Ocean Tides from TOPEX/Poseidon Sea Level Data

APRIL 2001 CHERNIAWSKY ET AL. 649

J. Y. CHERNIAWSKY,M.G.G.FOREMAN,W.R.CRAWFORD, AND R. F. HENRY Fisheries and Oceans Canada, Institute of Ocean Sciences, Sidney, British Columbia, Canada

(Manuscript received 15 June 1999, in ®nal form 18 July 2000)

ABSTRACT Tidal constants are computed from TOPEX/Poseidon sea level data in the northeast Paci®c Ocean for the period of September 1992±December 1997. The method used is harmonic analysis. Tidal constituents are also calculated at track crossover locations, where twice as many observations are available. The crossover constituents are compared with nearest alongtrack constituents and with constituents at 57 pelagic sea level stations in the northeast Paci®c. Examples of small-scale features in the alongtrack constituents are presented. These include a strong evidence of internal M2 tides propagating away from the Aleutian Islands and of K1 and O1 shelf waves in the northern Gulf of Alaska. Good agreement was found between the shelf waves observed in altimetry data and those calculated by a numerical tidal model.

1. Introduction Satellite altimetry data provide a global view of var- Ocean sea level data from satellite altimetry sensors iability of tidal amplitudes, including smaller-scale fea- are used increasingly to describe seasonal and inter- tures such as internal tides (Ray and Mitchum 1996, annual changes in sea level and ocean circulation, during 1997; Kantha and Tierney 1997), or tides in coastal episodes of El NinÄo, or arising from longer-period cli- water, including tidally driven diurnal shelf waves mate change. TOPEX/Poseidon (T/P) is the ®rst global (Foreman et al. 1998, hereafter FCCHT). Barotropic system of satellite altimetry that was explicitly designed tidal models may not simulate baroclinic shelf waves to study ocean dynamics by accurately measuring sea accurately (Foreman and Thomson 1997; Cummins et surface height relative to the center of the earth (Fu et al. 2000) and are not capable of producing internal tides. al. 1996). On the other hand, internal tides are simulated in high- In order to decipher the low-amplitude (of the order resolution baroclinic models (Cummins and Oey 1997; of several centimeters) changes in sea level, it is ®rst Crawford et al. 1998; Kang et al. 2000; Cummins et al. necessary to apply a full suite of geophysical corrections 2001). to the altimetry data. Ocean tides are by far the largest Global models do a good job of detiding the T/P of these corrections, as they account for up to 80% of altimeter data in deep ocean but, because of coarse res- the sea surface height variability of the world ocean. olution or inaccurate parameterization of unresolved Thus considerable effort is directed toward removal of physical processes, small errors in deep ocean tides tend tides from the altimetry signal, either using large-scale to be ampli®ed in shallow seas. Thus most of the global hydrodynamic models (e.g., Le Provost et al. 1994), or models are not adequate near the coastlines (Andersen satellite-altimeter-based empirical models (Schrama and et al. 1995; Shum et al. 1997). There, the tides can be Ray 1994; Ray et al. 1994; Desai and Wahr 1995). more effectively removed using constituents from lim- Tidal constants derived from ocean altimetry are also ited-area high-resolution tidal models (e.g., Foreman et used to construct global inverse solutions to ®t ocean al. 1998), or from local analysis of long time series of hydrodynamics and satellite data (Egbert et al. 1994), altimeter sea level. for direct assimilation in numerical tidal models (Le As the time period of observations increases, har- Provost et al. 1998), to compute seasonal geostrophic monic analysis of T/P data becomes more reliable, as currents using coastal circulation models (Foreman et there is less contamination of detided sea level from al. 1998), or for specifying tidal elevations on open aliased tidal constituents and the accuracy of the cal- boundaries of regional models (Foreman et al. 2000). culated coef®cients improves, especially for the long- period ocean tides, such as MM and MF (Desai and Wahr 1997). Nevertheless, it is still desirable to know, Corresponding author address: Dr. Josef Cherniawsky, Institute of at each T/P data location, the expected errors in the Ocean Sciences P.O. Box 6000, Sidney, BC V8L 4B2, Canada. calculated constituents and the level of covariance be- E-mail: [email protected] tween them due to aliasing. These local estimates are

