Ocean Tides from TOPEX/Poseidon Sea Level Data

Transcription

1 APRIL 2001 CHERNIAWSKY ET AL. 649 Ocean Tides from TOPEX/Poseidon Sea Level Data 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 final form 18 July 2000) ABSTRACT Tidal constants are computed from TOPEX/Poseidon sea level data in the northeast Pacific Ocean for the period of September 1992 December 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 Pacific. 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 Ocean sea level data from satellite altimetry sensors are used increasingly to describe seasonal and interannual changes in sea level and ocean circulation, during episodes of El Niño, or arising from longer-period climate change. TOPEX/Poseidon (T/P) is the first global system of satellite altimetry that was explicitly designed to study ocean dynamics by accurately measuring sea surface height relative to the center of the earth (Fu et al. 1996). In order to decipher the low-amplitude (of the order of several centimeters) changes in sea level, it is first necessary to apply a full suite of geophysical corrections to the altimetry data. Ocean tides are by far the largest of these corrections, as they account for up to 80% of the sea surface height variability of the world ocean. Thus considerable effort is directed toward removal of tides from the altimetry signal, either using large-scale hydrodynamic models (e.g., Le Provost et al. 1994), or satellite-altimeter-based empirical models (Schrama and Ray 1994; Ray et al. 1994; Desai and Wahr 1995). Tidal constants derived from ocean altimetry are also used to construct global inverse solutions to fit ocean hydrodynamics and satellite data (Egbert et al. 1994), for direct assimilation in numerical tidal models (Le Provost et al. 1998), to compute seasonal geostrophic currents using coastal circulation models (Foreman et al. 1998), or for specifying tidal elevations on open boundaries of regional models (Foreman et al. 2000). Corresponding author address: Dr. Josef Cherniawsky, Institute of Ocean Sciences P.O. Box 6000, Sidney, BC V8L 4B2, Canada. CherniawskyJ@pac.dfo-mpo.gc.ca Satellite altimetry data provide a global view of variability of tidal amplitudes, including smaller-scale features such as internal tides (Ray and Mitchum 1996, 1997; Kantha and Tierney 1997), or tides in coastal water, including tidally driven diurnal shelf waves (Foreman et al. 1998, hereafter FCCHT). Barotropic tidal models may not simulate baroclinic shelf waves accurately (Foreman and Thomson 1997; Cummins et al. 2000) and are not capable of producing internal tides. On the other hand, internal tides are simulated in highresolution baroclinic models (Cummins and Oey 1997; Crawford et al. 1998; Kang et al. 2000; Cummins et al. 2001). Global models do a good job of detiding the T/P altimeter data in deep ocean but, because of coarse resolution or inaccurate parameterization of unresolved physical processes, small errors in deep ocean tides tend to be amplified in shallow seas. Thus most of the global models are not adequate near the coastlines (Andersen et al. 1995; Shum et al. 1997). There, the tides can be more effectively removed using constituents from limited-area high-resolution tidal models (e.g., Foreman et al. 1998), or from local analysis of long time series of altimeter sea level. As the time period of observations increases, harmonic analysis of T/P data becomes more reliable, as there is less contamination of detided sea level from aliased tidal constituents and the accuracy of the calculated coefficients improves, especially for the longperiod ocean tides, such as MM and MF (Desai and Wahr 1997). Nevertheless, it is still desirable to know, at each T/P data location, the expected errors in the calculated constituents and the level of covariance between them due to aliasing. These local estimates are 2001 American Meteorological Society

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

3 APRIL 2001 CHERNIAWSKY ET AL. 651 FIG. 1. Domain of the study in the northeast Pacific 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 2 D x b 2, (2) where the unknown vector x {C 0, C 1, S 1,...,C K, S K }. The right-hand side consists of the sea level observations, b h, while D is a design matrix with M 2K 1 columns and N rows, M N (e.g., Press et al. 1992, hereafter P92). Each row i of D is made of either cosine (even columns), or sine (odd columns) basis functions, calculated at a time of observation t i cos(jt i), j 2k 1 Dij (3) sin( t ), j 2k j i for k 1,...,K. As N gets larger, the columns of D become nearly orthogonal and more of the Rayleigh criteria are satisfied (see the section on aliasing below). D i0 1, for the mean sea level C 0. We use the common notation Z0 for C 0, even though (because of the difficulty in calculating the exact shape of the geoid) the mean sea level is not as well determined in the deep ocean, compared to that measured with coastal tide gauges. In conventional harmonic analysis, it is customary to calculate a nodal correction to account for close minor constituents [also called satellite modulation as in Foreman et al. (1995)] after solving for major constituent amplitudes and phases. The central time of the observations is often chosen to specify a single correction for the time period in question (e.g., F77; Pugh 1987; Tierney et al. 1998). This is fine if this period is of the order of, or less than a year. But T/P observations now span more than five years, during which time the relative error due to using the central time value may exceed several percent. For example, between 1992 and 1997, the amplitude modulation factor f(t) varies from to for K1, from to for O1, and from to for M2 (Schureman 1958). We therefore modify (3) by inserting slowly varying nodal corrections, f ij f j (t i ) and u ij u j (t i ), directly into the design matrix cos(jti u ij), j 2k 1 Dij f (4) ij sin( j t i u ij ), j 2k. The computation of each f(t) and u(t) is as in Godin (1972). The columns of D from (4) are somewhat less orthogonal than in (3). But f(t) and u(t) are slowly varying functions of time and f(t) is close to unity, so this is of little concern. We solve (1), subject to (2), using the singular value decomposition (SVD) method (Golub and Van Loan 1983; P92). The N M design matrix is written as D U W V T, where U is an N M column-orthogonal matrix, V is an M M orthogonal matrix and W diag(w j )isanm M diagonal matrix with positive or zero elements w j (the singular values). The solution vector is then x V [diag(1/w j )] (U T b), (5)

