig. 4.

It is clear from the spectrum of eigenvalues that essentially all the energy in the ensemble is captured byonly 17 empirical eigenfunctions. This degree of compression is an artifact however, a result of the low-passfiltering and smoothing implicit in the spline interpolation. In fact, prior to median filtering, there are only17 degrees of freedom (corresponding to the 17 splinefunctions used in the interpolation) in the spline interpolated image set, and one would expect to find only17 nonzero eigenvalues. The median filtering of thespline-interpolated images distributes some of the energy from the first 17 eigenfunctions to the remainingones. We shall concentrate on the information providedin the first four eigenfunctions, noting that over 99% ofthe energy is captured by these modes.

In Table 1, the characteristics of the primary EOFsobtained in this study are compared with those of theAVHRR studies cited in the introduction. Also includedin this table are the number of input images, the areacovered by each image as well as the number of pixels,the period covered, and the general geographic locationof the study. It is clear from the table that the fractionof variance explained by the first eigenfunction variessignificantly from study to study, with the value (80%)obtained in this study near the middle of the range.Except for the Gallaudet and Simpson (1994) results,the fraction of the variance explained by the first EOFtends to decrease with the increase in the number oftemporal degrees of freedom as one would expect. Alsoas one might expect, the fraction of variance explainedby the first EOF is much higher for semienclosedregions: the Gulf of California (Paden et al. 1991) andthe Santa Barbara Channel (Lagerloef and Bernstein1988) as compared with open ocean areas. Our resultsare consistent with these trends.

a. Empirical eigenfunction 1

We now examine each of the first four eigenfunctionsin detail. As indicated in Table 1, on average 80% ofthe energy in the fluctuations is accounted for by thefirst eigenfunction. As can be seen in Fig. 4, ψ1(x) > 0everywhere, so the eigenfunction therefore represents awarming or cooling of the entire region. The modalcoefficients an(t) are plotted in Fig. 5. The coefficienta1 increases almost linearly with time, indicating a general warming with the advancing season (recall that thetime series of images covers the period from Marchthrough June), and we deduce that this eigenfunctionrepresents the seasonal warming of the sea surface. Thefirst EOF in all of the EOF analyses performed to dateon temporally demeaned AVHRR SST fields have beeneverywhere positive and have been associated with seasonal warming/cooling.

Temperature time series (Fig. 6, obtained from the244 manually declouded and spline-interpolated images) for the Sargasso Sea (the location marked with ana in Fig. 4a), for the Gulf Stream (point b), for thecoastal water off New Jersey (point c), and for the Gulfof Maine (point d) all show a steady increase in temperature with season. The uppermost curve in this figurecorresponds to the Gulf Stream (point b in Fig. 4a) andthe least squares fit straight line to this curve has thesmallest slope; that is, the increase in temperature forthe Gulf Stream was smaller than that for the other threeregions considered. This is consistent with the first eigenfunction (Fig. 4a) in which the Gulf Stream has thesmallest values. This is not surprising given the sourceof Gulf Stream water. In particular, Gulf Stream surfacewater leaving the continental shelf at approximately35°N comes from nearly equal parts of water flowingthrough the Florida Straits (the Florida Current) andwater flowing north of the Indies (the Antilles Current),both joining the western boundary current south of 30°N(Cornillon 1992). Between 30° and 35°N the westernboundary current entrains less than 1% of its mass fromthe more seasonally variable water shoreward of thecurrent (Stommel 1966). This, coupled with the shorttransit time for the surface water to move from 30° to35°N, means that the surface water in the stream downstream of the point of separation (35°N) warms at a ratecloser to that at 30°N than at 35°N. This is consistentwith what we observe and with climatological data. Atlases of surface water properties obtained from hydrographic and from ship surface data (Weare 1977; Böhnecke 1991) show the increase between March and Junein SST in the Sargasso Sea immediately to the south ofthe Gulf Stream (∼35°N between 65° and 70°W) to beapproximately 1°C greater than the increase in the samelongitudinal range at 30°N. Our data (Fig. 6) show thewater immediately to the south of the stream (point ain Fig. 4a) to warm by approximately 1°C more thanwater in the Gulf Stream (point b in Fig. 4a) over thesame time interval although our data suggest warmingof approximately 2°C less in both regions during 1985compared with the climatological values.