᭧ 2001 American Meteorological Society

Unauthenticated | Downloaded 09/30/21 01:58 PM UTC 650 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 provided by covariance matrices in the solution pro- It is not removed by our current detiding procedure cedure presented below. The accuracy of the constitu- (though we intend to do so in a future), introducing a ents and of the detided signal is improved by using small error in the derived annual harmonic (SA). The selective data trimming before the analysis, and by plac- 14-month period has no obvious tidal aliases. Thus the ing time-varying nodal corrections inside the solution pole tide adds somewhat to the level of noise in the matrix. detided signal. We describe the T/P altimeter data in the next section, The domain of the study is in the northeast Paci®c while section 3 gives an outline of the analysis method. Ocean, extending from 30Њ to 62ЊN and from 180Њ to An example of detided alongtrack sea level time series the west coast of North America (116ЊW). The T/P al- is in section 4. Aliasing is reviewed in section 5, in the timeter tracks in this area are shown in Fig. 1. context of the constituent covariance matrix. Results from analyses at track crossovers are described in section 6. Examples of small-scale features in tidal con- 3. Harmonic analysis stituents are shown in section 7. These include clear The method adopted here is harmonic analysis (e.g., evidence of internal tides off the Aleutian Islands and Godin 1972; Pugh 1987; Foreman 1977, hereafter F77) new observations of diurnal shelf waves near Kodiak of the T/P altimeter data at tidal frequencies, that is, a Island, Alaska. Comparison with tidal constituents from least squares ®t for tidal constituent amplitudes and pelagic sites is in section 8. phases. Unlike the response method (Munk and Cart- wright 1966; Cartwright and Ray 1990), harmonic anal- 2. TOPEX/Poseidon sea level data ysis computes amplitudes and phases of each constituent independently, with no assumption of smooth admit- TOPEX/Poseidon sea level data in the current study tance across each tidal frequency band. This may be were obtained from the National Aeronautics and Space considered a drawback, especially for the weak con- Administration (NASA) Ocean Altimeter Path®nder stituents, but is an advantage for observations of small- Project (courtesy of Richard Ray and Brian Beckley) at er-scale variability, such as internal tides, or near-ocean the Goddard Space Flight Center (GSFC). These include coastlines where nonlinear interactions, resonances, and cycles 1±194, from 22 September 1992 to 31 December shallow water tides are important and the assumption 1997, or about 5.3 yr. of smooth admittance may not be valid (Tierney et al. GSFC applied standard geophysical corrections, ex- 1998). cept for the ocean tides, ocean load tides, and pole tides. Ray (1998) concludes that for speci®c sites with These corrections include computation of precise sat- strong signals at aliased frequencies (e.g., K1 and SSA; ellite ephemerides, solid-earth body tide, cross-track ge- section 5) the response method is preferable, though oid correction, inverted barometer loading, and media judging from his Table 4, there is little difference be- (dry and wet atmosphere, ionospheric refraction) and tween the two methods in the detided energy in the low instrument (sea state bias, calibration adjustments) cor- frequency (0±18.4 cycles yrϪ1) band. We use 21 con- rections. Explanations of their algorithms can be found stituents, compared to 11 in Ray (1998), and derive at the GSFC homepage (http://neptune.gsfc.nasa.gov/ speci®c error estimates for each constituent. Perhaps we ocean.html) or in Koblinsky et al. (1999). could state that for shorter periods, the response method The standard global tide models work quite well in is preferable, but for the 5-yr and longer T/P time series, the deep ocean (e.g., Ray et al. 1994; Le Provost et al. harmonic analysis offers more ¯exibility in the calcu- 1994), but are not as good on continental shelves (An- lation of individual constituent amplitudes. Harmonic dersen et al. 1995; Shum et al. 1997), where the relevant analysis is therefore more robust at locations where ad- length scales are much shorter and the dominant pro- mittance is not a smooth function of frequency, for ex- cesses, such as resonance and dissipation, may not be ample, where internal tides, shelf waves, or shallow- resolved by coarse-resolution grids. Thus, in order to water tides are present. The effect of aliasing on com- use T/P data near coastlines it is necessary to derive puted constituents can be estimated by comparing along- better tidal constants, either from high-resolution mod- track complex amplitudes to their values at a nearby els, as in Foreman et al. (1998), or from local tidal crossover location (section 6). analysis of the altimeter data, as is done here. We write the observations at time ti as hi ϭ h(ti)(i The presence of ocean load tides is of no consequence ϭ 1,...,N), with estimated errors ␴ , and seek a least- when detiding altimetry with harmonic analysis. It is i squares ®t for the tidal constituent amplitudes Ak and only necessary to add ocean load tides when comparing phases ␾ at select frequencies ␻ , k ϭ 1,...,K (Table the analyzed tidal constituents with those obtained from k k 1). We rewrite Ak and ␾ k in terms of the cosine and in situ data, or from hydrodynamic models. In this case sine coef®cients C and S , A ϭ (CS22ϩ )1/2 and ␾ we add load tides calculated from the global model of k k k kk k ϭ arctan(Sk/Ck) (F77), and solve the overdetermined Schrama and Ray (1994; courtesy of Richard Ray). system of equations The pole tide has two dominant periods, annual and 14-month, and is relatively small (of the order of 1 cm). D ´ x ϭ b, (1)

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FIG. 1. Domain of the study in the northeast Paci®c Ocean showing bathymetric contours (every 2000 m, plus 200 m) and TOPEX/Posedon tracks, ascending toward northeast and descending from northwest. Results from tracks 27, 48, 52, 62, 116, and 117 (thick lines) and comparison to pelagic station (solid squares) data are discussed below. by minimizing a merit function 1987; Tierney et al. 1998). This is ®ne if this period is of the order of, or less than a year. But T/P observations ␹ 2 ϭ |D ´ x Ϫ b| 2, (2) now span more than ®ve years, during which time the where the unknown vector x ϭ {C 0, C1, S1,...,CK, relative error due to using the central time value may SK}. The right-hand side consists of the sea level ob- exceed several percent. For example, between 1992 and servations, b ϭ h, while D is a design matrix with M 1997, the amplitude modulation factor f(t) varies from ϭ 2K ϩ 1 columns and N rows, M Ͻ N (e.g., Press et 1.015 to 0.883 for K1, from 1.024 to 0.808 for O1, and al. 1992, hereafter P92). Each row i of D is made of from 1.000 to 1.038 for M2 (Schureman 1958). either cosine (even columns), or sine (odd columns) We therefore modify (3) by inserting slowly varying basis functions, calculated at a time of observation ti nodal corrections, f ij ϭ f j(ti) and uij ϭ uj(ti), directly into the design matrix cos(␻ t ), j ϭ 2k Ϫ 1 D ϭ ji (3) ij cos( t u ), j 2k 1 Άsin(␻jit ), j ϭ 2k ␻jiϩ ij ϭ Ϫ Dijϭ f ij (4) for k ϭ 1,...,K. As N gets larger, the columns of D Άsin(␻jit ϩ u ij), j ϭ 2k. become nearly orthogonal and more of the Rayleigh The computation of each f(t) and u(t) is as in Godin criteria are satis®ed (see the section on aliasing below). (1972). The columns of D from (4) are somewhat less Di0 ϭ 1, for the mean sea level C0. We use the common orthogonal than in (3). But f(t) and u(t) are slowly notation Z0 for C 0, even though (because of the dif®- varying functions of time and f(t) is close to unity, so culty in calculating the exact shape of the geoid) the this is of little concern. mean sea level is not as well determined in the deep We solve (1), subject to (2), using the singular value ocean, compared to that measured with coastal tide decomposition (SVD) method (Golub and Van Loan gauges. 1983; P92). The N ϫ M design matrix is written as D In conventional harmonic analysis, it is customary to ϭ U ´ W ´ VT, where U is an N ϫ M column-orthogonal calculate a nodal correction to account for close minor matrix, V is an M ϫ M orthogonal matrix and W ϭ constituents [also called ``satellite modulation'' as in diag(w )isanM ϫ M diagonal matrix with positive or Foreman et al. (1995)] after solving for major constit- j zero elements wj (the singular values). The solution vec- uent amplitudes and phases. The central time of the tor is then observations is often chosen to specify a single correc- T tion for the time period in question (e.g., F77; Pugh x ϭ V ´ [diag(1/wj)]´(U ´ b), (5)

