Thoughts on Sun-Synchronous* Altimetry

R. D. RayNASA Goddard Space Flight Center

14 March 2007

* Yes, a sun-synchronous wide swath is still sun-synch!

Whatʼs so bad about sun-synchronous altimetry?For many applications, absolutely nothing. For these, T/P “solved” the tide problem.But for others....

I. It prevents serious tide studies.

II. It maps “diurnal” errors to undesirable periods.

Shallow-water tides; tides polewards of 66°; open-ocean internal tides

— into the mean sea surface.— into long (climate-like) periods.— into the annual and semi-annual cycles.

Question: How large are the effects in (II.) ?Answer: Generally ~1 cm or smaller at basin scales (excl. polar seas); ~ 5 cm at short scales in shallow water. ~ 5 cm—but usually smaller—at internal-tide scales; BUT.....

Itʼs not just tides! ... diurnal ionospheric delay errorsdiurnal pressure oscillations (IB / dry trop)diurnal rain contamination (ITCZ)thermal effects on spacecraft & tracking stationsatm. drag / radiation pressure errors

Tidal Alias Periods for a Sun-Synch. Altimeter

Speed Amplitude @ Alias PeriodTide (°/h) CresCity (cm) (days)P1 14.959 12 365S1 15.000 1 ∞K1 15.041 39 365T2 29.959 1 365S2 30.000 18 ∞K2 30.082 5 183

Lunar-tide aliases depend on orbit.Solar-tide aliases are always:

Note: 0.041°/h = 1 cpy

Attempts to use sun-synch altimetry for tides reviewed in:Ray, R. D., “Tidal analysis experiments with sun-synchronous

satellite altimeter data,” Journal of Geodesy, in press, 2006.

– data analysis (Chabert d’Hières and Le Provost 1979;Le Provost 1991; Ponchaut et al. 2001).

Owing to the difficult challenges posed by the nonlinear-ities in the equations of motion, Le Provost’s analyticalanalyses proved to be of crucial importance to hissubsequent successful physical and numerical modeling.In this paper, we address shallow-water tides in the

context of another of Le Provost’s longstanding interests—the tidal analysis of satellite altimeter data and theassimilation of altimetry into numerical tidal models (LeProvost et al. 1994; Le Provost 2001). Tidal data assim-ilation is now routinely applied at both global and regionalscales but it was not previously used for nonlinear tides.Considering that numerical modeling is intrinsically morechallenging for nonlinear tides than for linear tides—forexample, spatial scales are shorter and the forcing is notknown a priori—the use of data assimilation appears quiteattractive at first sight. Unfortunately, nonlinear tidespresent their own special difficulties for assimilation. Thework here represents a first step in this direction. The readerwill discover that the problem is by no means solved.

After a brief review of nonlinear shallow-water tides,this paper focuses on mapping the M4 barotropic tide overthe northwest European Shelf, the region of Le Provost’skeen interest for so many years. M4 is the primary overtideof M2 and it reaches an amplitude exceeding 30 cm inseveral parts of the English Channel. Our focus is primarilyon tidal elevations rather than currents and the models weemploy are exclusively 2-D barotropic. The use ofgeneralized inverse methods with 3-D tidal models is apromising approach for more detailed investigations oftidal currents and internal tides, but such work is in itsinfancy.

2 Shallow-water tides

In coastal regions, the tidal range is generally larger than inthe open ocean and the tidal waves are considerably morecomplex. The patterns of the tidal waves shorten as thewave speed reduces. Because long waves propagate aspðgHÞ whereH is the water depth and g is normal gravity,tidal wavelengths shorten dramatically in shallows, for

Table 1 Principal nonlinear tides on the Northwest European Shelf

Tide Origin Doodson number Frequency! (�/hr)

