Residual Tide analysis in shallow waTeR - ConTRibuTions of enVisaT and eRs alTimeTRy

Roman savcenko, wolfgang bosch

Deutsches Geodatische Forschungsinstitut, 80539 Munchen, Germany, E-mail: savcenko@dgfi.badw.de

absTRaCT

Global ocean tide models perform rather well in open ocean but tides in shallow water are significant less well known. Here residual tidal analysis with multi-mission altimeter data is performed to improve a global ocean tide model. The altimeter data of ERS and ENVISAT al-lows to increase the spatial resolution for tides empirical-ly estimated from TOPEX and JASON-1 data. The long repeat periods of ERS and ENVISAT lead to severe alias effects making it difficult or even impossible to separate some tides from each other. Due to the sun-synchronous orbit of the S2 tide is frozen and cannot be estimated from ERS and ENVISAT data alone. A discrete multi-mission crossover analysis is performed to homogenize the data of different missions. Using combined sea level anomaly time series at geographical coordinate grid nodes the bet-ter separation of major tidal constituents is achieved.

1. inTRoduCTion

The FES2004 ocean tide model [5] has become a de-facto standard for computing ocean tide correction for altimetry data. This model performs rather well in open ocean but tides in shallow water area are significant less well known – although the grid resolution of 7.5’ is capa-ble to describe small scale variation of tidal water level. By a case study for the North-West European shelf (see Fig. 1) it is demonstrated that sufficient long time series of multi-mission satellite altimeter data can significantly improve global ocean tide models (like FES2004) for shallow water areas. Empirical ocean tide analyses most-ly rely on TOPEX/Poseidon (TP) and Jason-1 data as these missions provide the longest time series with high temporal resolution. They allow to resolve and separate all major tidal constituents. However, the ground track spacing for these missions is large and the spatial resolu-tion is poor. On the other hand, the much denser ground track pattern of ERS-1/2 and ENVISAT can’t be fully ex-ploited as the sun-synchronous orbits do not at all allow estimating e.g. the S2 tidal constituent. For other con-stituents to be resolved and separated the 35 day repeat period of the ESA missions requires a particular long time series. M2 and N2, for example, can be separated only after 8.5 years.

The combination of data with different sampling char-acteristics allows overcoming the obstacles of all these missions. It is shown that ENVISAT and ERS data, com-plementing data from TOPEX, Jason1 and GFO allows resolving and separating all dominant tidal constituents with high spatial resolution. Combining altimeter data from different mission in a common tide analysis re-quires, however, a careful harmonisation and cross cali-bration. This pre-processing includes a global crossover analysis based on nearly simultaneous single- and dual satellite crossover differences. In a final step ocean bot-tom pressure records are used to validate the improve-ment, achieved by the residual tide analysis.

2. alTimeTeR daTa PRe-PRoCessing

Unfortunately the present situation for multi-mission al-timeter data is rather heterogeneous. Therefore the data was carefully harmonised and cross calibrated before

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Figure 1 The area of investigation with the sub-satellite tracks of TOPEX/Poseidon and Jason1 (T/P/J in red), the shifted ground tracks of TOPEX/Poseidon extended mission (T/P-EM in purple), of ERS-1/2 and ENVISAT (ERS/ENVISAT in green) and of GFO in blue.

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Proc. ‘Envisat Symposium 2007’, Montreux, Switzerland 23–27 April 2007 (ESA SP-636, July 2007)

performing the tidal analysis. Table 1 summarises the al-timeter data, their observation periods and the versions of the data taken for the present analysis. ENVISAT “Geo-physical Data Records” (GDRs) are available by ESA/CNES. The “Ocean Product Record” (OPR) for ERS-1 and ERS-2 were delivered by CERSAT. TOPEX/Po-seidon data has been taken from the “Geophysical Data Record” (GDR-M), Version C, as distributed by AVISO. GFO data were kindly provided by NOAA’s Laboratory for Satellite Altimetry and the Jason1 GDR data were taken from the ftp server of AVISO/JPL. It should be emphasized: The data of the ERS-1 “geodetic” phases E and F is still based on old-fashion software and tracking algorithm. From cycle 38 on ENVISAT data is processed by improved software and new orbits not yet applied to cycle 9 – 37. Finally, the Jason-1 data is just being re-processed (GDR, version B) and there are still a few cy-cles (22-94) left that have not been re-processed by the new software.