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TABLE 1. Tidal constituents frequency (cycles day Ϫ1), mean, standard deviation, minimum and maximum of amplitudes (cm), from harmonic analysis of 194 cycles of T/P sea level data at 26 449 locations in the northeast Paci®c Ocean. Constituents Frequency Mean Std dev Min Max 1 Z0 0.00000 Ϫ15.2 2.45 Ϫ26.1 14.2 2 SA 0.00274 5.10 1.66 0.043 15.4 3 SSA 0.00548 1.27 0.86 0.020 10.6 4 MM 0.03629 0.68 0.45 0.003 3.31 5 MF 0.07320 1.28 0.67 0.008 5.74 6 Q1 0.89324 3.55 1.24 0.059 10.2 7 O1 0.92954 19.3 5.96 0.307 53.8 8 NO1 0.96645 1.41 0.51 0.080 6.76 9 P1 0.99726 9.78 3.14 0.472 28.3 10 S1 1.00000 1.75 1.14 0.003 13.5 11 K1 1.00274 30.4 9.70 2.813 89.0 12 J1 1.03903 1.89 0.65 0.017 5.64 13 OO1 1.07594 1.25 0.60 0.029 6.23 14 2N2 1.85969 1.33 0.83 0.004 6.54 15 MU2 1.86455 1.30 0.79 0.001 12.2 16 N2 1.89598 9.62 5.89 0.494 49.6 17 NU2 1.90084 1.92 1.19 0.030 9.94 18 M2 1.93227 43.9 28.9 4.890 225.2 19 L2 1.96857 1.06 0.65 0.003 9.28 20 T2 1.99726 1.04 0.59 0.003 5.03 21 S2 2.00000 14.5 9.63 0.203 83.7 22 K2 2.00548 4.04 2.43 0.024 23.7

2 and variance estimates ␴ (xj) are given by diagonal el- anomaly was diminished (long-dashed line) when using ements of the covariance matrix all input data, and some of its energy (as well as energy

M from peaks in the end of 1992 and in late summer of VVjl kl cov(x , x ) ϭ ␴ 2(␩) . (6) 1994) was projected onto the annual harmonic, produc- jk ͸ 2 lϭ1 ΂΃wl ing excessive negative anomalies (ringing) in summers We replaced individual observation error estimates of 1993, 1995, and 1996. The detided signal, when using only data up to April 1997 (short-dashed line), is not ␴ (h ) with a standard deviation ␴(␩), where ␴ 2(␩) ϭ i i signi®cantly different from the second curve (solid line). ⌺N ␩ 2/(N Ϫ M) is computed from the detided signal iϭ1 i It is worthwhile noting here that this anomaly is a local ␩ ϭ ␩(t ) i i signature of a strong anticyclonic eddy, which was trav- K eling west-southwest along the Aleutian Trench during ␩ ϭ h Ϫ C Ϫ fAcos[␻ t Ϫ ␾ ϩ u ]. (7) ii 0 ͸ ik k k i k ik 1995±97 (Crawford et al. 2000). kϭ1 The use of ␴(x ) as an error estimate for x assumes j j 4. Coastal example a normal distribution for ␩i (P92). But this is often not the case, especially when the altimeter track crosses Figure 3 is a time-distance (HovmoÈller) plot of de- steep fronts, or large eddies. To get around this problem, tided T/P sea level for descending track number 52 (see we repeat the above calculation using a trimmed dataset Fig. 1), including the mean sea level (Z0) and the sea- that excludes from b the observations hi that give ␩i sonal cycle (SA and SSA). The color density scale is outside of Ϯ2␴(␩). This reduces the size of b by less shown below the plot (in mm), while the two curves in than 5%. the top panel give a time series of mean and rms de- In fact, when a strong nontidal anomaly is present in viation (in cm) of sea level at each alongtrack location. the input data, part of it is projected onto the eigen- A 5-point median alongtrack ®lter was applied to the functions of D (the tidal constituents), thus also reducing detided data before plotting, in order to remove small- ␴ 2(␩). Thus selective trimming of b helps to restore scale noise. Also, a time ®lter [one application each of much of the original anomaly. a (1, 2, 1) ®lter and of a 3-point median ®lter, while An example of a strong anomaly in the detided sea excluding missing data] was used to smooth over in- level is shown in Fig. 2 for a single location in the tercycle synoptic variability. Alaskan Stream. The ®rst curve (long-dashed line) was This track extends through the Gulf of Alaska and produced using all input data in b, the second (solid crosses Queen Charlotte and Vancouver Islands along line) shows the detided signal after trimming was per- the British Columbia coast (Fig. 1). It passes through formed, while the third curve (short-dashed line) was northern Hecate Strait at about 53.5ЊN, where the stron- produced when the harmonic analysis was performed gest semidiurnal tides (M2 amplitudes approach 2 m) using only data up to April 1997. Clearly, the original along this coast are observed. Despite this, very little,

