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02-08-2005 Conference Proceedings (not refereed) I

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Assimilation of Altimeter Wave Measurements into Wavewatch III

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6. AUTHOR(S) 5d. PROJECT NUMBERPaul Wittman and James Cummings

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Oceanography Division NRL/PP/7320--04-0003

Stennis Space Center, MS 39529-5004

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14. ABSTRACTAssimilation of altimeter measured significant wave heights (SWH) into a global implementation of the Wavewatch III model was performed for March 2004, usingSWH data obtained from ENVISAT and JASON satellites. The wave model is forced by 3-hourly Navy Operational Global Atmospheric Prediction System(NOGAPS) marine surface winds. A 6-hour time window about the synoptic time is used to select the altimeter SWH data for the assimilation. The satellitemeasurements are quality controlled and bias corrected before being used in the analysis. An Optimum Interpolation (01) scheme is used to compute the SWHincrement field from the altimeter SWH innovations. The "first guess: 6-hr. model forecast directional wave spectra are then corrected by the ratio of the analysiswave height over the first guess wave height. This correction is distributed uniformly over the wave model spectra. Prior to the March 2004 assimilation run, a 6-month analysis-only run (no forecast model update) was performed. Wavewatch III prediction errors at the 6-hr. forecast period, and spatial covariance functions.Observation errors are found to vary with satellite , prediction errors are found to vary with position, and a second-order autoregressive function is found to be anadequate fit to the bin-averaged spatial autocorrelation estimates. Spatial correlation analysis of the analysis residuals shows that the analysis is effectively extractingall of the information in the altimeter SWH measurements.

15. SUBJECT TERMS

altimeter, wave heights, assimilation

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(831) 656-5021

Standard Form 298 (Rev. 8/98)Prescribed by ANSI Std. Z39.18

Assimilation of Altimeter Wave Measurements into Wavewatch III

Paul A. WittmannFleetNumerical Meteorology and Oceanography Center7 Grace Hopper Ave, Stop 1, Monterey, CA 94943-5501

James A. CummingsOceanography Division

Naval Research Laboratory7 Grace Hopper Ave, Stop 2, Monterey CA 93943-5502

Abstract. Assimilation of altimeter measured significant wave heights (SWH) into a global implementation of theWavewatch III model was performed for March 2004, using SWH data obtained from ENVISAT and JASON satellites.The wave model is forced by 3-hourly Navy Operational Global Atmospheric Prediction System (NOGAPS) marinesurface winds. A 6-hour time window about the synoptic time is used to select the altimeter SWH data for theassimilation. The satellite measurements are quality controlled and bias corrected before being used in the analysis.An Optimum Interpolation (01) scheme is used to compute the SWH increment field from the altimeter SWHinnovations. The "first guess" 6-hour model forecast directional wave spectra are then corrected by the ratio of theanalysis wave height over the first guess wave height. This correction is distributed uniformly over the wave modelspectra. Prior to the March 2004 assimilation run, a six-month anialysis-only run (no forecast model update) wasperformed. The SWH innovations from the analysis-only run are used to compute the statistical parameters required inthe 01; observation errors, Wavewatch III prediction errors at the 6-hour forecast period, and spatial covariancefunctions. Observation errors are found to vary with satellite, prediction errors are found to vary with position, and asecond-order autoregressive function is found to be an adequate fit to the bin-averaged spatial autocorrelationestimates. Initial testing of the assimilation system shows a decrease in wave model SWH forecast mean and root meansquare errors when compared to selected deep-water wave buoys and yet-to-be-assimilated altimeter SWHobservations. Spatial correlation analysis of the analysis residuals shows that the analysis is effectively extracting all ofthe information in the altimeter SWH measurements.