In order to harmonize the altimeter data as far as possible the best mission specific corrections were applied and common geophysical models for the mean sea surface and ocean tide models were used. For the computation of sea level anomalies the standard corrections were applied with the following modifications: • the mean sea surface CLS01 [5] was taken as a com-

mon reference surface for all missions • for all altimeter systems the ocean tide corrections

were computed using the FES2004 tide model [6] • for the sea state bias of TOPEX (side A and side B) the

model of [3] was used • the TOPEX microwave radiometer drift was corrected

using the results of [9] • all corrections described in [10] were applied to the

ERS-1/2 data sets • the orbit data of ERS-1/2 was replaced by DEOS pre-

cise orbits [8] • for ENVISAT the ultra stable oscillator (USO) drift

(GDR-A) and anomalies (GDR-B) are corrected as suggested by ESA

• for Jason1 GDR version A altimeter data the radiom-eter wet troposphere correction are replaced using re-placement product provided by Jet Propulsion Labora-tory (JPL) [4].

It is important to mention that the inverted barometer (IB) correction was not replaced and taken as provided in the original data. The reprocessed Jason1 GDR-B data and the ENVISAT data from cycle 38 onwards provide an IB-correction based on the hydrodynamic model MOG2D while the other data sets still include the instantaneous IB –correction proportional to the air pressure. This is an inconsistency to be resolved in future analyses.

For the cross calibration a discrete crossover analysis [2] was performed in order to estimate radial errors for all al-timeter systems for the operating period of TP and TP-EM (October 1992 to October 2005). This cross calibration automatically captures relative range biases and ensures that the radial component of all altimeter systems is con-sistent. The estimated radial error components have been corrected before performing the ocean tide analysis.

3. Tidal aliasing and de-CoRRelaTion

Empirical ocean tide analysis faces two difficulties – the alias effect and the problem of de-correlating tidal signals with periods very close to each other. Alias effects emerge if the altimeter systems sample high frequency tide sig-nals with periods of some 12 and 24 hours only every few days – the satellite repeat period. In this case the tides ap-pear as signals with periods much longer than the sam-pling interval. These periods – called alias periods – are different for the tidal constituents and depend on the repeat period of the altimeter satellite, see tabulated alias periods in [11] and [1]. The capability to separate neighbouring periods from each other is expressed by the Rayleigh cri-terion. In case of the empirical tide analysis by altimetry the Rayleigh criterion must be applied to the alias periods. The minimal time span needed for the accurate separation of two tides is called Rayleigh period. For the tidal analy-sis of altimeter data these periods can again become very large – even infinite if one of the tidal signals cannot be de-aliased at all. A comprehensive discussion on the alias and Rayleigh periods can be found in [11].

With some thirteen years the altimeter time series avail-able on the TOPEX ground tracks is long enough to de-alias all major tidal constituents and also fulfils the Rayleigh criterion for their de-correlation. The time se-

Mission (Phase) Cycles Period Source ReplacementsTOPEX/Poseidon 001-481 1992/09/23-2005/10/08 MGDR-C AVISO Chambers SSB correction, FES2004Jason1 001-135 2002/01/15-2005/09/14 GDR-A/B PODACC FES2004

ERS-1 (C & G) 083-101 1992/04/14-1993/12/20

OPR-V6 CERSAT DEOS orbits, FES2004, pole tide, 1.5ms time bias144-155 1995/03/24-1996/04/28ERS-1 (D, E & F) 102-143 1993/12/25-1995/03/21 OPR-V3 CERSAT DEOS orbits, FES2004, pole tide, 1.5ms time biasERS-2 000-085 1995/04/29-2003/07/02 OPR-V6 CERSAT DEOS orbits, FES2004, pole tide, 1.3ms time biasENVISAT 009-040 2002/09/24-2005/09/19 GDR ESA/CNES FES2004GFO 037-159 2000/01/07-2005/10/04 GDR NOAA FES2004

Table 1. Altimeter mission data used for the present analysis

ries of 35-day repeat periods of the sun synchronous mis-sions ERS and ENVISAT are, however, very problematic in resolving and separating several tidal constituents. The solar tide constituents S2, K1 and P1 are affected by se-vere correlation problems and cannot be estimated using the data of these missions alone. The M2 and N2 tides can be separated from each other only if at least a nine years time series of data is available.