4 652 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 TABLE 1. Tidal constituents frequency (cycles day 1 ), mean, standard deviation, minimum and maximum of amplitudes (cm), from harmonic analysis of 194 cycles of T/P sea level data at locations in the northeast Pacific Ocean Constituents Frequency Mean Std dev Min Max Z0 SA SSA MM MF Q1 O1 NO1 P1 S1 K1 J1 OO1 2N2 MU2 N2 NU2 M2 L2 T2 S2 K and variance estimates 2 (x j ) are given by diagonal elements of the covariance matrix M VV jl kl 2 j k 2 l1 wl cov(x, x ) (). (6) We replaced individual observation error estimates i (h i ) with a standard deviation (), where 2 () N 2 /(N M) is computed from the detided signal i1 i i (t i ) K i i 0 ik k k i k ik k1 h C f A cos[ t u ]. (7) The use of (x j ) as an error estimate for x j assumes a normal distribution for i (P92). But this is often not the case, especially when the altimeter track crosses steep fronts, or large eddies. To get around this problem, we repeat the above calculation using a trimmed dataset that excludes from b the observations h i that give i outside of 2(). This reduces the size of b by less than 5%. In fact, when a strong nontidal anomaly is present in the input data, part of it is projected onto the eigenfunctions of D (the tidal constituents), thus also reducing 2 (). Thus selective trimming of b helps to restore much of the original anomaly. An example of a strong anomaly in the detided sea level is shown in Fig. 2 for a single location in the Alaskan Stream. The first curve (long-dashed line) was produced using all input data in b, the second (solid line) shows the detided signal after trimming was performed, while the third curve (short-dashed line) was produced when the harmonic analysis was performed using only data up to April Clearly, the original anomaly was diminished (long-dashed line) when using all input data, and some of its energy (as well as energy from peaks in the end of 1992 and in late summer of 1994) was projected onto the annual harmonic, producing excessive negative anomalies (ringing) in summers of 1993, 1995, and The detided signal, when using only data up to April 1997 (short-dashed line), is not significantly different from the second curve (solid line). It is worthwhile noting here that this anomaly is a local signature of a strong anticyclonic eddy, which was traveling west-southwest along the Aleutian Trench during (Crawford et al. 2000). 4. Coastal example Figure 3 is a time-distance (Hovmöller) plot of detided T/P sea level for descending track number 52 (see Fig. 1), including the mean sea level (Z0) and the seasonal cycle (SA and SSA). The color density scale is shown below the plot (in mm), while the two curves in the top panel give a time series of mean and rms deviation (in cm) of sea level at each alongtrack location. A 5-point median alongtrack filter was applied to the detided data before plotting, in order to remove smallscale noise. Also, a time filter [one application each of a (1, 2, 1) filter and of a 3-point median filter, while excluding missing data] was used to smooth over intercycle synoptic variability. This track extends through the Gulf of Alaska and crosses Queen Charlotte and Vancouver Islands along the British Columbia coast (Fig. 1). It passes through northern Hecate Strait at about 53.5N, where the strongest semidiurnal tides (M2 amplitudes approach 2 m) along this coast are observed. Despite this, very little,