Introduction. The assimilation of radar spectral components. Since that time, SEASATaltimeter wave heights into numerical wave and GEOSAT have failed, but other altimetermodels has progressed over the last 15 years satellites have been launched. Currently,with the deployment of altimeters on a number JASON-I, GFO and ENVISAT satelliteof satellites orbiting the earth. The significant altimeters provide wave height measurements towave height (SWH) is estimated from the a number of operational weather centers (Bidlotbackscatter of the altimeter pulse.. The narrow. and Holt, 1999). Greeenslade (2001) looked atfootprint gives high resolution along tract, but the effect of the spectral adjUstment method andsparse data coverage between tracks. Two main the error correlation length. She found that theissues need to be considered: 1) the method of results were more sensitive to the length scaleinterpolation of the wave height corrections, and than the choice of spectral adjustment method.2) the method used to modify the first guessdirectional wave spectraof the model based on More recent studies have focused on the_the wave height analysis. sensitivity of the wave model to the

simultaneous assimilation of data from severalThe first attempts to assimilate altimeter altimeters (Skandrani et al., 2003), and themeasured wave heights in numerical wave choice of the spatial autocorrelation functionsmodels were made by Esteva (1998) and used in the 01 method (Greenslade, 2004).Lionello et al. (1992), using SEASAT and Unlike NWP models, wave models are stronglyGEOSAT data. Both of these studies used forced by surface winds, so the impact of thestandard optimum interpolation (OI) techniques assimilation is often diminished over forecastto create wave height analysis. Esteva scaled the time, particularly in the wind sea portion of thewave model spectra by the ratio of the first guess directional wave spectra. However, it has beenSWH to the analyzed SWH, while Lionello et al. shown that corrections to the low frequencyused a more sophisticated method using the local portion of the spectra retain the corrections for awind velocity to modify the sea and swell longer time (Bender and Glowacki, 1996). In

12005084* 027

general, these studies have found that time domains of the Wavewatch forecast modelassimilation of altimeter data into the operational grid and update cycle. The forward operator inwave models has a positive effect on the wave NCODA is simply a spatial interpolation of themodel bias in the short term (0-36 hour) forecast, forecast model grid to the observation location

performed in two dimensions. Thus, HPbIHT isWave Model Configuration. The Wavewatch approximated directly by the background errorIII version 2.22 (Tolman, 1990) configuration covariance between observation locations, andused for the assimilation test is identical to that PbIHT directly by the error covariance betweenof the Fleet Numerical Meteorology and observation and grid locations. For the purposesOceanography Center (FNMOC) operational of discussion, the quantity [y - H(Xb)] is referredglobal model. The model is run on a 0.5-degree to as the innovation vector, [y - H(xa)] is theresolution spherical grid, using an ice analysis to residual vector, and Xa - Xb is the increment (ormask points under the ice. The model is correction) vector.initialized by the 6-hour forecast, or first guess,spectra from the previous run. The wind forcing Specification of the background and observationtime step is 3 hours. The spectral resolution of error covariances in the analysis is verythe wave model is 24 directions (15 deg angular important. The NCODA background errorresolution) and 25 frequencies, ranging from covariances are separated into a background0.42 to 0.04 hertz (Wittmann, 2002). error variance and a correlation. In two-

dimensional mode only the horizontal correlationAssimilation Method. The wave model data component needs to be specified. The horizontalassimilation is performed by the Naval Research correlation is modeled as a second order auto-Laboratory (NRL) Coupled Ocean Data regressive (SOAR) function of the form,Assimilation (NCODA) system. NCODA is afully three-dimensional multivariate optimum Ch =(1 s.)exp(-s.)interpolation system developed as part of theOffice of Naval Research (ONR) sponsored where Sh is the horizontal distance between twoNavy coupled modeling project (Cummings, locations (observations or observation and a grid2003). In this study, NCODA is executed intwo dim nsi nal mod to pro ide upd ted S W Hpoint). The distance is norm alized by thetwo-dimensional mode to provide updated Sfra geometric mean of the horizontal correlationfields for the Wave Watch III wave forecast length scales prescribed a priori at the twomodel using a sequential incremental update locations. NCODA allows the correlation lengthcycle. The analysis background field, or first scales to vary with location, but in theguess, is generated from a short-term wave assimilation experiment reported here the SWHmodel forecast. In the wave model data error correlation length scale is set to a constantassimilation runs described here a six-hour value (223 km). This value was computed usingupdate cycle is used. NCODA computes 'the . innovation correlation methodcorrections to the first-guess SWH field using all (Hollingsworth and Lonnberg, 1986) from a non-of the altimeter SWH observations that have assimilative JASON-I altimeter SWH innovationbecome available since the last analysis was time series created in a six-month run of themade. The forecast model with the new initial analysis from June through December 2003.conditions is then run forward in time to produce Statistical analysis of the innovations is the mostthe next forecast. -- common,--and-the-most-accurateytechniquefor-