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FIG. 2. Detided sea level from T/P altimeter data at 52.136ЊN, 167.747ЊW: long-dashed line was produced using all input data in harmonic analysis, solid line, when Ϯ2␴(␩) trimming was performed, short-dashed line, without this trimming, but using only data up to Apr 1997. A weak time ®lter was used to suppress intercycle variability. if any, aliased signal with a near-60-day period (see next signals whose amplitudes are of interest. When required, section) is visible in Fig. 3. It can be compared to an SA and SSA harmonics are added back to ␩. analogous plot along track 52 (up to cycle 154, Fig. 4) It is well known that certain tidal constituents may that was decided with constituents from a global GSFC be aliased to other constituents because the repeat cycle model. While there is little difference between the two of the T/P orbit, ⌬t ϭ 9.9156 days, is longer than diurnal plots in the deep ocean, a clear near-60 day signal is and semidiurnal tidal periods. Indeed, this ⌬t was cho- present in Hecate Strait in Fig. 4. (In fairness to the sen by the T/P design team to minimize aliasing of major GSFC model, we must add that it was designed for constituents with one another (Parke et al. 1987), but it optimum performance in the deep ocean and, because was not possible to do so for all the constituents. In of spatial smoothing, is not expected to do well close theory, for each ␯k, there is an in®nite set of possible to the ocean margins.) Virtually all global tidal models aliasing frequencies have similar errors near the coastlines (e.g., Shum et al. j 1997) and are obviously not adequate in the shallow ␯aiϭϮ␯ Ϯ ,(1Յ i Յ K, j Ն 0) (8) waters. Thus locations where there is signi®cant dis- ⌬t agreement between tidal models are ¯agged in the (e.g., Parke et al. 1987; Yanagi et al. 1997). This leads GSFC-detided global sea level data (see explanation of to a revised Rayleigh criterion the T/P ¯ag word in http://neptune.gsfc.nasa.gov/ocean. html). j R (T) ϭ ␯ ϪϮ␯ Ϯ T Ͼ 1, Unfortunately, because of the aliasing (Ray 1998; also kij k΂΃ i ⌬t next section), we may have removed along with the tides Η Η any geophysical signal with a period of near 60 days. (1 Յ i Յ K, j Ն 0), (9) We do not think this is a serious problem, since, as far that can be used to infer a minimum time period Tmin as we know, such signals are not commonly observed required to resolve any two tidal frequency components. in this part of the world. Table 2 lists candidate alias pairs from the analyzed

constituents for Tmin Ͼ 1 yr. In practice, this time period 5. Aliasing and noise may need to be somewhat longer than Tmin if input data are missing or corrupted by excessive noise. For ex-

The tidal frequencies chosen for our analysis, ␻k ϭ ample, for the K1±SSA pair, where SSA has a broad 2␲␯k (k ϭ 1,..., K), were selected out of 45 astro- spectral peak due to variations in the weather, the time nomical tidal constituents (F77), based on decreasing period may be much longer. magnitudes within each of the main constituent groups: Figures 5a,b show two examples of a covariance ma- low-frequency, diurnal, and semidiurnal. We selected K trix and of correlation coef®ents, rjk ϭ cov(xj, xk)/ ϭ 22 constituents (Table 1), including Z0. Some of these [␴(xj)␴(xk)], at two locations alongtrack 52. The number have a predominantly nontidal character, for example, of available data is N ϭ 178 for Fig. 5a and N ϭ 138 the annual SA, semiannual SSA, solar diurnal S1. But for Fig. 5b, out of the possible 194. The effect of missing they were included because they are reasonably regular data is apparent for the latter location (in Queen Char-

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FIG. 3. Detided sea level (in mm, including Z0, SA, and SSA) after harmonic analysis of descending track 52, Sep 1992±Dec 1997 (up to cycle 194). This track crosses two islands (shown as white bars) off the British Columbia coast, Graham Island (at about 54ЊN) and Vancouver Island (near 50ЊN; see Fig. 1). Mean (purple curve; in cm) and standard deviation (blue) of sea level over the track are shown on top. lotte Sound, just north of Vancouver Island). In addition which is a particular case of Eq. (9) when one of the to the visible correlations for the expected alias pairs constituents in the pair is Z0 (␯i ϭ 0). Thus, when SSA±K1, P1±K2, and N2±T2, we also observe in Fig. altimeter data are analysed with inadequate removal of 5b enhanced correlations at most of the other alias pairs tidal signals, this aliasing introduces spurious features from Table 2, as well as at other pairs, though at reduced at the aliased frequencies levels. These are likely due to the data dropout rate of 29%. j Aliasing also occurs as each tidal frequency is folded ␯ ϭ ␯ Ϫ (10) about the T/P cycle Nyquist frequency (Ray 1998), kjΗΗ k ⌬t

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FIG. 4. As in Fig. 3, except detided with a global tidal model, Sep 1992±Nov 1996 (up to cycle 154; the color range was scaled as in Fig. 3).