5 APRIL 2001 CHERNIAWSKY ET AL. 653 FIG. 2. Detided sea level from T/P altimeter data at N, W: long-dashed line was produced using all input data in harmonic analysis, solid line, when 2() trimming was performed, short-dashed line, without this trimming, but using only data up to Apr A weak time filter was used to suppress intercycle variability. if any, aliased signal with a near-60-day period (see next section) is visible in Fig. 3. It can be compared to an analogous plot along track 52 (up to cycle 154, Fig. 4) that was decided with constituents from a global GSFC model. While there is little difference between the two plots in the deep ocean, a clear near-60 day signal is present in Hecate Strait in Fig. 4. (In fairness to the GSFC model, we must add that it was designed for optimum performance in the deep ocean and, because of spatial smoothing, is not expected to do well close to the ocean margins.) Virtually all global tidal models have similar errors near the coastlines (e.g., Shum et al. 1997) and are obviously not adequate in the shallow waters. Thus locations where there is significant disagreement between tidal models are flagged in the GSFC-detided global sea level data (see explanation of the T/P flag word in html). Unfortunately, because of the aliasing (Ray 1998; also next section), we may have removed along with the tides any geophysical signal with a period of near 60 days. We do not think this is a serious problem, since, as far as we know, such signals are not commonly observed in this part of the world. 5. Aliasing and noise The tidal frequencies chosen for our analysis, k 2 k (k 1,..., K), were selected out of 45 astronomical tidal constituents (F77), based on decreasing magnitudes within each of the main constituent groups: low-frequency, diurnal, and semidiurnal. We selected K 22 constituents (Table 1), including Z0. Some of these have a predominantly nontidal character, for example, the annual SA, semiannual SSA, solar diurnal S1. But they were included because they are reasonably regular signals whose amplitudes are of interest. When required, SA and SSA harmonics are added back to. It is well known that certain tidal constituents may be aliased to other constituents because the repeat cycle of the T/P orbit, t days, is longer than diurnal and semidiurnal tidal periods. Indeed, this t was chosen by the T/P design team to minimize aliasing of major constituents with one another (Parke et al. 1987), but it was not possible to do so for all the constituents. In theory, for each k, there is an infinite set of possible aliasing frequencies j a i, (1 i K, j 0) (8) t (e.g., Parke et al. 1987; Yanagi et al. 1997). This leads to a revised Rayleigh criterion j R kij(t) k i T 1, t (1 i K, j 0), (9) that can be used to infer a minimum time period T min required to resolve any two tidal frequency components. Table 2 lists candidate alias pairs from the analyzed constituents for T min 1 yr. In practice, this time period may need to be somewhat longer than T min if input data are missing or corrupted by excessive noise. For example, for the K1 SSA pair, where SSA has a broad spectral peak due to variations in the weather, the time period may be much longer. Figures 5a,b show two examples of a covariance matrix and of correlation coeffients, r jk cov(x j, x k )/ [(x j )(x k )], at two locations alongtrack 52. The number of available data is N 178 for Fig. 5a and N 138 for Fig. 5b, out of the possible 194. The effect of missing data is apparent for the latter location (in Queen Char-

6 654 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 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 54N) and Vancouver Island (near 50N; 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 to the visible correlations for the expected alias pairs SSA K1, P1 K2, and N2 T2, we also observe in Fig. 5b enhanced correlations at most of the other alias pairs from Table 2, as well as at other pairs, though at reduced levels. These are likely due to the data dropout rate of 29%. Aliasing also occurs as each tidal frequency is folded about the T/P cycle Nyquist frequency (Ray 1998), which is a particular case of Eq. (9) when one of the constituents in the pair is Z0 ( i 0). Thus, when altimeter data are analysed with inadequate removal of tidal signals, this aliasing introduces spurious features at the aliased frequencies j kj k (10) t

7 APRIL 2001 CHERNIAWSKY ET AL. 655 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 j 19 one gets kj day 1,or1/ kj 62 days. This type of aliasing shows up in Fig. 4. Interconstituent aliasing is, obviously, not the only factor that affects the accuracy and reliability of the computed constituents, especially the minor ones. Serious limitations are also imposed by nontidal signals of interest to oceanographers (a.k.a. noise to a tidal analyst), which are broadband in nature and thus can be aliased to all of the constituents. For example, OO1 and S1 amplitudes are about 2 cm each on track 52 north of Vancouver Island, near 51.4N. These values are only marginally higher than their error estimates at this location, (x j ) 1.2 cm (diagonal elements in Fig. 5a). On the other hand, the semidiurnal constituents 2N2, MU2, L2, and T2 are also relatively small, ranging between 1.5 and 3 cm at this location. However, their (x j ) in Fig. 5a are smaller than 1 cm. This supports inclusion of the minor constituents in our analysis, especially since their amplitudes increase significantly in

8 656 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 TABLE 2. Candidate tidal alias pairs ( ) ki, sign in front of the alias frequency, index j, R kij k a T (where a i j/t, t , T 1914 days), and minimum duration T min (days). This table contains entries for T min 1yr. many coastal locations (maximum amplitudes are listed in Table 1). In general, the (x j ) values are quite similar to small-scale variations in the alongtrack amplitudes. Alias pair Sign of i j R kij T min SSA K1 Q1 NU2 Q1 M2 Q1 S2 O1 N2 O1 T2 NO1 2N2 P1 K2 MU2 L2 N2 T2 NU2 M2 NU2 S2 M2 S Analyses at crossovers The aliasing problems noted in the previous section are reduced at track crossover locations (Fig. 1), where twice as many observations are available. Therefore analyzed constituents at these locations can be used for checking the nearest alongtrack values, and possibly provide large-scale corrections. We interpolated raw alongtrack sea level data, separately for ascending and descending tracks, to each 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 coefficients at a crossover between tracks 99 and 52, at N, W (where FIG. 5. Two examples of absolute values of the covariance matrix (below and including the diagonal, in mm 2 ) and of the correlation coefficient (above the diagonal) for track 52: (a) at N, W, where valid N 173, out of possible 194, and (b) at N, W (just north of Vancouver Island), where N 138.