estimating observation and forecast errorcovariances. Fig. 1 shows the bin-averagedformulated in NCODA as, autocorrelation estimates as a function of

a =b distance, and a non-linear least squares fit of thex = xb + pbHT(HPbHT + R)-[y - H(Xb)] SOAR model. As can be seen in Fig. 1, a SOAR

function accurately models the long positive tailwhere xa is the analysis, Xb is the background, Pb of the estimated correlations. In comparison, theis the background error covariance, H is the spatial autocorrelation analyses and SOARforward operator, R is the observation error models fit to the JASON-1 and ENVISATcovariance, and y is the observation vector. The altimeter SWH innovations from the March 2004observation vector contains all of the synoptic assimilation run are shown in Fig. 2. TheSWH observations within the geographic and correlation length scale derived from the non-

2

assimilative SOAR model is almost twice as errors, and it is only possible to obtain alarge as the length scale computed from the horizontally homogeneous (domain-averaged)assimilation run. However, the functional form estimate of the background error variance usingof the SOAR models is very similar between the this method. Observation and background errorstwo innovation time series. A longer innovation for JASON-1 and ENVISAT computed using thetime series from an assimilation run is needed to innovation correlation method are shown indetermine if these estimated differences in Table 1 for both the assimilation and non-correlation length scales are real. assimilation control runs of the wave model.

The background error variances in NCODA Quality Control and Observation(Eb2) vary with location and evolve with time. Preprocessing. All altimeter SWH observationsThe error variances are computed from a time are subject to quality control (QC) procedureshistory of the analyzed increment fields and prior to assimilation. The primary purpose of theupdated at the end of each update cycle. A QC system is to identify observations that areclimate error growth rate parameterization is obviously in error, as well as the more difficultused to account for the inherent sampling process of identifying measurements that falllimitations of the altimeters. In the long-term within valid and reasonable ranges, butabsence of altimeter SWH observations, the nevertheless are erroneous. The need for qualitybackground error variances are slowly restored control is fundamental to any data assimilationto climate variability values using a climate system. Accepting erroneous data can cause andecorrelation time scale of -96 hours. The incorrect analysis, while rejecting extreme, butclimate decorrelation time scale is calculated valid, data can miss important events. The SWHfrom observations and assumes a zero mean QC procedures include land/sea boundarySWH climate field. In practice, the background checks, shallow water retrieval checks,. anderror variances reflect the long-term average background field checks against Wavewatch IIIprediction error variances of the model forecast model forecast fields using 6-hour predictionat the analysis update time. To initialize the error variances. Cross validation checks are alsoassimilation run the background error variances performed between the altimeter SWHare computed from the time history of the non- observations and sea ice concentration to checkassimilative analyzed increments (Fig. 3). Note for impossible SWH retrievals. Sea ice analysesthat because of the assumption of a zero mean are performed at the same time as the SWHSWH climate field, the background error analysis to provide the QC procedure with avariances in Fig. 3 computed using the climate contemporaneous sea ice concentration field.error growth scheme are likely to be inflated. SSM/I sea ice retrievals from the DMSP series of

satellites are used in the sea ice analysis. TheThe observation errors and the background errors QC processes result in the assignment of a

............. ..... are assumed . to be uncorrelated, and errors probability of gross error to each altimeter SWHassociated with observations made at different retrieval. The magnitude of an acceptable grosslocations and at different times are also assumed error probability is a user-defined parameter into • be uncorrelated. As a result of these NCODA, and thus an integral component of theassumptions, the observation error covariance space/time queries performed on the QC datamatrix R is set equal to 1 + e.2 along the files when gathering SWH observations fordiagonal and zero elsewhere. Note that e assimilation.represents observation error variances that havebeen normalized by the background error A "super observation" algorithm is used to thinvariances interpolated to the observation location the data prior to the analysis. Thinning of the(e.2 = Eo2 / Eb2). Observation error variances are relatively high volume altimeter SWHcomputed from the non-assimilative innovation observations is a necessary step in the analysis intime series using the innovation correlation order to remove redundancies in the data andmethod. The SOAR correlation function that is minimize horizontal correlations amongfit to the bin-averaged observed covariances is observations. NCODA uses an adaptiveextrapolated to zero distance and the background algorithm to computes super-observations byerror variance is computed. The difference averaging SWH retrieval innovations into binsbetween this value and the innovation variance is dependent on grid resolution and observationthe observation error variance. The method data type (satellite). The algorithm is adaptive inassumes horizontally uncorrelated observation that as the model grid resolution increases the

