Jun 15, 1989 - An input sampling interval 5t -- I day (with input data only accepted ..... one another. ... rms scatter about the line of perfect agreement is 7 m.
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 94, NO. C6, PAGES 8040-8052,JUNE 15, 1989
Producing_Accurate Maps of the Gulf StreamThermalFront Using ObjectiveAnalysis D. RANDOLPHWATTS, KAREN L. TRACEY, AND AMY I. FRIEDLANDER Graduate Schoolof Oceanography,University of RhodeIsland, Narragansett
The objectiveanalysis(OA) methodis adaptedin orderto maptheGulfStreamthermalfront, andresulting outputmapsareverifiedagainstindependent data. Thisextension oftheOA method involvesremovingthe meanfieldandnormalizingthe varianceprior to performingthe objective analysis;both of thesefieldsare restoredafterward.In this way,the mappedfield decaysto an appropriate meanfieldfar awayfromthe measurement sites,andthe statistics of thenormalized perturbationfieldare homogeneous, asis requiredby the OA method.We test the sensitivityof the adaptedOA methodto variations in fourcontrolparameters: meanfield,correlation function, standarddeviationfield, and input samplinginterval. Of these,specification of the meanfield has the most influenceupon the mappedfields,whereasthe correlationfunctionhas more influence
uponthe estimatederrorfields. Usinga time-averaged meanfield produces the bestmapsof the thermalfield. The space-timecorrelationfunctionis determinedempiricallyfrom 5 yearsof
inverted echo sounder data collected in the Gulf Stream. Interestingly, even in this frontal region
the observedcorrelationsdo not differ significantlyin the cross-stream and downstreamdirections.
An input samplinginterval5t -- I day (with input data onlyaccepted withinT - :t:1day) provedto be best,as powerspectraconfirmthat all significant varianceis preserved.Within a reasonablevariation of the control parameters, this adapted OA method is robust, and accurate
mapsof the thermal field are obtained. The OA mapsof the Gulf Streamthermalfield are verifiedagainstnearly500independent measurements fromexpendable bathythermographs and air-droppedexpendable bathythermographs. The rmsdifference is determinedto be 47 m, which is lessthan the contourinterval of 50 m, and which representslessthan 7% of the dynamic range
(>700m) foundacross theGulfStream.Thustheadapted OA methodcanbeappliedto frontal
regions,suchas the Gulf Stream,wherethe traditionalmethodcannotbe used. 1.
INTRODUCTION
regionsin areas that were away from strong boundary cur-
Objectiveanalysis(OA) is a techniquethat statistically determinesthe optimal estimate of a quantity from a lim-
rents.
Wattsand Tracey[1985]extendthe useof the OA method
to the Gulf Stream thermal front region, where homogeneous ited number of measurements of that quantity. The optistatistics are not found. To overcome this difficulty we inmal estimate is the one, of all possiblelinear combinations troduce the technique of "preconditioning" the input data of the input data, that has the least error; the relative prior to performing the objective analysis: First, the spaweightingamongthe inputs is determinedfrom knowledge tially variable mean field is removed from the observations, of the space-timecorrelation function of the quantity being and the residual perturbation field is then normalized by estimated. The first use of objective analysis was in methe field of standard deviation. In this way, the perturbateorologywhen Gandin [1965] analyzedatmosphericpres- tion field has homogeneousstatistics and is appropriate for sure and wind fields. The technique was later introduced to objective analysis. The preconditioning procedure is consisoceanography by Brethertonet al. [1976],whodemonstrated tent with the method of determining the correlation funcits use with simulated temperature and velocity data. Adtion, which inherently removesthe mean and divides by the
ditionally, Freelandand Gould [1976] applied the method to velocitymeasurementsobtained in the MODE region to produce stream function maps.
Several generalizationsof the OA technique were pre-
variance.
Both
the mean
and standard
deviation
fields are
restored to the output field after running the OA, thus producing maps of the Gulf Stream thermal field. We use the modified OA technique to produce a seriesof
sentedby Carter [1983]and Carter and Robinson[1987]. mapsof the 12øCisothermdepth (Z12) from a setof inverted They extendedits application to include spatial anisotropy
by definingthe correlationsasfunctionsof both z and y distances. Additionally, they introduced correlation functions that were dependenton the time lag as well. This allowed estimatesof a field to be generatedfrom data collectedover a (correlated)period of time. In theseapplications,a necessaryconditionfor performing objectiveanalysishas been that the fieldsbeing estimated have homogeneous statistics, i.e., zero mean and uniform variance. To satisfy these requirements,the techniquehas earlier been applied only to data obtained from mid-ocean
echosounder(IES) observationstaken in the Gulf Stream region near Cape Hatteras, North Carolina. In previous
investigations[Watts and Rossby,1977; Watts and Johns, 1982],IESs have been shownto be reliable instrumentsfor monitoring the thermocline depth within the water column with high temporal resolution. However, the instrumentsare typically located at only a few sites, separated by relatively large distances. Since observationswhich provide both high temporal and spatial resolution are desired, in this paper we apply the OA method to the IES thermocline depth data to produce daily maps of the Gulf Stream thermocline depth
field on a highly resolved(20-kin spacing)horizontalgrid.