9 APRIL 2001 CHERNIAWSKY ET AL. 657 N 376). These can be compared to track 52 values in Figure 5a, at a location (where N 178) that is 1.6 km away from the crossover. Diagonal elements on Fig. 6 are of the order of 50 mm 2, 3 4 times smaller than on Fig. 5a. Thus expected errors in the sine and cosine coefficients are less than 8 mm, a factor of 2 smaller than on track 52 alone. What is more important, the off-diagonal covariance values are significantly smaller than the diagonal values and the corresponding r values on Fig. 6 are of the order, or less than 0.2. We note that similar correlation matrices were shown (though only for six constituents) in Andersen and Knudsen (1997), who used data from multiple satellites to reduce the aliasing problem between K1 and SSA. Table 3 lists amplitudes A k and expected amplitude errors (A k ) at the same crossover point and at a nearby location on track 52. Also listed are magnitudes of vector amplitude differences and ratios between expected amplitude errors. As there are almost twice as many data at the crossover point, the latter ratios are close to the theoretical value of 2 1/2. Notably, this ratio is almost constant, ranging between 0.64 and 0.66, except for SSA, K1, P1, and K2, for which it is between 0.58 and Therefore, as expected, relative improvement at crossovers is somewhat more pronounced for the aliased constituent pairs than for the other constituents. Similar relations also hold at other crossover locations, except that the departure from the theoretical ratio varies somewhat [sometimes exceeding 2 1/2 ], depending on relative data quality at a crossover and at a nearest alongtrack location. For some constituents the vector difference in amplitudes (column 6 in Table 3) exceeds the (A k ) estimates (here, for SA, O1, L2, T2, and S2), which may be a sign that these particular estimates are slightly too low. But this table is for a single location. Examination of analogous tables for varying crossover locations shows such isolated differences for other constituents as well. It is quite likely that, despite data editing [excluding from vector b the observations h i that give i outside of 2()], there is still some projection of nontidal anomalies onto the tidal constituents. Such influence of broadband geophysical noise is not in- FIG. 5.(Continued)

10 658 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 FIG. 6. As on Fig. 5a, except for a crossover at N, W (tracks 99 and 52), where N 376. Note the change in the covariance scale from Fig. 5a. cluded in the formal error estimates given by (A k ). However, any serial correlation in data (e.g., due to correlated errors in geophysical corrections) would make the matrix D not as well conditioned and increase the values of some (A k ). 7. Small-scale features a. Internal tides propagating away from Aleutian Islands It is interesting to see smaller-scale features in the alongtrack and crossover constituents. Indeed, internal (especially M2) tides are observed in many tracks, with prominent examples around Hawaii shown in Ray and Mitchum (1996, 1997), or Kang et al. (2000). Global distribution of semidiurnal internal tidal energy from T/P altimetry was presented by Kantha and Tierney (1997). They identified four areas of enhanced M2 internal tide energy in this region: (a) the already mentioned area northeast of the Hawaian Ridge, (b) around Mendocino Escarpment near California, (c) east and south of Kodiak Island in the Gulf of Alaska, and (d) around Aleutian Islands between 165W and 180. A relatively clear example of the M2 internal tide signature south of the Aleutian Islands is shown on Fig. 7. The top two panels show M2 amplitude and phase (thin lines) on T/P descending track 117 (see Fig. 1), including those from crossover analyses (marked as triangles). Third and fourth panels show amplitude and phase for residual M2 tide, computed, similar to Ray and Mitchum (1997), by taking a difference between the alongtrack complex amplitude and its polynomial fit (bold line on Figs. 7a,b). The latter fit is used here instead of the barotropic model values, which, in effect, gets around a problem of likely inclusion of regional model bias (large-scale difference) in the residual tide. An 8th degree polynomial was used for this 2850-km track segment, based on a 400-km length scale. The less certain phase values for small residual amplitudes (less than 0.6 cm) are shown as dots in Fig. 7d. At least four internal tide cycles are visible on Fig. 7 between 46.5N and 52N. Their wavelength projec-