The steepest curve in Fig. 6 corresponds to a point(point c in Fig. 4a) on the continental shelf east of theentrance to the Delaware Bay. The temperature time series at this location shows rapid warming. This is becausea strong seasonal thermocline forms in the shelf waterconcentrating the heating to a relatively shallow surfacelayer when compared with the Sargasso Sea (Cornillonet al. 1987b). [Rapid warming of shelf waters comparedto adjacent deep water has been observed by others (Kelly1985; Lagerloef 1992; Gallaudet and Simpson 1994;Fang and Hsieh 1993) (although in the second mode); (Paden et al. 1991).] The rate of warming observedhere, approximately 15°C over the 4-month period, issimilar to the climatological warming of 15°C observedby Bunker (1976) for 1 April through 30 June. From Fig.4a it is clear that this is true for all shallow water regionsexcept for Georges Bank, point a in Fig. 4a. The difference in the water on Georges Bank results from the strongtidal mixing, which does not allow the development ofa strong seasonal thermocline such as seen at other shallow water locations.

Figure 4a also shows that the shelf warms more rapidly on average than the slope water, although we didnot attempt to quantify the change in the latter becauseof the significantly greater spatial variability evident inthe first EOF for this water mass when compared withthe four water masses represented by points a to d. Thelarge spatial variability results from the presence ofwarm core rings, which “drag” warm water off the GulfStream and cold water from the shelf as they propagateto the west (Morgan and Bishop 1977; Bisagni 1983;Garfield and Evans 1987). The more rapid warming ofthe shelf water in comparison with the slope water isconsistent with observations of Bunker (1976) and ofZheng et al. (1984). As indicated above, Bunker observed warming of 15°C on the shelf, while Zheng etal. observed warming of 11°C in the slope water for thesame monthly interval.

The coldest curve in Fig. 6 corresponds to point d ofFig. 4a located in the Gulf of Maine off Cape Cod. Thewarming of 8°C observed here for 1 April through 30June 1985 is identical to that observed by Bisagni andSano (1993) from AVHRR-derived SST values for thesame region but in 1987.

b. Empirical eigenfunction 2

The second eigenfunction, the most subtle of those contributing significantly, is a correction to the first eigenfunction; that is, this eigenfunction provides for a slightchange in the warming trend defined by the first eigenfunction. To show this, a linear trend was removed fromthe first modal coefficient (of the original decomposition)and a quadratic was fit to the residual signal. The residualsignal (thin solid curve) and the quadratic fit (dash-dottedcurve) are shown in Fig. 7 along with the second modalcoefficient (dashed curve) and a quadratic fit to it (thicksolid curve of negative curvature). Also shown in thisfigure is the quadratic fit to the second modal coefficientmultiplied by −1 (thick solid curve with positive curvature). The quadratic fits to the detrended signals are almostidentical in magnitude and phase, differing only in sign;hence our conclusion that the second eigenfunction is acorrection to the first eigenfunction.

This correction appears to result from several differentprocesses. These are identified by region, with the help ofFig. 8, which shows only those areas of the second eigenfunction for which its absolute value exceeds 0.025,that is, only those areas contributing significantly to thecorrection. Values from −0.025 to 0.025 are colored grayin the figure. Superimposed on the second eigenfunctionin this figure are the continental outline in red, the 200-misobath in white, and, crosshatched in black, regions ofpersistently high SST gradient. The persistent gradientregions were obtained by applying a Sobel gradient operator (Duda and Hart 1973) to each of the images in thetime series, averaging the resulting gradient images andthen marking locations in the figure where this averageexceeded a given threshold.

The different areas contributing significantly to thecorrection term are 1) a narrow band (yellow) huggingthe coast that extends south from the mouth of the Delaware Bay (∼39°N) to Cape Hatteras (∼35°N), 2) amoderately large region (yellow) seaward of the southern edge of the Gulf Stream off Cape Hatteras, 3) a band(yellow) that follows the northern edge of the GulfStream, 4) a band (blue/green) that follows the 200-misobath from approximately the entrance of ChesapeakeBay (∼37°N) to the middle of Georges Bank (∼67°W)and along most of the extent of this isobath in the Gulfof Maine, and 5) alternating bands (yellow and blue/green/violet) between regions 3 and 4. The processesinvolved in the correction differ from one area to another. The correction due to region 1 clearly results fromcoastal runoff from the Delaware and the ChesapeakeBays (Ketchum and Keen 1955; Wright and Parker1976). The region becomes wider to the south both because the Chesapeake adds to the runoff and becausethe plume is mixed to greater distances off shore as itmoves to the south. The modification here results inmore rapid warming early in the period. This occursbecause the shallow, fresher water in the Delaware Bayand the Chesapeake Bay is warmed more rapidly thanthe shelf water. Fig. 9a shows the contribution to thetemperature at (36.9°N, 75.8°W), the mouth of the Chesapeake Bay, as a function of time for the first EOF (thin)and for the sum of the first two EOFs (bold).