Alias period (days) Amplitude (cm)at Dover, UKT/P ERS GFO

Long periodMSf M2�S2 073.555 1.0159 30.2 94.5 110.2 2.2SemidiurnalMNS2 M2þN2�S2 227.655 27.4238 77.3 3,166.1 98.7 2.62MS2 M2þM2�S2 237.555 27.9682 20.3 135.1 81.8 5.8SNM2 S2þN2�M2 263.655 29.4556 21.0 129.5 35.4 4.02MN2 M2þM2�N2 265.455 29.5285 20.6 349.2 39.2 7.0MSN2 M2þS2�N2 283.455 30.5444 51.9 129.5 60.8 3.62SM2 S2þS2�M2 291.555 31.0159 19.9 94.5 66.7 4.2TerdiurnalMK3 M2þK1 365.555 44.0252 96.8 127.5 392.7 1.5FourthdiurnalMN4 M2þN2 445.655 57.4238 244.5 3,166.1 62.3 9.4M4 M2þM2 455.555 57.9682 31.1 135.1 158.6 25.5ML4 M2þL2 465.455 58.5126 30.9 74.4 34.9 3.0MS4 M2þS2 473.555 58.9841 1,083.9 94.5 361.0 16.4MK4 M2þK2 475.555 59.0662 219.8 195.8 121.3 5.0S4 S2þS2 491.555 60.0000 29.4 1 84.4 1.6Sixth diurnal2MN6 M2þM2þN2 645.655 86.4079 83.3 91.7 77.5 3.8M6 M2þM2þM2 655.555 86.9523 20.7 314.5 105.7 6.5MSN6 M2þS2þN2 663.655 87.4238 47.4 3,166.1 45.5 1.72MS6 M2þM2þS2 673.555 87.9682 65.9 135.1 2,608.1 6.52MK6 M2þM2þK2 675.555 88.0503 48.4 77.6 196.4 1.82SM6 S2þS2þM2 691.555 88.9841 55.7 94.5 115.0 1.4EighthdiurnalM8 M2þM2þM2þM2 855.555 115.9364 27.4 72.6 79.3 2.03MS8 M2þM2þM2þS2 873.555 116.9523 32.0 314.5 282.7 2.9

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Nonlinear Shallow-Water Tides

Sun-Synch Aliasing of “Diurnal” Errors

Maps into Cause Example (tide) Example (non-tide)1. Mean Dʼl errors constant

in time S2 errors Diurnal pressures in IB (not smart)

2. Long periods Dʼl errors vary slowly in time

S2 temporal variationsS2 internal tides

Errors in iono correction at 11-yr solar cycle

3. Annual / semi Dʼl errors vary with season P1, K1, K2 errors Errors in in iono correction with

solar declination

Consequences

1. Errors in absolute dynamic topography & currents (Will grow in importance; GOCE) Constant SSH discrepancies at x-over points Location-dependent biases between missions (cf. temperatures from microwave sounders)

2. Diurnal errors look like climate signals

3. Corrupts studies of seasonal cycle

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Mean Jason – Topex Sea-surface Height Differencesas Function of Local Time

Jason-1 cal/val period Feb–July 2002

NOTE: Results are independent of tide corrections!

Could cause be related to spacecraft thermal effects?

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Mean Jason – Topex SSH Differences

as Function of Local Time

In T/P and Jason, these errors map mostly into ~ 60-day periods.Sun-synch maps them into long periods.

Jason - Topex Ionosphere ComparisonsJason-1 Cal-Val Campaign

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Joint PDFMean Jason-Topex ionosphere correction

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Differences in corrections are seen to be small.Are differences in true ionospheric delays small?

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Jason-1 SLR Residuals vs. Local TimeJason cycles 1 – 165

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Amplitude of S2 Ocean Tide

TPXO7 formal error

Tide Errors: S2FES04 / GOT00 differences

FES04 / TPXO7 differences GOT4 / TPXO7 differences

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Note: A sun-synchronous altimeter cannot fix these errors.

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Temporal Variability of Tidal “Constants”

Year-to-year variability from analyses of hourly tide-gauge data

Neah BaySitkaTarawa

Yearly amplitudes in cm.

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Secular Trends in S2

Gulf of Maine

In contrast to decreasing S2, M2 is increasing—rapidly (13 cm/century at Eastport)

R Ray, Continental Shelf Research, 26, 422, 2006.

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Correlation of MSL and M2 – Hilo, Hawaii

Monthly M2 Amplitude in black

Mitchum & Chiswell, ����������������� 2000.Ray & Mitchum, ���������������� 1997.

Monthly MSL in red

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Variability of Yearly S2 Estimates40 tide-gauge stations

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BarometerECMWFGuam

Atmospheric Tide Errors

Dry_trop (mm) = ~2.3 Pressure (mb)

Mearim River30 km inland

Amazon River

RIVER TIDES

Washington, DC6th & Water St, SW

Tidal Bores in Brazil

SUMMARY1. Constant diurnal errors -> mean sea surface

2. Slowly varying diurnal errors -> false climate-like signals

3. Seasonal modulations of diurnal errors -> contaminates true seasonal cycle.

The causes of these errors are unlikely to be “fixed” by using the sun-synch data — sun-synch altimetry wonʼt fix tide errors.

Source of #2 Errors

— Variability of S2 suface tide e.g., Gulf of Maine — few cm over 50 yrs— Variability of S2 internal tides ~1 cm over 100 km (mode 1) ~few mm over < 50 km (mode 2+)— Errors in ionospheric corrections unknown -- strong 1-y, 11-y periodicity

— Variability of S1 air tide (IB, dry trop) ~1 cm, “tide-like” spatial pattern all temporal scales, inc. El Nino— Radiational / drag errors in POD << 1 cm at 1 cpr— Unknown errors e.g. Topex-Jason bias vs. local time