11 APRIL 2001 CHERNIAWSKY ET AL. 659 TABLE 3. Constituent amplitudes and expected errors (in cm) for a crossover at N, W (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. Constituents A X X X X k (A k ) A k (A k ) A k A k (A k )/(A k ) Z0 SA SSA MM MF Q1 O1 NO1 P1 S1 K1 J1 OO1 2N2 MU2 N2 NU2 M2 L2 T2 S2 K tion onto this track is estimated to be about 180 km. The internal tide propagates away from the Aleutian Islands, as is evident from the residual phase increasing with distance from the islands (Fig. 7d). Similar calculation was done for an ascending track 75 (not shown), which crosses track 117 near 50N at an angle of about 72 and shows analogous wavelength projection of about 200 km. Combining these in a simple trigonometric relation yields the compass direction of propagation for the internal tide at this location to be around 170T. The estimated wavelength is about 153 km, while phase speed is near 3.5 m s 1. Essentially the same value (160 km) was obtained for the wavelength of a first-mode baroclinic internal wave at M2 frequency (P. Cummins 1999, personal communication), based on the calculation of the vertical eigenmodes for a climatological density profile in this area. Examination of residual phase plots from other neighboring, descending and ascending tracks (Cummins et al. 2001) reveals the presence of similar southwardpropagating wave packets at varying distances from the islands, all the way south to 41N. This is approximately 1100 km from the suspected source region in the Aleutian Islands, specifically, the Amutka Pass near 172W, where there is significant tidal exchange with the deep part of the Bering Sea. b. Shelf waves in the Gulf of Alaska As mentioned above, analogous internal tides were also observed near Hawaii. However, we are not aware of previously published altimetric observations of shelf waves in the Gulf of Alaska. Figures 8 11 show examples of a shelf wave signature in the alongtrack and crossover K1 and O1 complex amplitudes, also compared to the amplitudes from the Northeast Pacific model (FCCHT). In general, the alongtrack values (thin lines) agree quite well with the crossover values (triangles) and with the Northeast Pacific model-generated constituents (thick lines), except for some possible aliasing with geophysical noise (or between SSA and K1) at certain parts of the track. The first example (Fig. 8) shows that amplitudes and phases of K1 and O1 change rapidly along ascending track 48 near Vancouver Island. These changes are due to shelf waves, generated at the mouth of Juan de Fuca Strait, which at this location destructively interfere with the longer-wavelength barotropic Kelvin waves. This observed behavior (Crawford and Thomson 1984) was confirmed in the high-resolution modeling studies of Foreman and Thomson (1997), Cummins et al. (2000), and FCCHT. The shelf near Juan de Fuca is narrower and shorter than that off Cook Inlet in the northern Gulf of Alaska. Also, K1 and O1 amplitudes reach their maxima in Cook Inlet (e.g., FCCHT). We therefore expect to see more pronounced shelf waves off Cook Inlet and near Kodiak Island. This is, indeed, what is observed in Figs Figure 9 shows K1 and O1 along the descending track 116, which begins from Kodiak Island at about 57.2N (see Fig. 1 for locations of the tracks). This plot is analogous to that in Fig. 8, since this track is also downstream of a strait, that is, Cook Inlet, except that amplitude dips are about twice as large. Figure 10 is for track 27, which runs parallel to track 116, but originates in the mouth of Cook Inlet at about 59.1N. Similar dips

12 660 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 FIG. 7. M2 (a) amplitude and (b) phase along T/P track 117 from alongtrack harmonic analysis (thin line), from large-scale polynomial fit to the alongtrack complex amplitudes (thick line) and from harmonic 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 fit. (e) The alongtrack water depth. in amplitudes of K1 and O1 are observed on this wide shelf, while the phase changes more rapidly over short distances than would be expected for a large-scale Kelvin wave. Further evidence for short-wavelength behavior is visible in the plot for track 62 (Fig. 11), which runs almost parallel to and over the shelf break off the Cook Inlet. Model output shows alongtrack variations to have an amplitude of about 1 cm and a wavelength of the order of 120 km. Similar scale variations also show up over this shelf break in the model-derived barotropic energy flux (FCCHT). However, this is not observed in FIG. 8. (a, b) K1 and (c, d) O1 amplitudes and phases along T/P track 48 from alongtrack harmonic analysis (thin line), harmonic analysis at crossovers (triangles), and from northeast Pacific model [thick line; from FCCHT. (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, we may wish to examine analytical k dispersion curves for barotropic shelf wave modes over varying bottom topography. These were calculated using a numerical procedure for solving two coupled first-order ordinary differential equations for sea level modal amplitude and its spatial derivatives (R. F. Henry, unpublished report, 1984). This procedure was also used in Crawford and Thomson (1984) and in Foreman (1987). Figure 12 shows the first two barotropic modes for shelf topography under tracks 27 and 116 (see Figs. 9e and 10e), except we have terminated the depth profile for track 27 at about 58.5N, where a chain of islands and