Regions 2 and 5 are related to rapid changes in SSTcaused by warm water being advected off the GulfStream by a cold core ring for region 2 and by twowarm core rings for region 5. Early in the time seriesthe cold core ring–stream interaction is clearly visible(not shown here) as a thin band of Gulf Stream waterbeing drawn off of the stream as the ring begins tointeract with the it. Toward the middle of the intervalthe ring–stream interaction is much more vigorous andlarge quantities of water are drawn off of the streaminto the adjacent Sargasso Sea. The interaction betweenthe stream and the warm core rings occurs as large eastward propagating meanders touch the rings again drawing off warm Gulf Stream water.

The remaining regions, 3 and 4, are, we believe, associated with vertical motion or mixing in the water column, which tends to modify the warming rate when compared to regions with relatively little vertical motion. Analternative hypothesis is that regions 3 and 4 in this EOFresult from the simple horizontal displacement of strongfronts during the course of the study. We discount thishypothesis, however, because the bands in regions 3 and4 are offset relative to the bands containing the persistentfronts (Fig. 8) and because motion of the fronts wouldrequire another EOF in quadrature with this one and noneis apparent. Large vertical motion, on the other hand, canresult in a temporal change in the SST that may be explained by one EOF. Such motion is often related tostrong fronts associated with steep gradients in bathymetry (the shelf break near the 200-m isobath) (Voorhis etal. 1976; Houghton and Marra 1983; Houghton et al.1988) and steep gradients in dynamic topography (thenorthern edge of the Gulf Stream) (Bower et al. 1985;Bower and Rossby 1989; Meyers 1994). The relationshipbetween the second EOF and persistent fronts associatedwith bathymetry (region 3) and persistent fronts associated with the northern edge of the Gulf Stream (region4) is presented in Fig. 8.

Figure 9b shows the time series of the first EOF (thin)and the sum of the first two EOFs (dash-dot) for a location in the shelf/slope front band (39.3°N, 72.6°W)and the sum of the first two EOFs (bold) for a locationin the Gulf Stream frontal band (38.3°N, 70.6°W). Thefirst EOF at the Gulf Stream location is almost identicalto that at the shelf/slope front hence is not shown. Thecorrection of the second EOF provides for more rapidwarming in the shelf/slope frontal band early in the timeseries and less rapid warming later in the time series.The correction for the band along the northern edge ofthe Gulf Stream is of opposite sign but approximatelyof the same magnitude.

Given that several different processes contribute to thesecond EOF, the question naturally arises as to whetheror not some dominate. In particular, are the features associated with the possibly more interesting processessuch as vertical motion in the vicinity of strong frontsrobust? In order to demonstrate that the two bands in thesecond eigenfunction associated with persistent fronts(regions 3 and 4) constitute a significant contribution tothis mode, the Sargasso Sea, the slope water region, thecontinental shelf, and the Gulf of Maine were coveredwith a mask and the EOF procedure was applied to theresulting time series of images. The first four eigenfunctions obtained are shown in Fig. 10 and the modal coefficients in Fig. 11. The first eigenfunction again corresponds to a general seasonal warming, the second andthird to propagating meanders (discussed below), and thefourth to the previous second mode. Aside from a reordering of the eigenfunctions and a slight change in theshape of the eigenfunctions and their modal coefficients,the information content is largely the same as obtainedfrom the unmasked time series; the signal associated withthe two isolated bands is robust.