3

actual number of innovations averaged into a a=(Ha/H)super-observation decrease until, eventually, the a (f, I)=original data are directly assimilated. The F"(f,®) = aF (f,O)resolution of the altimeter SWH retrievals is -7km along track, and the analysis is performed on The assimilation run and a non-assimilativea global 0.5-degree spherical grid. This control run are compared to independent buoydiscrepancy in resolution between the and yet-to-be assimilated altimeter SWHobservations and the model grid results in SWH measurements. The 18 moored buoy locationssuper-observations being formed, typically, from are shown in Fig. 5. The buoy SWH-7 altimeter SWH retrievals, measurements are plotted against collocated

model forecast SWH fields from the assimilationThe altimeter SWH bias corrections' of Cotton and non-assimilation control runs of the(2002) for GFO, ERS2, and Topex are applied to Wavewatch model (Fig. 6). Fig. 6 shows a 32%the SWH retrievals prior to assimilation. Bias reduction in bias and a 15% reduction in rootcorrections do not exist for JASON-1 and mean square error for the assimilation run at theENVISAT at the time of the wave model data 6-hour forecast period. Further impacts of theassimilation runs, so these satellite data are not assimilation can be seen from individual buoybias corrected. Bias corrections are applied prior time series. For example, National Data Buoyto the QC and prior to the data thinning Center (NDBC) buoy 44004 is located 200procedures. nautical miles east of Cape May, New Jersey, in

3124 meters of water. The time series of theValidation and Verification. Simple bulk buoy 44004 SWH measurements show a 8.5 mmeasures of root-mean-square (RMS) error and wave event on day 70, under predicted by almostmean bias of the innovations are computed every 2 m in the control run, that is closely predicted inupdate cycle. These statistics are used to assess the 6-hour forecast from the assimilation runthe quality of the analysis. Spatial (Fig. 7). Fig. 8 shows similar, improvedautocorrelation analysis of the SWH analysis agreement of the 6-hour wave model SWHresidual vectors [y - H(x.)] is used to determine forecast from the assimilation with the buoythe fit of the analysis to the altimeter SiWH SWH trace for NDBC buoy 46059, located in theobservations. In theory, the analysis residuals North Pacific, as compared to the non-should be uncorrelated at all spatial lags greater assimilative control run of the wave model. Fig.than one. Any spatial correlation remaining in 9 shows the time series of altimeter SWHthe residuals represents information that has not innovations and residuals at each update cycle.been extracted by the analysis (Hollingsworth The 6-hour Wavewatch SWH forecasts at theand Lonnberg 1989). Fig. 4 shows the residual altimeter observation locations from theautocorrelation analyses of JASON-I and operational free run of the model are also shownENVISAT altimeter SWH observations from the .. in Fig. 9. The stability and the effect of theassimilation run. As expected, autocorrelations assimilation system is seen in the unbiasedat all spatial lags greater than one are close to residuals and in the consistent reduction in errorzero, which indicates an effective analysis . of the innovations from the control run. The

average 6-hour forecast RMS error over the 30-In the Wavewatch analysis update cycle, day period is 0.61 m in the control run, and 0.46innovations of the ENVISAT and JASON-i m in the assimilation run.altimeter tracks synoptic about the analysis timeare computed and processed through the Discussion. The experiment described here is aNCODA analysis scheme to produce the first attempt to assimilate altimeter SWH into theanalyzed increments. The analyzed increment FNMOC global Wavewatch III model. Futurefield is added to the Wavewatch 6-hour SWH work will include testing the sensitivity of theforecast (HR) valid at the analysis time, to spectral modification method and the effect ofproduce the corrected SWH analysis field (H'). the assimilation on the wave model forecast atThe analyzed wave model spectrum (F') as a forecast periods longer than the 6-hour updatefunction of frequency (f) and direction (0) is then cycle. Also, work is underway to look at spatialobtained from the forecast spectrum (Ff) using a dependence of the horizontal correlation lengthsimple scaling strategy, scales used in the assimilation. A real-time

operational test of the FNMOC wave modelassimilation system is planned for the 2004-2005

4

northern hemisphere winter. Once the Background Errors in a Global Wave Model. Journalassimilation method is verified it will be of Atmospheric and Oceanic Technology, Submitted.included in the FNMOC operational wave modelludd iHollingsworth, A. and P. Lonnberg (1986). Thestatistical structure of short-range forecast errors as

determined from radiosonde data. Part I: The windReferences field. Tellus 38A:111-136.