13 APRIL 2001 CHERNIAWSKY ET AL. 661 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 Peninsula would, presumably, block the shelf waves. According to Fig. 12a, the wide shelf under track 27 can support both long prograde and retrograde (d/dk 0) diurnal barotropic shelf waves with wavelengths (2/k) of about 100 km. On the other hand, stratification tends to move the dispersion curve peak to higher wavenumbers and lift its descending branch toward and, possibly, above the diurnal band, thus diminishing the likelihood of the short-wave retrograde shelf waves. This was the conclusion reached in Crawford and Thomson (1984) and Cummins et al. (2000) for the Vancouver Island shelf. So, it is possible that the model-generated 120-km shelf waves shown in Fig. 11 would disappear if stratification was included in this model. However, the shelf under track 27 is significantly wider and may be able to support such a short-wave behavior. Nevertheless, there is a good agreement in Figs between the barotropic model and altimeter data amplitudes and phases of K1 and O1 over the shelf. (This excludes the aliasing between K1 and SSA as a possible explanation of the variations in K1, as was suggested by one of the reviewers.) In particular, the nearshore dips in amplitudes in Figs. 9 and 10, similar to those seen off Vancouver Island (Fig. 8), suggest destructive interference between the long wavelength Kelvin waves and the shorter locally generated shelf waves. Analyses of ocean current data and numerical experiments with a baroclinic model, such as that used in Cummins et al. (2000), will be helpful to learn more about the characteristics of shelf waves in this area. 8. Comparison to pelagic stations Table 4 presents a summary of a comparison between complex amplitudes of major constituents from 57 pelagic bottom pressure records and from T/P sea level data interpolated from the three nearest crossovers. Lo-

14 662 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 FIG. 11. As in Fig. 8, but for track 62. cations of the deep-ocean stations are shown on Fig. 1. Some were located close to each other, but deployed during different years. A number of stations in this observing network are used extensively for monitoring and predicting tsunamis in the Pacific Ocean (Eble and González 1991; González et al. 1998; González 1999; Pacific Marine Environment Laboratory (PMEL) tsunami FIG. 12. First (solid line) and second (dashed line) barotropic shelf wave modes for tracks (a) 27 and (b) 116. The corresponding depth profiles are shown in Figs. 10e and 9e. Web page Station tidal constituent data were obtained from PMEL, courtesy of Marie Eble. Some of these are also listed in Crawford et al. (1981) and in Smithson (1992). Only deep ocean stations with at least 180 days of data were included in the comparison. p p TABLE 4. Mean amplitudes (A k) and standard deviations [(A k)] of 8 major constituents from 57 pelagic bottom pressure records in the p x northeast Pacific; mean magnitudes of vector amplitude differences (A Ak A k ) between these stations and T/P crossover locations (interpolated to pelagic locations), their standard deviations and minimum and maximum values (cm). A p k (A p k) A (A) (A) min (A) max Q1 O1 P1 K1 N2 M2 S2 K