(cf. Schlax and Chelton 1994). For example, for M2 and S1 amplitudes are about 2 cm each on track 52 Ϫ1 and j ϭ 19 one gets ␯kj ϭ 0.0161 day ,or1/␯kj ϭ 62 north of Vancouver Island, near 51.4ЊN. These values days. This type of aliasing shows up in Fig. 4. are only marginally higher than their error estimates at

Interconstituent aliasing is, obviously, not the only this location, ␴(xj) ഠ 1.2 cm (diagonal elements in Fig. factor that affects the accuracy and reliability of the 5a). On the other hand, the semidiurnal constituents computed constituents, especially the minor ones. Se- 2N2, MU2, L2, and T2 are also relatively small, ranging rious limitations are also imposed by nontidal signals between 1.5 and 3 cm at this location. However, their of interest to oceanographers (a.k.a. ``noise'' to a tidal ␴(xj) in Fig. 5a are smaller than 1 cm. This supports analyst), which are broadband in nature and thus can inclusion of the minor constituents in our analysis, es- be aliased to all of the constituents. For example, OO1 pecially since their amplitudes increase signi®cantly in

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TABLE 2. Candidate tidal alias pairs ( )ki, sign in front of the alias many coastal locations (maximum amplitudes are listed frequency, index Ϯj, R ϭ ␯ Ϫ ␯ |T (where ␯ ϭϮ␯ Ϯ j/⌬t, ␦t ϭ kij k a a i in Table 1). In general, the ␴(x ) values are quite similar 9.9156, T ϭ 1914 days), and minimum duration T (days). This j min to small-scale variations in the alongtrack amplitudes. table contains entries for Tmin Ͼ 1yr.

Alias pair Sign of ␯i ϮjRkij Tmin SSA±K1 ϩ Ϫ10 0.562 3404 6. Analyses at crossovers Q1±NU2 Ϫ ϩ10 1.747 1096 Q1±M2 ϩ Ϫ28 3.247 589 The aliasing problems noted in the previous section Q1±S2 Ϫ Ϫ11 3.247 384 are reduced at track crossover locations (Fig. 1), where O1±N2 ϩ Ϫ28 3.247 589 twice as many observations are available. Therefore an- O1±T2 ϩ Ϫ29 4.069 470 NO1±2N2 ϩ Ϫ28 4.431 432 alyzed constituents at these locations can be used for P1±K2 Ϫ ϩ10 0.562 3404 checking the nearest alongtrack values, and possibly MU2±L2 ϩ Ϫ38 1.500 1276 provide large-scale corrections. N2±T2 Ϫ ϩ1 0.822 2328 We interpolated raw alongtrack sea level data, sep- NU2±M2 ϩ Ϫ38 1.500 1276 NU2±S2 Ϫ ϩ1 3.233 592 arately for ascending and descending tracks, to each M2±S2 ϩ Ϫ39 1.733 1104 crossover location. These were combined into a single time series and harmonically analyzed, as explained in section 3. Figure 6 shows an example of the covariance matrix and correlation coef®cients at a crossover between tracks 99 and 52, at 51.428ЊN, 128.979ЊW (where

FIG. 5. Two examples of absolute values of the covariance matrix (below and including the diagonal, in mm2) and of the correlation coef®cient (above the diagonal) for track 52: (a) at 51.416ЊN, 128.964ЊW, where valid N ϭ 173, out of possible 194, and (b) at 50.887ЊN, 128.307ЊW (just north of Vancouver Island), where N ϭ 138.

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N ϭ 376). These can be compared to track 52 values constant, ranging between 0.64 and 0.66, except for in Figure 5a, at a location (where N ϭ 178) that is 1.6 SSA, K1, P1, and K2, for which it is between 0.58 and km away from the crossover. 0.60. Therefore, as expected, relative improvement at Diagonal elements on Fig. 6 are of the order of 50 crossovers is somewhat more pronounced for the aliased mm2, 3±4 times smaller than on Fig. 5a. Thus expected constituent pairs than for the other constituents. errors in the sine and cosine coef®cients are less than Similar relations also hold at other crossover loca- 8 mm, a factor of 2 smaller than on track 52 alone. What tions, except that the departure from the theoretical ratio is more important, the off-diagonal covariance values varies somewhat [sometimes exceeding 2Ϫ1/2], depend- are signi®cantly smaller than the diagonal values and ing on relative data quality at a crossover and at a nearest the corresponding |r| values on Fig. 6 are of the order, alongtrack location. or less than 0.2. We note that similar correlation matrices For some constituents the vector difference in am- were shown (though only for six constituents) in An- plitudes (column 6 in Table 3) exceeds the ␴(Ak) esti- dersen and Knudsen (1997), who used data from mul- mates (here, for SA, O1, L2, T2, and S2), which may tiple satellites to reduce the aliasing problem between be a sign that these particular estimates are slightly too K1 and SSA. low. But this table is for a single location. Examination

Table 3 lists amplitudes Ak and expected amplitude of analogous tables for varying crossover locations errors ␴(Ak) at the same crossover point and at a nearby shows such isolated differences for other constituents location on track 52. Also listed are magnitudes of vec- as well. It is quite likely that, despite data editing [ex- tor amplitude differences and ratios between expected cluding from vector b the observations hi that give ␩i amplitude errors. As there are almost twice as many outside of Ϯ2␴(␩)], there is still some projection of data at the crossover point, the latter ratios are close to nontidal anomalies onto the tidal constituents. Such in- the theoretical value of 2Ϫ1/2. Notably, this ratio is almost ¯uence of broadband geophysical ``noise'' is not in-