Copyright1989 by the AmericanGeophysicalUnion.
A crucial input to the OA mapping technique is the correlation function. Knowledge of its shape at short lags determines how well resolved the objective maps can be. The
Paper number 89JC00471.
0148-0227/89/89JC-00471$05.00 8040
WATTS
ET AL.:
77 ø
OBJECTIVE
75 ø
ANALYSIS
7 •o
OF THE GULF
71ø
I
I
STREAM
THERMAL
69* I
I
FRONT
67* I
8041
65*
I
I
I
i
I .? .:.,........:I 40ø
' '•
•
•- 20O
I000
.•• 2000 oOø
38 ø
36 ø
34 ø
I
Fig. 1. Map of IES array. IES locations are shown by the solid circles and squares;the box outlines the 240 km by 460 km region mapped by objective analysis.
Appendix describes how we empirically determined a correlation function for the Gulf Stream frontal region from nearly 5 years of IES measurementson grids of various spacings. An analytic correlation function, obtained by fitting a temporally and spatially decaying and propagating cosine function to these observed correlations, is used to produce
of the thermocline depth field, which are extracted from
approximately500 expendablebathythermograph(XBT) and AXBT
records
taken
in the
Gulf
Stream
frontal
re-
gion throughout a 3-year period. At each available XBT or AXBT site, the corresponding Z•2 estimate is determined from the OA maps that were generated from the IES arthe objective maps of the thermocline depth field. ray. The differences between the observed and estimated The output maps of the Gulf Stream Z•2 field are gen- values are compared for different regions that have differerated after specifying several control parameters for the ent levels of estimated mapping errors. Section 7 presents objective analysis. The most important of these are the an illustrative sequenceof OA maps using our preferred set choiceof (1) the mean field, (2) the correlationfunction, of control parameters and superimposes a set of isopycnal
(3) the standarddeviationfield, and (4) the temporalsam- float ("RAFOS float," Bower and Rossby[1989])tracksand pling interval. These control parameters could be specified current vectors from independent measurements. Section 8 or approximatedin variousways (e.g., determinedfrom dif- summarizes the important results.
ferenttime periods)for the study region.The main goalsof this study are to test (1) how robustthe output maps are to varying the choice between reasonable representations of
thesecontrolparametersand (2) whetherthe OA mapsaccurately represent the "true"Z•2 field. In this paper, we summarize the important results of an extensive set of tests that we conducted and reported in the work of Tracey et al.
2. 2.1.
IES
DESCRIPTION
OF THE
DATA
Measurements
For the period July 1982 through May 1985, thermocline depth records of the Gulf Stream downstream of Cape Hatteras were collectedusing an array of inverted echosounders. The array, shownin Figure 1, was configuredsuch that severaJ hnes of instruments were approximately normal to the
[1987]. In the sectionsthat follow, after describingthe data (section 2), we test the sensitivityof the OA technique(sec- historical mean axis of the current. The IESs were located tions 3 and 4) to variationsin the selectionof the above on various subsets of the lines during several deployments four control parameters. For these tests, objective maps are produced from data obtained from air-dropped expendable
throughout the 3-year period. The number of IESs in the water at any given time varied from eight to 20, depending
bathythermograph (AXBT) surveysand from mooredIES
on the deploymentand recoveryschedules[Tracey et al., 1985; Tracey and Watts, 1986b; Friedlander et al., 1986].
arrays. For each test, we check the output maps for "internal consistency"by comparing them with the actual observations.
Then
we examine
the differences
between
the OA
Typically, the instruments were placed about 60 km apart across the Gulf Stream and 65 km apart downstream.
Usingthe techniquesdescribedby Watts and Johns[1982] ters ("robustness"and "intercomparison tests"), repeating and Traceyand Watts [1986a],the travel times measuredby
maps produced with different choicesof the above parame-
these comparisons for a representative selection of maps. The preferred set of control parameters are summarized in section
5.