Bender L.C. and T Glowakci (1996). The assimilation Hollingsworth, A. and P. Lonnberg (1989). Theof altimeter data into the Australian wave model. Aust. verification of objective analyses: Diagnostics ofMet. Mag. 45, 41-48. analysis system performance. Meteor. Atmos. Phys.

40:3-27.Bidlot, J.R. and M.W. Holt (1999). Numerical wavemodeling at operational weather centres, Coastal Lionello, P., H. Gunther, and P. Janssen, (1992).Engineering 37, 209-429. Assimilation of altimeter data in a global third

generation wave model. ECMWF Tech. Report No.Cotton, P.D. (2002). Satellite Observing Systems, 67.Ltd, UK

Skandrani C., J.M. Lefevre, L.Aouf, P. QueffeulouCummings, J.A. (2003). Ocean Data Assimilation, In (2003). Impact of multi-sources of altimeter dataCOAMPS: version 3 model description, pp 21-28. (ERS2, ENVISAT, JASON) on wave forecasts. CD-NRL Publication NRL/PU/7500-03-448. ROM proceedings of EGS-AGU-EUG Joint

Esteva, D.C. (1988). Evaluation of preliminary Assembly, Vol. 5, NICE, France, 06-11 Apr. 2003.

experiments assimilating Seasat significant wave Tolman, H.L. (1990). A third-generation model forheights into a spectral wave model. J. Geophys. Res. wind waves on slowly varying, unsteady and93, 14099-14106. inhomogeneous depths and currents. Journal of

Greenslade D.J.M (2001). The assimilation of ERS-2 Physical Oceanography, 21, 782-787.

significant wave height data in the Australian region. Wittmann, P.A. (2002). Implementation ofJournal of Marine Systems, 28, 141-160. Wavewatch III at Fleet Numerical Meteorology and

Oceanography Center. Conf. Proceedings: MTS/IEEE:Greenslade D.J.M and I.R. Young (2004). The Impact Conference and Exposition. Nov 5-8, 2001 Honolulu,of Altimeter Sampling Patterns on Estimates of HI (sponsored by the Marine Technology Society and

IEEE), 1474-1479.

5

Table 1. Altimeter SWH observation and Wavewatch IH SWH prediction errors (m) estimated fromthe spatial autocorrelation functions computed from the non-assimilation (June-December2003) and assimilation (March 2004) innovations time series.

Non-assimilative Control Run Assimilation RunSatellite Observation Prediction Observation PredictionGFO 0.30 0.45ERS2 0.43 0.48ENVISAT 0.30 0.37JASON-1 0.40 0.60 0.43 0.44

1.0

0.6

-0 0, 4 -

30U X

0.0

--.2

! I ! II I I I I

O 1.00 200 300 400 500 000 700 800 900 1000

Figure 1. Bin averaged correlations (x) for JASON-1 altimeter SWH observations estimated from anon-assimilative run of the analysis system from June-December 2003. The solid line is a leastsquares fit of a SOAR function to the bin averaged correlation estimates. A total of 775,500altimeter innovations are used in the calculations. The correlation length scale is estimated to be-~223 km.

6

MMkI•T Cycle Envisat SfH b ASON M Cyc JasonSYa bm-Inuv Spatl Correlation b -rhnr Spaal Correlation

1o 58 km nest 01 Mar 0.1 Apr 2004 6 5I knn nest 01 Mar - 01 Apr 2004

GA6 .3c96

0.80.

0.2 US

0.00.

0 100 ODD 3001 •A, 000 W00 700 801 900 1 0 iOO W 8 400 DW NO 700 BUD 900 10W

Visrtarnce (kin) Dlistance (kmn)

Figure 2. ENVISAT (a) and JASON-1 (b) SWH autocorrelation functions computed from theMarch, 2004 assimilation innovation time series. The bin-averaged correlation estimates are markedwith an x, and the non-linear least squares fit of a SOAR function to the correlation estimates isshown as solid curves. A total of 287,072 JASON-1 and 188,898 ENVISAT innovations are used inthe calculations. The correlation length scales are estimated to be 110 kin for JASON-1 and 114 kmfor ENVISAT.