FIG.5.(Continued)

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FIG. 6. As on Fig. 5a, except for a crossover at 51.428ЊN, 128.979ЊW (tracks 99 and 52), where N ϭ 376. Note the change in the covariance scale from Fig. 5a.

cluded in the formal error estimates given by ␴(Ak). and south of Kodiak Island in the Gulf of Alaska, and However, any serial correlation in data (e.g., due to (d) around Aleutian Islands between 165ЊW and 180Њ. correlated errors in geophysical corrections) would A relatively clear example of the M2 internal tide make the matrix D not as well conditioned and increase signature south of the Aleutian Islands is shown on Fig. the values of some ␴(Ak). 7. The top two panels show M2 amplitude and phase (thin lines) on T/P descending track 117 (see Fig. 1), 7. Small-scale features including those from crossover analyses (marked as triangles). Third and fourth panels show amplitude and a. Internal tides propagating away from phase for residual M2 tide, computed, similar to Ray Aleutian Islands and Mitchum (1997), by taking a difference between It is interesting to see smaller-scale features in the the alongtrack complex amplitude and its polynomial alongtrack and crossover constituents. Indeed, internal ®t (bold line on Figs. 7a,b). The latter ®t is used here (especially M2) tides are observed in many tracks, with instead of the barotropic model values, which, in effect, prominent examples around Hawaii shown in Ray and gets around a problem of likely inclusion of regional Mitchum (1996, 1997), or Kang et al. (2000). model bias (large-scale difference) in the residual tide. Global distribution of semidiurnal internal tidal en- An 8th degree polynomial was used for this 2850-km ergy from T/P altimetry was presented by Kantha and track segment, based on a 400-km length scale. The less Tierney (1997). They identi®ed four areas of enhanced certain phase values for small residual amplitudes (less M2 internal tide energy in this region: (a) the already than 0.6 cm) are shown as dots in Fig. 7d. mentioned area northeast of the Hawaian Ridge, (b) At least four internal tide cycles are visible on Fig. around Mendocino Escarpment near California, (c) east 7 between 46.5ЊN and 52ЊN. Their wavelength projec-

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TABLE 3. Constituent amplitudes and expected errors (in cm) for a crossover at 51.428ЊN, 128.979ЊW (columns 2 and 3) and for a track 52 location that is 1.6 km away (columns 4 and 5). Last two columns give magnitudes of vector amplitude differences (in cm) and ratios between expected error values.

X X X X Constituents Ak ␴(Ak ) Ak ␴ (Ak)|Ak Ϫ Ak| ␴(Ak )/␴(Ak) Z0 Ϫ10.48 0.40 Ϫ9.84 0.61 0.64 0.656 SA 4.20 0.57 3.27 0.87 1.19 0.655 SSA 3.26 0.59 3.53 1.00 0.82 0.589 MM 0.82 0.52 0.94 0.79 0.11 0.655 MF 1.57 0.74 2.38 1.13 1.20 0.658 Q1 5.59 0.66 4.94 1.00 0.76 0.653 O1 26.05 0.64 24.10 0.98 2.13 0.655 NO1 2.13 0.52 1.73 0.78 0.91 0.662 P1 14.34 0.58 14.45 0.96 0.22 0.600 S1 2.05 0.79 3.16 1.21 1.17 0.653 K1 45.20 0.64 45.61 1.11 0.76 0.576 J1 3.69 0.66 3.88 1.00 0.32 0.659 OO1 1.92 0.80 2.93 1.21 1.14 0.664 2N2 3.08 0.59 3.15 0.90 0.07 0.655 MU2 2.54 0.56 2.79 0.85 0.39 0.656 N2 23.28 0.55 23.07 0.86 0.22 0.641 NU2 5.10 0.57 4.49 0.87 0.71 0.649 M2 113.80 0.56 114.05 0.87 0.35 0.645 L2 2.38 0.56 2.43 0.87 1.16 0.643 T2 1.59 0.57 1.83 0.87 1.59 0.652 S2 34.64 0.57 33.78 0.88 1.65 0.643 K2 9.19 0.71 10.27 1.18 1.21 0.598 tion onto this track is estimated to be about 180 km. amples of a shelf wave signature in the alongtrack and The internal tide propagates away from the Aleutian crossover K1 and O1 complex amplitudes, also com- Islands, as is evident from the residual phase increasing pared to the amplitudes from the Northeast Paci®c mod- with distance from the islands (Fig. 7d). Similar cal- el (FCCHT). In general, the alongtrack values (thin culation was done for an ascending track 75 (not lines) agree quite well with the crossover values (tri- shown), which crosses track 117 near 50ЊN at an angle angles) and with the Northeast Paci®c model-generated of about 72Њ and shows analogous wavelength projec- constituents (thick lines), except for some possible al- tion of about 200 km. Combining these in a simple iasing with geophysical noise (or between SSA and K1) trigonometric relation yields the compass direction of at certain parts of the track. propagation for the internal tide at this location to be The ®rst example (Fig. 8) shows that amplitudes and around 170ЊT. The estimated wavelength is about 153 phases of K1 and O1 change rapidly along ascending km, while phase speed is near 3.5 m sϪ1. Essentially track 48 near Vancouver Island. These changes are due the same value (160 km) was obtained for the wave- to shelf waves, generated at the mouth of Juan de Fuca length of a ®rst-mode baroclinic internal wave at M2 Strait, which at this location destructively interfere with frequency (P.Cummins 1999, personal communication), the longer-wavelength barotropic Kelvin waves. This based on the calculation of the vertical eigenmodes for observed behavior (Crawford and Thomson 1984) was a climatological density pro®le in this area. con®rmed in the high-resolution modeling studies of Examination of residual phase plots from other neigh- Foreman and Thomson (1997), Cummins et al. (2000), boring, descending and ascending tracks (Cummins et and FCCHT. al. 2001) reveals the presence of similar southward- The shelf near Juan de Fuca is narrower and shorter propagating wave packets at varying distances from the than that off Cook Inlet in the northern Gulf of Alaska. islands, all the way south to 41ЊN. This is approximately Also, K1 and O1 amplitudes reach their maxima in Cook 1100 km from the suspected source region in the Aleu- Inlet (e.g., FCCHT). We therefore expect to see more tian Islands, speci®cally, the Amutka Pass near 172ЊW, pronounced shelf waves off Cook Inlet and near Kodiak where there is signi®cant tidal exchange with the deep Island. This is, indeed, what is observed in Figs. 9±11. part of the Bering Sea. Figure 9 shows K1 and O1 along the descending track 116, which begins from Kodiak Island at about 57.2ЊN b. Shelf waves in the Gulf of Alaska (see Fig. 1 for locations of the tracks). This plot is analogous to that in Fig. 8, since this track is also down- As mentioned above, analogous internal tides were stream of a strait, that is, Cook Inlet, except that am- also observed near Hawaii. However, we are not aware plitude dips are about twice as large. Figure 10 is for of previously published altimetric observations of shelf track 27, which runs parallel to track 116, but originates waves in the Gulf of Alaska. Figures 8±11 show ex- in the mouth of Cook Inlet at about 59.1ЊN. Similar dips