In section 6, we confirm the accuracy of the IES OA maps by comparing them with independent measurements
the IESs were scaled to thermocline depths. Briefly sum-
marizing those procedures,for many practical purposesthe main thermocline depth can be represented by the depth
of an individual isotherm,suchas the 12øC isotherm(Z•2) situated near the highest temperature gradient of the main
8042
WATTS ET AL.' OBJECTIVE ANALYSIS OF THE GULF STREAM THERMAL FRONT 3.
thermocline. The Za2 measurements for each instrument are
METHODS
smoothedusinga 40-hourlow-passfilter to removethe tidal 3.1. OA Mapping and Control Parameters and internal wavesignals;the estimatedaccuracyof Za2 is For all of our tests and comparisons, objective maps of 25 m [Traceyand Watts,1986a]. The observed standarddeviations(a) of the IES Zaamea- the Gulf Stream thermocline depth field are made varying surementsvariedsystematically,dependingon the proximity only one control parameter at a time; when not otherwise of the instrument
sites to the mean Gulf Stream location.
specifiedthe following "default" suite of control parameters
Highervariancewaslocatednear the meanGulf Streamcen- are used: (1) the meanfield is the temporalmean ZT (deter as a result of the steep thermocline slope in that re- scribedin section4.1), (2) the correlationfunction is the gion. Lowervariancewasfoundboth to the north and to analyticexpressionpwa, (describedin section4.2), (3) the the south, where the thermochneslopeis smaller. standarddeviation field is the Gaussianfunction •ra, (deObjectivemapsof the thermochnefield in the array area scribedin section4.3), and (4) the subsampling intervalis wereproducedfrom theseZaa recordsat daily intervalsfor 2 days(describedin section4.4). In addition to the above control parameters, the OA mapJuly 1982 to May 1985 [Tracey et al., 1985; Traceyand Watts, 1986b; Friedlanderet al., 1986]. The 240 km by ping technique selectsfrom all the available data within a 460 km boxedregionin Figure I hasbeenmapped,with its specifiedmaximumtime lag (T) and maximumradial disbaselying along064øT and output pointson a 20-kmsquare tance(R) the numberof points(N) whichhavethe highest correlations;it then makes an interpolation weighted by the correlationsto estimate the value at each output grid point.
grid. 2.2.
AXBT
By usingan assumednoiselevel (fractionalvarianceE), the
Measurements
During 1984, J. Bane flew sevenAXBT surveyflights mappingareasof the Gulf Streamthermalfield from Cape Hatterasto 65øW. For six of these(June 1, 4, 6, and 12; October 17; and November13), there was sufficientoverlap in the coverageto allow us to comparethe AXBT Zaa measurements with those obtained from the IESs.
The AXBT probes, manufactured by Sippican Corporation, measuredthe temperature structure to a depth of
760m with an accuracyof 11-15 m [Boyd,1987].The depth of the 12øC isotherm
was extracted
from each record.
If
OA map is not required to fit the observationsexactly, and the field is smoothed. Variations in each of these parameters also produce small differencesin the output Z•2 maps and in their
associated
error fields.
For a given array of measurementswe test severaldifferent combinations of T, R, N, and E in order to determine the most appropriate values. The choice of T, I to 4 days, affects the amount of temporal smoothing, as discussedlater. We use R - 120 kin, which is double the typical IES spacing of about 60 km, and within which the measurementsare significantlycorrelated. Variations in the value of E mainly
estimated (signal/noise) -• fieldrather temperatureinversions (associated with lower-salinity Slope affecttheobjectively than the OA maps themselves. We use E - 0.05 because it Water) resultedin morethan oneZ• value,the deepestone is the best estimate of the true error associated with these waschosento representthe main thermochne. Temperature
probeswereexcludedfrom this study if they werelocatedin
IES measurements. Having selectedT, R, and E, we choose
Approximately100 probeswere launchedper flight, with spacingsof about 18 km cross-streamand 55 km downstream. Sincethe primary purposewas to map the detailed
the determinant of this matrix becomes too small, the matrix is singular, and the mapping quality is poor. The output
regionswherethe profileswereeitherall warmer(Sargasso the maximumN for whichthe OA mapping(whichdepends uponinvertingthe matrix of correlations)remainsstable. If Sea)or all colder(SlopeWater) than 120C.
thermal front structure associated with the Gulf Stream, the
surveyregionchangedwith eachflight, dependingon the location of the current. As a result, the survey region had
varyingamountsof overlapwith the regionmappedby the
mapsshownin section4 (producedwith a subsamplinginterval of 2 days) result from specifyingT = -4-4days and
N = 9, whereas the mapsusedin section 6 (subsamphng intervalof 1 day) are producedusingT = =kl day and N = 7.
fixed IES array. 3.2.
The Estimated
Error
Fields
The Gulf Stream Z•a field is estimated over a large continuous area both within and extrapolated somewhat beyond Five research cruises were conducted to the study area the IES array. However,if an output point is too far removed between1982 and 1985 (July 1982, April 1983, September from the measurement sites, the quality of the estimate there 2.3.