15 APRIL 2001 CHERNIAWSKY ET AL. 663 The vast majority of the station amplitudes agree quite well with the T/P-derived amplitudes. For example, mean difference A in M2 complex amplitude is 1.15 cm, with standard deviation of 0.69 cm. For K1 these values are somewhat larger, 1.72 and 0.84 cm, respectively. There are several suspect disagreements, especially in O1, K1, M2, and S2, which show up as excessive (A) max values in Table 4. For example, the diurnal tides appear to be about 10% too strong at the five BEMPEX [Barotropic, Electromagnetic, and Pressure Experiment (Filloux et al. 1991)] sites, located in the southwest part of the study area (Fig. 1), when compared to nearby T/P crossovers. This is not related to distances to crossovers; in fact, the BEMPEX site at 43.3N, 160.1W, is only 40 km away from the nearest crossover, but it is the one with the largest (A) max for K1 and O1, 4.63, and 3.99 cm, respectively. The reason behind these particular disagreements is not as yet known, though this detracts little from the fairly good general agreement in Table 4 between the two sets of measurements. 9. Conclusions We used harmonic analysis, in combination with singular value decomposition (SVD) method, to calculate tidal constituents from alongtrack TOPEX/Poseidon (T/P) sea level data. The SVD covariance matrix has provided us with estimates of errors in the calculated coefficients, while analysis at track crossovers gave the means for large-scale correction of alongtrack constituents. Additional improvements were introduced into the harmonic analysis by using time-varying nodal corrections and selective data trimming. The latter helps to reduce projection of strong sea level anomalies onto calculated constituents and thus improves the accuracy of the detided time series. This was convincingly demonstrated in the case of a large eddy crossing a T/P track south of the Aleutian Islands. The problem with the aliasing arising from the T/P day repeat cycle was discussed in terms of the SVD error-covariance matrix. We do not find this aliasing to be serious enough to prevent us from using the harmonic analysis, as its advantages (local estimates and more degrees of freedom) relative to the response method outweigh the minor loss of accuracy due to possible aliasing of K1 with SSA and P1 with K2. As the T/P time series gets longer, we expect these aliasing problems to diminish. Comparison between alongtrack and crossover constituents showed that the SVD-calculated error estimates are quite realistic. As expected, the errors at crossovers were reduced by a factor that is close to the theoretical value of 2 1/2, while the problem with the aliasing was sufficiently small to allow large-scale corrections to the alongtrack constituents, for example, by using a tidal model for dynamical interpolation (FCCHT). Similarly, a comparison between the T/P crossovers and 57 pelagic sites in Northeast Pacific confirmed our confidence in the calculated coefficients. Unmistakable signatures of internal M2 tides are visible in the alongtrack amplitude and phase values. We found a clear example of these waves propagating south (about 170T) from the Aleutian Islands, with wavelengths of about 153 km and phase speeds near 3.5 m s 1. They were measurable in a number of T/P tracks as far as 1100 km away from the suggested source region. Examination of alongtrack and crossover K1 and O1 constituent amplitudes and phases on tracks that descend from Cook Inlet into the Gulf of Alaska showed sea level signatures of shelf waves, in good agreement with the numerical model results of FCCHT. In particular, nearshore dips in amplitudes, similar to those seen off Vancouver Island, suggest destructive interference between the long wavelength Kelvin waves and the shorter locally generated shelf waves. Acknowledgments. We are grateful to Richard Ray and Brian Beckley of the NASA Pathfinder Team for providing us with the tidalist version of T/P altimeter data. Comments from Richard Ray and anonymous reviewers were very helpful in improving the paper. This work was supported in part by the Canadian Panel for Energy Research and Development (PERD Project 24110) and by the Department of Fisheries and Oceans Climate Program Sea Level Altimeter Analysis. REFERENCES Andersen, O. B., and P. Knudsen, 1997: Multi-satellite ocean tide modelling The K1 constituent. Progress in Oceanography, Vol. 40, Pergamon, , P. L. Woodworth, and R. A. Flather, 1995: Intercomparison of recent ocean tide models. J. Geophys. Res., 100, Cartwright, D. E., and R. D. Ray, 1990: Oceanic tides from Geosat altimetry. J. Geophys. Res., 95, Crawford, W. R., and R. E. Thomson, 1984: Diurnal-period continental shelf waves along Vancouver Island: A comparison of observations with theoretical models. J. Phys. Oceanogr., 14, , W. J. Rapatz, and W. S. Huggett, 1981: Pressure and temperature measurements on seamounts in the North Pacific. Mar. Geod., 5, , J. Y. Cherniawsky, P. F. Cummins, and M. G. G. Foreman, 1998: Variability of tidal currents in a wide strait: A comparison beween drifter observations and numerical simulations. J. Geophys. Res., 103, ,, and M. G. G. Foreman, 2000: Multi-year meanders and eddies in the Alaskan Stream as observed by TOPEX/Poseidon altimeter. Geophys. Res. Lett., 27, Cummins, P. F., and L.-Y. Oey, 1997: Simulation of barotropic and baroclinic tides off northern British Columbia. J. Phys. Oceanogr., 27, , D. Masson, and M. G. G. Foreman, 2000: Modeling diurnal tides and currents off Vancouver Island. J. Phys. Oceanogr., 30, , J. Y. Cherniawsky, and M. G. G. Foreman, 2001: North Pacific internal tides from the Aleutian Ridge: Observations and modelling. J. Mar. Res., in press. Desai, S. D., and J. M. Wahr, 1995: Empirical ocean tide models