The difficulties on alias and Rayleigh periods apply if the data of a single mission at a particular point is consid-ered. Using data on crossing or adjacent tracks already improves the temporal resolution and can in general miti-gate the alias effect. The advantage of an complementary sampling on single satellite crossover points depends of the tidal constituent and the latitude [11]. The most effi-cient solution to the correlation problems can be achieved by combining time series of missions with different sam-pling characteristics. Because the sampling of combined time series strongly depends on the spatial distribution of tracks there are no simple rules to examine the separation proprieties. In this case a careful analysis of correlations after ocean tide analysis is essential (see below).

4. Tidal analysis

For the tidal analysis the classical least squares harmonic approach is preferred over the response method [7] – due to uncertainties in a smooth admittance function. As the altimeter data is already corrected by the FES2004 ocean tide model a residual ocean tide analysis is performed by estimating simultaneously mean value, trend, seasonal variations (with annual and semi-annual periods), correc-tions to eight major tidal constituents (M2, S2, K2, N2, Q1, P1, K1, O1), and to the shallow water constituent M4. The least squares approach was applied using the following observation equation.

(1) [ ]1 2

1 2

ˆ ˆ

cos( ) sin( )

cos( ) sin( )

i i i i i i ii

j j j jj

v m d tf h t u h t u

a t a t

z + = + ⋅D +

+ w D + + w D + +

+ W D + W D

∑∑

where

z estimated sea level anomaly v estimated residual m mean value d trend Dt time since reference epoch h1i; h2i cosine or sine coefficients a1i; a2i cosine or sine coefficients of seasonal variation wiDt astronomical argument of tidal constituent Wj angular frequency for seasonal variations fi; ui nodal corrections to amplitude and phase

To mitigate the correlation problem the analysis is per-formed on the nodes of a regular geographical 7.5‘x7.5‘ grid. For every grid node normal equations are accumu-lated using all measurements inside a spherical radius of 1.125°. A Gauss function with half weight width of 0.375° is applied for weighting inverse proportional to the distance. The choice of spherical radius and the half weight width is critical and must ensure that observations from the TOPEX and Jason1 missions contribute to every grid node in order to achieve the necessary de-correlation for all constituents.

Figure 2 presents the mean correlation matrix for the whole area of investigation indicating with maximum correlation of ±0.3 that in general the de-correlation is rather successful. For grid nodes inside the diamond shaped areas not covered by the tracks of TOPEX and Jason1 the de-correlation is clearly degraded. As a meas-ure for this the correlation between S2 and the mean is expressed by the geometric mean rS2,m, defined by

( )2 22, 2 , 2 ,

12S m S c m S s mr = r +r (2)

where r2S2c,m and r2

S2s,m are the correlation coefficients of cosine- and sine- coefficients of S2 and mean sea level respectively. Figure 3 shows the geographical distribu-tion of rS2,m – reproducing the diamond shaped areas in-side TOPEX and Jason1 tracks. The maximum correla-tion remains below 0.5 and does not seriously derogate the separation of the two constituents. The same pattern appears in corresponding plots (not shown here) for the correlations between K2 and the semi-annual signal and between P1 and K1.

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Figure2. Mean correlation matrix for the solve-for pa-rameters of the harmonic analysis.

5. ResulTs

Figures 4, 5, and 6 show the residual amplitude and phases of the tidal constituents solved for in the area of the North-West European Shelf. As the sea level anoma-lies were already corrected by the FES2004 model the results indicate significant local improvements over the global FES2004 ocean tide model. The most outstand-ing improvements are obtained for M2 and M4 with re-sidual amplitudes of up to 10 cm distributed over several

patches. The correlation analysis shows that neither M2 nor M4 is correlated with any other tidal constituent. S2 and K2 show single patches with similar amplitude. The residual amplitudes for the diurnal constituents remain below 5 cm and are distributed over one or two patches only. None of the amplitude pattern shows any depend-ency on the sub-satellite ground tracks.

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Figure 4. Residual tides of major semidiurnal constituents (M2, S2, K2, N2). The panels of the upper row show residual amplitudes (cm), the lower row shows the corresponding phases (degree).

Figure 3. Geographical distribution of the correlation between S2 constituent and the mean sea level. 45˚

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Figure 5. Residual tides of the shallow water constituent M4. The panel on top show residual amplitudes (cm), the lower panel shows the corresponding phases (degree).