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FIG. 8. (a, b) K1 and (c, d) O1 amplitudes and phases along T/P track 48 from alongtrack harmonic analysis (thin line), harmonic FIG. 7. M2 (a) amplitude and (b) phase along T/P track 117 from alongtrack harmonic analysis (thin line), from large-scale polynomial analysis at crossovers (triangles), and from northeast Paci®c model ®t to the alongtrack complex amplitudes (thick line) and from har- [thick line; from FCCHT. (e) The alongtrack water depth. monic analysis at crossovers (triangles). (c) Amplitude and (d) phase of residual M2 tide, computed by taking a difference between alongtrack complex amplitudes and their polynomial ®t. (e) The alongtrack water depth. the altimeter-derived amplitude variations in Fig. 11, which may be masked by alongtrack noise level. In light of the short-wavelength variations in Fig. 11, in amplitudes of K1 and O1 are observed on this wide we may wish to examine analytical ␻±k dispersion shelf, while the phase changes more rapidly over short curves for barotropic shelf wave modes over varying distances than would be expected for a large-scale bottom topography. These were calculated using a nu- Kelvin wave. merical procedure for solving two coupled ®rst-order Further evidence for short-wavelength behavior is ordinary differential equations for sea level modal am- visible in the plot for track 62 (Fig. 11), which runs plitude and its spatial derivatives (R. F. Henry, unpub- almost parallel to and over the shelf break off the Cook lished report, 1984). This procedure was also used in Inlet. Model output shows alongtrack variations to have Crawford and Thomson (1984) and in Foreman (1987). an amplitude of about 1 cm and a wavelength of the Figure 12 shows the ®rst two barotropic modes for shelf order of 120 km. Similar scale variations also show up topography under tracks 27 and 116 (see Figs. 9e and over this shelf break in the model-derived barotropic 10e), except we have terminated the depth pro®le for energy ¯ux (FCCHT). However, this is not observed in track 27 at about 58.5ЊN, where a chain of islands and

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FIG. 9. As in Fig. 8, but for track 116. FIG. 10. As in Fig. 8, but for track 27. shallow depths between Kodiak Island and Kenai Pen- phases of K1 and O1 over the shelf. (This excludes the insula would, presumably, block the shelf waves. aliasing between K1 and SSA as a possible explanation According to Fig. 12a, the wide shelf under track 27 of the variations in K1, as was suggested by one of the can support both long prograde and retrograde (d␻/dk reviewers.) In particular, the nearshore dips in ampli- Ͻ 0) diurnal barotropic shelf waves with wavelengths tudes in Figs. 9 and 10, similar to those seen off Van- (2␲/k) of about 100 km. On the other hand, strati®cation couver Island (Fig. 8), suggest destructive interference tends to move the dispersion curve peak to higher wave- between the long wavelength Kelvin waves and the numbers and lift its descending branch toward and, pos- shorter locally generated shelf waves. Analyses of ocean sibly, above the diurnal band, thus diminishing the like- current data and numerical experiments with a baro- lihood of the short-wave retrograde shelf waves. This clinic model, such as that used in Cummins et al. (2000), was the conclusion reached in Crawford and Thomson will be helpful to learn more about the characteristics (1984) and Cummins et al. (2000) for the Vancouver of shelf waves in this area. Island shelf. So, it is possible that the model-generated 120-km shelf waves shown in Fig. 11 would disappear 8. Comparison to pelagic stations if strati®cation was included in this model. However, the shelf under track 27 is signi®cantly wider and may Table 4 presents a summary of a comparison between be able to support such a short-wave behavior. Never- complex amplitudes of major constituents from 57 pe- theless, there is a good agreement in Figs. 8±10 between lagic bottom pressure records and from T/P sea level the barotropic model and altimeter data amplitudes and data interpolated from the three nearest crossovers. Lo-