16 664 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 estimated from TOPEX/POSEIDON altimetry. J. Geophys. Res., 100, , and, 1997: Error analysis of empirical ocean tide models estimated from TOPEX/POSEIDON altimetry. J. Geophys. Res., 102, Eble, M. C., and F. I. González, 1991: Deep-ocean bottom pressure measurements in the Northeast Pacific. J. Atmos. Oceanic Technol., 8, Egbert, G. D., A. F. Bennett, and M. G. G. Foreman, 1994: TOPEX/ POSEIDON tides estimated using a global inverse model. J. Geophys. Res., 99, Filloux, J. H., D. S. Luther, and A. D. Chave, 1991: Update on seafloor pressure and electric field observations from the north-central and northeastern Pacific: Tides, infratidal fluctuations and barotropic flow. Tidal Hydrodynamics, B. B. Parker, Ed., Wiley, Foreman, M. G. G., 1977: Manual for tidal heights analysis and prediction. Pacific Marine Science Rep , Institute of Ocean Sciences, Sidney, British Columbia, Canada, 66 pp. [Available online at tidpack.htm.], 1987: An accuracy analysis of selected finite difference methods for shelf waves. Contin. Shelf Res., 7, , and R. E. Thomson, 1997: Three-dimensional model simulations of tides and buoyancy currents along the west coast of Vancouver Island. J. Phys. Oceanogr., 27, , W. R. Crawford, and R. F. Marsden, 1995: De-tiding: Theory and practice. Quantitative Assessment for Coastal Ocean Models, D. R. Lynch and A. M. Davies, Eds., Coastal and Estuarine Studies, Vol. 47, American Geophysical Union, ,, J. Y. Cherniawsky, J. F. R. Gower, L. Cuypers, and V. A. Ballantyne, 1998: Tidal correction of TOPEX/POSEIDON altimetry for seasonal sea surface elevation and current determination off the Pacific Coast of Canada. J. Geophys. Res., 103, ,,, R. F. Henry, and M. Tarbotton, 2000: A highresolution assimilating tidal model for the Northeast Pacific Ocean. J. Geophys. Res., 105, Fu, L.-L., C. J. Koblinsky, J.-F. Minster, and J. Picaut, 1996: Reflecting on the first three years of TOPEX/POSEIDON. Eos, 77, 109, 111, 117. Godin, G., 1972: The Analysis of Tides. University of Toronto Press, 264 pp. Golub, G. H., and C. F. Van Loan, 1983: Matrix Computations. Johns Hopkins University Press, 476 pp. González, F. I., 1999: Tsunami! Sci. Amer., 280, , H. B. Milburn, E. N. Bernard, and J. Newman, 1998: Deepocean assessment and reporting of tsunamis (DART): Brief overview and status report. Proc. Int. Workshop on Tsunami Disaster Mitigation, Tokyo, Japan, Japan Meteorological Agency and Science and Technology Agency, Kang, S. K., M. G. G. Foreman, W. R. Crawford, and J. Y. Cherniawsky, 2000: Numerical modeling of internal tide generation along the Hawaiian Ridge. J. Phys. Oceanogr., 30, Kantha, H. L., and C. C. Tierney, 1997: Global baroclinic tides. Progress in Oceanography, Vol. 40, Pergamon, Koblinsky, C. J., R. Ray, B. D. Beckley, Y.-M. Wang, L. Tsaoussi, A. Brenner, and R. Williamson, 1999: NASA Ocean Altimeter Pathfinder Project, Report 1: Data Processing Handbook. Tech. Memo. NASA/TM , 55 pp. Le Provost, C., M. L. Genco, F. Lyard, P. Vincent, and P. Canceil, 1994: Tidal spectroscopy of the world ocean tides from a finite element hydrodynamic model. J. Geophys. Res., 99, , F. Lyard, J. M. Molines, M. L. Genco, and F. Rabilloud, 1998: A hydrodynamic ocean tide model improved by assimilating a satellite altimeter-derived data set. J. Geophys. Res., 103, Munk, W. H., and D. E. Cartwright, 1966: Tidal spectroscopy and prediction. Philos. Trans. Roy. Soc. London A, 259, Parke, M. E., R. H. Stewart, D. L. Farless, and D. E. Cartwright, 1987: On the choice of orbits for an altimetric satellite to study ocean circulation and tides. J. Geophys. Res., 92, Press, W. H., S. A. Teukolsky, W. T. Vettering, and B. P. Flannery, 1992: Numerical Recipes in Fortran. 2d ed. Cambridge University Press, 963 pp. Pugh, D. T., 1987: Tides, Surges, and Mean Sea Level. John Wiley, 472 pp. Ray, R. D., 1998: Spectral analysis of highly aliased sea-level signals. J. Geophys. Res., 103, , and G. T. Mitchum, 1996: Surface manifestation of internal tides generated near Hawaii. Geophys. Res. Lett., 23, , and, 1997: Surface manifestation of internal tides in the deep ocean: Observations from altimetry and island gauges. Progress in Oceanography, Vol. 40, Pergamon, , B. Sanchez, and D. E. Cartwright, 1994: Some extensions to the response method of tidal analysis applied to TOPEX/PO- SEIDON. Eos, 75 (Suppl.), 108. Schlax, M. G., and D. B. Chelton, 1994: Aliased tidal errors in TO- PEX/POSEIDON sea surface height data. J. Geophys. Res., 99, Schrama, E. J. O., and R. D. Ray, 1994: A preliminary tidal analysis of TOPEX/POSEIDON altimetry. J. Geophys. Res., 99, Schureman, P., 1958: Manual of harmonic analysis and prediction of tides. Special Publication 98, U.S. Department of Commerce, Coast Guard and Geodetic Survey, Washington, DC, 317 pp. Shum, C. K., and Coauthors, 1997: Accuracy assessment of recent ocean tide models. J. Geophys. Res., 102, Smithson, M. J., 1992: Pelagic Tidal Constants. Vol. 3. IAPSO Publication Scientifique 35, IUGG, 191 pp. [Available from Dr. M. Smithson, PSMSL, Bidston Observatory, Birkenhead, Merseyside CH4 7RA, United Kingdom.] Tierney, C. C., M. E. Parke, and G. H. Born, 1998: An investigation of ocean tides derived from along-track altimetry. J. Geophys. Res., 103, Yanagi, T., A. Morimoto, and K. Ichikawa, 1997: Co-tidal and corange charts for the East China Sea and the Yellow Sea derived from satellite altimetric data. J. Oceanogr., 53,