c. Empirical eigenfunctions 3 and 4

The third and fourth eigenfunctions (Figs. 4c and 4d)are large only in the Gulf Stream and slope waterregions. In the Gulf Stream region they have the approximate form of a pair of modes that are π/2 out of(spatial) phase with each other. This is particularly evident in Fig. 10 (modes 2 and 3 in this case). The corresponding modal coefficients (Fig. 5) are roughly sinusoidal oscillations one-quarter period out of phasewith each other. It can be seen that the period is approximately 40 days. The superposition
a3tψ3xa4tψ4x
is thus approximately a traveling wave, representing themeandering of the Gulf Stream. The 40-day periodagrees well with that of the most energetic meanders(46 days) obtained by Lee and Cornillon (1996) froman analysis of eight years of satellite-derived GulfStream paths. The wavelength of these meanders, measured from either the third or fourth eigenfunction, 375km, is comparable to that obtained by Lee and Cornillon,427 km, and can be regarded as the representative scaleof a traveling wave. From the period and wavelengthwe estimate the phase velocity as 9.4 cm s−1, a littlesmaller than the mean phase speed of ∼12 cm s−1 observed by Lee and Cornillon (1996).

4. Results: Partial data

We have also examined an ensemble of 73 fairlycloudless and complete images. This ensemble coversapproximately the same region as the previous ensemblebut was obtained by searching several thousand imagesfrom 1986 to 1991 to find those with putatively lessthan 10% cloud cover or missing regions. Nonetheless,these images are flawed by missing pixels and cloudcover. The EOF procedure was therefore modified tocope with these flaws (Everson and Sirovich 1995).

The spectrum of eigenvalues for the relatively cloud-free ensemble is less compact than for the declouded,smoothed and interpolated ensemble. That a greaternumber of eigenfunctions contribute significantly maybe attributed to two facts: 1) the data now record seasurface temperatures in four seasons over a number ofyears, rather than a single season and 2) as noted previously, the process of fitting a spline to the decloudeddataset constrains the number of degrees of freedom.Even so, for the multiyear dataset, only 32 modes arerequired to capture 90% of the energy and we estimatethat modes with indices greater than about 20 are dominated by noise. The first four EOFs account for approximately 70% of the variance (Table 1).

The first empirical eigenfunction, which has approximately the same form as Fig. 4a, carries the bulk ofthe variance (62%). The large change in variance between eigenvalues 1 and 2 is a feature in common withother studies (see Table 1) and indicates that the seasurface temperature is dominated by seasonal warmingand cooling. The fact that the form of this eigenfunctionis quite similar to that of the first eigenfunction derivedfrom the spline-interpolated data supports the idea thatthis EOF represents the gross seasonal warming andcooling of the ocean.

The modal coefficient corresponding to this EOF wasestimated by a least squares procedure (Kelly 1985; Everson and Sirovich 1995) from the unflawed observations [i.e., excluding missing pixels, those colder thanthe mean Φ(x) by 5°C, and those colder than 0.5°C].Figure 12 shows the coefficient as estimated from theensemble of 73 snapshots spanning 1986–1991 and usedto find the eigenfunctions. Choosing relatively unobscured snapshots means that a large proportion of datacomes from the months of July and August, when theskies are clear; this clustering of the data is obvious inthe figure. Although the data are irregularly spaced, theinterpretation of the first eigenfunction as a mode describing the seasonal warming is compelling.

In Fig. 13 the coefficient of seasonal warming for asingle year (54 snapshots at roughly weekly intervalsthroughout 1990) is shown. Each data point is derivedfrom a single snapshot and the eigenfunctions from therelatively cloud-free ensemble. The annual warming andcooling cycle, peaking in August, is apparent.

Empirical eigenfunctions corresponding to the meandering of the Gulf Stream are now not evident. Thisis not surprising both because the average separation intime between images is approximately 30 days, closeto the period of the dominant meanders, and becausethe stream moves laterally from year to year (Lee andCornillon 1995).

5. Summary

In the preceding we analyzed, using the EOF procedure, a series of sea surface temperature images covering the Gulf Stream and surrounding ocean from thepoint of separation (off Cape Hatteras), ∼75°W, to theNew England Seamounts, ∼65°W. These images werespatially complete, clouds having been replaced by aninterpolated value obtained from cubic splines fit at eachpixel location. The first mode, explaining 80% of thevariance, was shown to correspond to seasonal heating/cooling. This is typical of the results of other EOF analyses as is the fraction of the variance explained by thismode. Also consistent with previous analyses, waterson the continental shelf were shown to heat up morequickly than those in the open ocean as spring progresses into summer. Less typical are the different ratesof warming seen in the open ocean. Unlike other resultsobtained from EOF analyses, the next three higher orderEOFs were shown to correspond to a subtle correctionto the first mode and to propagating meanders. The average phase velocity and wavelength of the meanderswas obtained from the EOFs.