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FIG. 12. First (solid line) and second (dashed line) barotropic shelf FIG. 11. As in Fig. 8, but for track 62. wave modes for tracks (a) 27 and (b) 116. The corresponding depth pro®les are shown in Figs. 10e and 9e. cations of the deep-ocean stations are shown on Fig. 1. Some were located close to each other, but deployed Web page http://www.pmel.noaa.gov/tsunami/). Station during different years. A number of stations in this ob- tidal constituent data were obtained from PMEL, cour- serving network are used extensively for monitoring and tesy of Marie Eble. Some of these are also listed in predicting tsunamis in the Paci®c Ocean (Eble and Gon- Crawford et al. (1981) and in Smithson (1992). Only zaÂlez 1991; GonzaÂlez et al. 1998; GonzaÂlez 1999; Pa- deep ocean stations with at least 180 days of data were ci®c Marine Environment Laboratory (PMEL) tsunami included in the comparison.

pp TABLE 4. Mean amplitudes (Akk) and standard deviations [␴(A )] of 8 major constituents from 57 pelagic bottom pressure records in the px northeast Paci®c; mean magnitudes of vector amplitude differences (᭝A ϭ |AkkϪ A |) between these stations and T/P crossover locations (interpolated to pelagic locations), their standard deviations and minimum and maximum values (cm). Q1 O1 P1 K1 N2 M2 S2 K2

p Ak 4.54 25.27 12.65 39.95 14.04 67.25 22.16 6.01 p ␴(Ak) 0.52 2.98 1.76 5.11 3.21 15.42 3.55 0.73 ⌬A 0.49 1.22 0.55 1.72 0.57 1.15 0.74 0.60 ␴(⌬A) 0.33 0.77 0.26 0.84 0.33 0.69 0.60 0.40

(⌬A)min 0.04 0.47 0.09 0.73 0.10 0.14 0.09 0.07

(⌬A)max 1.21 3.99 1.39 4.63 1.36 3.59 3.30 1.71

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The vast majority of the station amplitudes agree quite a comparison between the T/P crossovers and 57 pelagic well with the T/P-derived amplitudes. For example, sites in Northeast Paci®c con®rmed our con®dence in mean difference ⌬A in M2 complex amplitude is 1.15 the calculated coef®cients. cm, with standard deviation of 0.69 cm. For K1 these Unmistakable signatures of internal M2 tides are vis- values are somewhat larger, 1.72 and 0.84 cm, respec- ible in the alongtrack amplitude and phase values. We tively. There are several suspect disagreements, espe- found a clear example of these waves propagating south cially in O1, K1, M2, and S2, which show up as ex- (about 170ЊT) from the Aleutian Islands, with wave- cessive (⌬A)max values in Table 4. For example, the di- lengths of about 153 km and phase speeds near 3.5 m urnal tides appear to be about 10% too strong at the sϪ1. They were measurable in a number of T/P tracks ®ve BEMPEX [Barotropic, Electromagnetic, and Pres- as far as 1100 km away from the suggested source re- sure Experiment (Filloux et al. 1991)] sites, located in gion. Examination of alongtrack and crossover K1 and the southwest part of the study area (Fig. 1), when com- O1 constituent amplitudes and phases on tracks that pared to nearby T/P crossovers. This is not related to descend from Cook Inlet into the Gulf of Alaska showed distances to crossovers; in fact, the BEMPEX site at sea level signatures of shelf waves, in good agreement 43.3ЊN, 160.1ЊW, is only 40 km away from the nearest with the numerical model results of FCCHT. In partic- crossover, but it is the one with the largest (⌬A)max for ular, nearshore dips in amplitudes, similar to those seen K1 and O1, 4.63, and 3.99 cm, respectively. The reason off Vancouver Island, suggest destructive interference behind these particular disagreements is not as yet between the long wavelength Kelvin waves and the known, though this detracts little from the fairly good shorter locally generated shelf waves. general agreement in Table 4 between the two sets of measurements. Acknowledgments. We are grateful to Richard Ray and Brian Beckley of the NASA Path®nder Team for 9. Conclusions providing us with the ``tidalist'' version of T/P altimeter data. Comments from Richard Ray and anonymous re- We used harmonic analysis, in combination with sin- viewers were very helpful in improving the paper. This gular value decomposition (SVD) method, to calculate work was supported in part by the Canadian Panel for tidal constituents from alongtrack TOPEX/Poseidon Energy Research and Development (PERD Project (T/P) sea level data. The SVD covariance matrix has 24110) and by the Department of Fisheries and Oceans provided us with estimates of errors in the calculated Climate Program ``Sea Level Altimeter Analysis.'' coef®cients, while analysis at track crossovers gave the means for large-scale correction of alongtrack constituents. REFERENCES Additional improvements were introduced into the Andersen, O. B., and P. Knudsen, 1997: Multi-satellite ocean tide harmonic analysis by using time-varying nodal correc- modellingÐThe K1 constituent. Progress in Oceanography, tions and selective data trimming. The latter helps to Vol. 40, Pergamon, 197±216. reduce projection of strong sea level anomalies onto , P. L. Woodworth, and R. A. Flather, 1995: Intercomparison of calculated constituents and thus improves the accuracy recent ocean tide models. J. Geophys. Res., 100, 25 261±25 282. Cartwright, D. E., and R. D. Ray, 1990: Oceanic tides from Geosat of the detided time series. This was convincingly dem- altimetry. J. 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