''"'"'"'" , .5 .75 1 1.2 1.8 175 2 a .......

Figure 3. Wave Watch III significant wave height 6-hour prediction error variances (M 2) computedfrom the June-December, 2003 non-assimilative innovation time series. See text for details on howthe background error variances are computed.

7

a b

+.6 O4

0,s as 4

OLD 0 0 4-

LI ~ +

0 LOD 200 900 400 N0O 000 700 50D 00 1000 0 LOD 200 30W 400 DO0 1100 700 8100 900 I=0

D!StancV fi~m) c fLn

Figure 4. Residual autocorrelation analyses of ENVISAT (a) and JASON-i (b) altimeter SWHinnovation time series from March 2004 assimilation run. The residual autocorrelation estimatesare marked with an o, and for reference purposes the innovation autocorrelation estimates are shownmarked with a +. The analysis residuals are essentially uncorrelated after one spatial lag.

6014 . ............. •. '.'.. ........ .• 4 ......

5 0O ;.. . .. . ... .. . . .. . . ............... ............. . .. . . . . .. . ........... . ....................... . .. .... .. .. ..

40N:

01001O0..0.. ... ..3•on: ................... .... ....:.".•" ' '•'•. ... .......... ................ 1 .............. ................ ...... AL• o :.... ! ." .. -......

00'.'51001.2ON; ~ ~ ....... ......................... ............ :........... .. .... "7 % 1 .........:_

-S00

E . . . . . .. . .. . . . . . .. ... . .. . ..... ., ! .. . ......... ....... .................................................. .......... . .170 1 70W 160W 150W 140W 130W 120W 110W 100W O 'w ..........

Figure 5. NDBC Buoy locations of the 18 buoys used to verify the control and assimilation runs.

8

AouýIllatlon Run +

Entries: 2520Bias: -0.15 +RMSE: 0.47 -+Scatter, 0.18 +1Cor Doea: 0.92 +Syrnetrie Slope: 0.94

+ ++++ + / / +

+ + + +

4- ++ +/÷+ +

+ + ++ /+/+ +

+ + ,+ + +

4++

0 2 4 6 aBuoy Ho (m)

Figure 6a. Wave height (IHs) measurements from 18 NDBC deep-water wave buoys plotted againstthe WW3 assimilation run, for March, 2002. Forecast time is 0.

Non-Aa-slmllatlon RunEntries: 2520 +Bias: -0.22RMSE: 0.55 + +Scatter 0.21Cor Coef: 0.89 + ISyrestric Slope: 0.91

/ + ++ + ++

+ ++ +I// +++ + +', 4 +1

+ ++-, ++

. . .. ... ... ... .. . . . . . .+ 4+

+ +4 + +

2 +- +- ----

+ +•.

2- ++

4 :

0 2 4 8Buoy He (m)

Figure 6b. Same as 6a, except model values are from the control run.

9

10 1 '3 I I '6 I ' ' I ' I '

Buoy 44004 Lot: 38.0 Lon: -70.0

iil

ES ~i-

+1+

2

60 65 70 75 so 85 90dulian Day for 2004

Figure 7. Time series plot of control run (solid line), assimilation run (dashed line) and wave heightmeasurements (crosses) from NDBC buoy 44004, located in the northwest Atlantic.

B j l , j I l l' I ' I I ' I ' ' I I I I

Buoy 4-5059 Lot: 38.0 Lon: -129.0

C +

" 4 "1++

+ +PI-

50 65 70 75 80 85 90Julian Day for 2004

Figure 8. Same as Figure 7, except for NDBC buoy 46059, located in the northeast Pacific.

10

0

I mar t p

Retiduml Inno•tLtan CG.

SWE fl~ta 6ututa

Figure 9. Verification of March 2004 assimilation and control runs using altimeter SWHobservations. In the top two frames, 6-hour forecast errors of the free run of the model (control) areshown in green and the errors of the assimilation run (innovation) are shown in red; analysisresiduals are shown in blue. (a) RMS error, (b) mean bias error, (c) data counts of ENVISAT andJASON-1 SWH super-observations used in each assimilation update. Each tick mark along the timeaxis represents an assimilation update cycle.

11