The technique was also applied to a number of imagescovering several years. These images were not spatiallycomplete, so the EOF procedure had to be modified tohandle gappy data. In this case the first mode explainedonly 60% of the variance and the higher order modescould not be associated with propagating meanders.These results are not surprising, since in the longer timeseries the images used were too far apart in time for agiven meander to appear in more than one or two, andthe irregular temporal spacing of the images did notallow for propagating meanders to be followed.

Acknowledgments

This paper was prepared with support from the Office of Naval Research, the Naval Research Laboratory, Stennis Space Center (N00014-93-1-G901); from the National Aeronautics and Space Administration (NAGW 3009); and from the State ofRhode Island and Providence Plantations through salarysupport to P. Cornillon. The satellite data processingsoftware used to process the imagery was developed byR. Evans, O. Brown, J. Brown, and A. Li of the University of Miami. Their continued support is greatlyappreciated. The comments of two anonymous reviewers as well as those of Gary Lagerloef have also helpedto improve this manuscript substantially. Their effort inthis regard is appreciated.

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Fig. 1.
Fig. 1.

AVHRR image (9 March 1985) of the western North Atlantic after spline interpolation and manual declouding. The color barshows the temperature in Celsius and the scales at the top and leftshow the longitude and latitude.

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 3.
Fig. 3.

Ensemble average sea surface temperature, Φ(x), derivedfrom the spline-interpolated ensemble.

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 4.
Fig. 4.

First four empirical eigenfunctions derived from the spline-interpolated ensemble. Projections onto these four eigenfunctions capture 99% of the variance on average.

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 2.
Fig. 2.

Eigenvalues derived from the spline-interpolated dataset.The 17th eigenvalue is indicated with an arrow.

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 5.
Fig. 5.

Modal coefficients, an(t), for the first four empirical eigenfunctions. The scale on the abscissa shows the day of the year 1985.The steady increase in a1 indicates the seasonal warming of the seasurface. The coefficients, a3 and a4, are a quarter period out of phase,which when combined with the spatial phase shift in eigenfunctionsψ3 and ψ4 produces a traveling wave.

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 6.
Fig. 6.

Temperature as a function of time averaged over 11.5 by11.5 km2 areas (5×5 pixels) centered on (34°N, 69°W) (solid);(37.5°N, 69°W) (dashed); (42°N, 69°W) (dotted); and (39°N, 74°W)(dash-dotted). Also indicated are the best fit, in a least square sense,straight lines. The letters next to the curves correspond to the pointslabeled in Fig. 4a.

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 7.
Fig. 7.

Coefficient of the first EOF after removing a linear trend(thin solid). Least square fit quadratic to the residual of the first EOF(dash–dot). Second EOF (dash). Least square fit quadratic to thesecond EOF (bold) and the negative of this fit (bold).

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 8.
Fig. 8.

Second EOF with regions of magnitude less that 0.025 masked (gray), 200-m isobath (white), continental outline (red), persistentfronts (crosshatched in black).

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 9.
Fig. 9.

(a) Time series of the first EOF (thin solid) and the sumof the first two EOFs (bold) at the entrance to the Chesapeake Bay(36.9°N, 75.8°W). (b) Time series of the first EOF (thin solid) andof the sum of the first two EOFs (bold) along the northern edge ofthe Gulf Stream (38.3°N, 70.6°W) and of the sum of the first twoEOFs (dash-dot) in the shelf/slope frontal band (39.3°N, 72.6°W).

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 10.
Fig. 10.

First four EOFs found in the regions associated with persistent fronts.

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 11.
Fig. 11.

Modal coefficients an(t) for EOFs in the regions associatedwith persistent fronts.

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 12.
Fig. 12.

The coefficient of the first empirical eigenfunction estimated from the clear regions of snapshots drawn from the years 1986–91. Though the data are irregularly spaced (and missing in the cloudyspring months) the annual cycle of seasonal warming is apparent.

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Fig. 13.
Fig. 13.

The coefficient of the first empirical eigenfunction estimated from weeklysnapshots in 1990.

Citation: Journal of Physical Oceanography 27, 3; 10.1175/1520-0485(1997)027<0468:AEEAOS>2.0.CO;2

Table 1.

Comparison of AVHRR SST EOF studies.

Table 1.
Table 1.

(Extended)

Table 1.
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