XBT
Measurements
1983, June1984, and January1985), during whichover400 XBT probeswere launched. All XBTs were T-7 probesmanufactured by Sippican Corporation; they measuredthe temperature structure to a depth of 800 m with an accuracy of 15 m. The depth of the 12'C isotherm was extracted for each probe in the same manner as described above for the AXBTs.
We have used these data in section 6 to test the accuracy of the IES OA maps. Since the XBT surveys were concentrated near the IES sites, the only probes excluded from this study were those for which the whole profile was warmer or colder than
12'C.
is of coursepoor. Statisticalestimates(percentstandarddeviation) of howwell the output Z•a field is determinedfrom the input data are produced during the objective analyses. The error associated with each output grid point depends only on the correlation function and the locations, in space and time, of the input data; it is independent of the measurementsthemselves. Figure 2 showsthe estimated error
field for the array shown.The errorsare low (_700 m) acrossthe Gulf
'
10.
O.
Level
Fig. 9. The number of comparisonsbetweenthe IES OA maps and individual XBTs and AXBTs for the five regions delimited
by estimated error level. The number of probes is a cumulative function
of error level.
of the XBTs fell within El5.
Stream.
The AXBT
sites were more
evenly distributed over the survey area.
For these comparisons,the estimated Z12 values are determined from the IES OA maps for the same locations as the probe sites, interpolating between output grid points. The Z12 value is extracted from the daily IES OA map that was closestin time to the probe launch. The differences AZ between the Z12 values from the IES OA maps and from the XBTs and AXBTs are calculated, and the mean and rms differencesare categorized according
to map region (Elo, Eis,E2o,E2s, and E3s) for each XBT cruise or AXBT flight. Since the navigation systems on the ship and aircraft are different, systematic offsets can occur in the absolute locations of the AXBT drop sites relative
6.2. Accountingfor the Sourcesof Error
Finally, we attempt to account for all the factors contributing to the observeddifferencesbetweenthe IES OA maps and the XBTs and AXBTs. Three sourcesof error contribute
to these differences.
Measurement errors of the probes. Sippican Corporation
reportsa depthaccuracyfor the T-7 XBTs (•5 m plus2% of the depth), which,whencombinedwith a 0.2øCtemperature accuracy and a typical vertical temperature gradient near
12øC (0.02øC m-i), yields an•XBT • (52q-10• + 102) «= 15 m rms XBT error in Z12 for a typical Gulf Stream Z12 of
400-500 m. The accuracyof the AXBTs (•^xB,) is similarly estimatedby Boyd [1987]to be 11 m, from the combination of a 0.2ø(2 temperature accuracy and -FS-m depth accuracy. Uncertainties in the 1•øC isotherm depth values of the ,
80.
to the IES array location. Thesenavigationoffsets(•2 km)
I
•
differ from flight to flight. If the position of the aircraft were
I
I
I
I
'
I
,
I
'
AXBT
in error, slightly to the north (south) of the array, the Z12 valuesfrom the AXBTs would be shallower(deeper) than those from the IESs, and the mean AZ would be positive
XBT
(negative).On four of the six AXBT flights,suchbiasesare inferredfrom mean IAZl biasesrangingfrom 20 to 90 m. To eliminate the effect of navigational biases,for each flight the mean AZ is removed prior to calculating the rms values for the five map regions. The observedrms differencesare plotted in Figure 10 for the above five map regions. For the comparisonswith the XBTs, only a slight increase,from 40 to 50 m, occursin the rms AZ as the map error increasesfrom 10% to 35%. The comparisonswith the AXBTs, however, show a steady in-
40.
\
-
30._
_
20._
-
_
_
crease from 34 m within El0 to 74 m within E3s. The AXBT
_
_
valueschangemore with region than those of the XBTs, because they are more evenly distributed throughout the five regions. Consequently,the AXBTs give better estimatesof the true differenceswithin the higher error regions. Our objective now is to choosethe largest geographicregion to map from our array without overly sacrificing ac-
0.
' 40.
I
'
,3O.
Estimated
20.
Error
10.
Level
Fig. 10. The rms difference between the estimated Z•2 values from IES OA maps and the observed Z•2 values from XBTs and AXBTs for the five map regions.
8050
WATTSET AL.' OBJECTIVEANALYSISOF THE GULF STREAMTHERMALFRONT
800.
DISTANCE (kin)
i,,I,,,I,,ll,,,I,,,I,,,I,,,l'''
0
+
100
200 '\ \ •'
300 \
400
...... '...... +
•,
+ +++•"•
600.
+^'$•'
+ A
4O0. MAY 16,1984 lO
200.
O.
O.
200.
400.
Zt2
600.
800.
of Probes
MAY 18,1984
Fig. 11. Comparison of the estimatedZ12 fromIES OA maps within the regionwheremappingerror is