6. ValidaTion

To validate the analysis results historical bottom pres-sure records for shallow water sites were used as kind-ly provided by the British Oceanographic Data Centre (BODC). In order to measure the improvements over the FES2004 ocean tide model, the time series of the bottom pressure records were first corrected by FES2004. In a second step the residual tide corrections of this analysis was subtracted. Figure 7 shows these two reduction steps for a sample time series of a bottom pressure gauge in the Irish See (marked by a star in Figure 8). Figure 8 opposes for all bottom pressure gauges the sea level vari-ability before applying the second step with the reduction in variance achieved after applying the second step. The percentage gain in variance is expressed by the formula

2 2

22 100%f f ta

f

+s − sDs = ⋅

s (3)

where s2f is the variance of sea surface heights corrected

with FES2004 and s2f+ta is the variance of sea surface

heights additionally corrected with results of tidal analy-sis. The significant variance reduction could be achieved in the areas with the largest residual amplitudes.

7. ConClusions

• ENVISAT and ERS altimetry data contribute essen-tially to ocean tide analysis – although the sun-syn-chronous orbits and the repeat cycle are sub-optimal.

• The case study for the North-West European shelf

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Figure 6. Residual tides of major diurnal constituents (Q1, P1, K1, O1). The top row shows residual amplitudes (cm), the lower row shows the corresponding phases (degree). Note, the amplitude colour scale differs from Fig. 4 and 5.

September, 1972

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Figure 7. Red: Water level time series (cm) of a bottom pressure gauge in the Irish See. Green: The time series after sub-tracting FES2004 tidal elevations. Blue: Time series after subtracting both, FES2004 and the residual tide corrections.

proves by comparison with independent bottom pres-sure records that sufficient long time series of multi-mission altimetry data can significantly improve state-of-the-art global ocean tide models.

• The combination of sea level data from different altim-eter missions allows a separation of tidal constituents with nearly always negligible correlation coefficients.

• Maximum correlation remain below 0.5 and appear between S2 and the mean value, K2 and the annual period, and P1 and K1 – a consequence of the sun-synchronous orbits of the ESA missions.

RefeRenCes

[1] Andersen O.B., 1999, Shallow water tides in the northwest European shelf region from TOPEX/POSEIDON altimetry. J. Geophys. Res., 104(C4): 7729-7741

[2] Bosch, W. 2006.Satellite Altimetry - Multi-Mission Cross Calibration. In: P. Tregoning and Ch. Rizos (Eds): Dynamic Planet – Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools. IAG Symposium, Vol. 130, 51-56, Springer, Berlin

[3] Chambers, D. P., S. A. Hayes, J. C. Ries, and T. J. Urban., 2003, New TOPEX sea state bias models and their effect on global mean sea level, J. Geophys. Res., 108(C10), 3305.

[4] Desai, S. personal communication

[5] Hernandez, F. and P. Schaeffer, 2000, Altimetric Mean Sea Surfaces and GravityAnomaly maps inter- comparisons AVI-NT-011-5242-CLS, 48 pp. CLS Ramonville St Agne.

[6] Lyard, F., F. Lefèvre, T. Letellier and O. Francis, 2006, Modelling the global ocean tides: a modern insight from FES2004. Ocean Dynamics, 56, 394-415.

[7] Munk, W., and Cartwright, D. (1966) “Tidal spectroscopy and prediction.” Philosophical Transactions of the Royal Society of London, 259, 553–581.

[8] R. Scharroo and P. N. A. M. Visser, 1998, Precise orbit determination and gravity field improvement for the ERS satellites. J. Geophys. Res., 103, C4, 8113-8127

[9] Scharroo, R., J. Lillibridge, and W.H.F. Smith, 2004, Cross-calibration and long-term monitoring of the Microwave Radiometers of ERS, Topex, GFO, Jason-1 and Envisat. Marine Geodesy, 97.

[10] Schrama E., R. Scharroo, and M. Naeije, 2000, Radar Altimeter Database System (RADS): Towards a generic multi-satellite altimeter database system, USP-2 report 00-11, BCRS/SRON, Delft, The Netherlands

[11] Smith A.J.E.,1999, Application of satellite altimetry for global ocean tide modelling. PhD. Delft Institute for Earth-Oriented Space Research. Delft University of technology. Faculty of Aerospace Engineering.

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Figure 8. Left: Standard deviation (cm) of sea level variability for the BODC bottom pressure gauges after subtracting the FES2004 tidal elevations. Right: The reduction in variance (in percent) achieved by applying the residual tide cor-rections.