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  • Ross, C. K., J. W. Loder, and M. J. Graca, 1988: Moored current and hydrographic measurements on the southeast shoal of the Grand Bank 1986 and 1987. Can. Data Rep. Hydrogr. Ocean Sci., No. 71, vi + 132 pp.

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  • Xu, Z., 1998: A direct inverse method for inferring open boundary conditions of a finite-element linear harmonic ocean circulation model. J. Atmos. Oceanic Technol.,15, 1379–1399.

  • View in gallery

    Model domain, triangular mesh, and topography. The bathymetry is based on the ETOPO5 dataset, complemented by finer-resolution (order 7 km) Canadian hydrographic charts in shallow water (⩽1000 m). The minimum water depth in the model is 10 m

  • View in gallery

    Distribution of the observational datasets. The T/P tracks are sequentially numbered with letters a and b indicating the southern and northern ends. The current stations referred to in the text are numbered in italic bold face

  • View in gallery

    Contours of M2 elevation amplitude (cm) from the observational data

  • View in gallery

    The specified b.c. (dashed lines) with the observations (circles) superimposed. (top) Amplitude and (bottom) phase lags (relative to Greenwich)

  • View in gallery

    Green’s function maps for a unit (100 cm) set up on a few selected boundary nodes illustrating (a) “influential” nodes and (b) nodes with limited influence on the interior solutions

  • View in gallery

    Controlling points and basis (piecewise linear and trigonometric) functions used to constrain the spatial structure on the open boundary segments A–G. For clarity, only the first mode of the trigonometric functions is plotted for the segment AG. For the separate HI section, only the piecewise linear function is used for the basis function

  • View in gallery

    Changes in the rms distance misfit between the full assimilative model solution and observations, as a function of the number of Fourier modes used in the b.c. spatial structure. (a) Rms elevation misfit; (b) rms current misfit

  • View in gallery

    Comparison between the specified b.c. (dashed curves), the inferred b.c. from full interior data assimilation (solid curves), and the along-boundary observations (circles). (top) Elevation amplitudes and (bottom) elevation phase lags

  • View in gallery

    Elevation amplitudes (solid dark lines in cm) and phase lags (dashed dark lines in degrees) from (a) the benchmark solution and (b) the full interior data assimilative solution. These are superimposed on the contoured amplitudes (shades with white contours and larger black labels) from the raw data (Fig. 3)

  • View in gallery

    Geographical distribution of the elevation amplitude misfits for (a) the benchmark and (b) the assimilative solutions, with circles indicating those ≥ 3 cm and dots indicating smaller misfits

  • View in gallery

    Histograms and statistics of the elevation misfits by (left) the specified b.c. and (right) full interior assimilative solutions. (top) The misfits in amplitude, (middle) the misfits in phase lag, and (bottom) for the distances on a complex plane. The misfit statistics shown are the mean, std dev (std), rms, minimum (min), and maximum (max) values

  • View in gallery

    Surface tidal ellipses in the Grand Banks region from the full interior assimilative solution (solid ellipses) and from available observations nearest to the surface (dotted ellipses). Station numbers are indicated for observed ellipses

  • View in gallery

    Comparison between the observed tidal ellipses (dashed curves) and the modeled ellipses (solid curves) from (left) the benchmark and (right) full interior assimilative solutions. The phases are indicated by the radial lines, and the station locations are shown in Fig. 12

  • View in gallery

    Histograms and statistics of the current misfits by (left) the specified b.c. and (right) full interior assimilative solutions. Proceeding from top to bottom, the panels are for the misfits in the major axis, minor axis, orientation, and distances on a complex plane

  • View in gallery

    Track-by-track comparisons of the observational data (circles), the benchmark solution (dashed lines), and the data assimilative solution (solid lines). Distances are measured from “a” ends to “b” ends of each track (cf. Fig. 2). The hw value shown for each track is an estimate of the average half-width of the 95% confidence interval. The in situ results are plotted against a sequential data index after being sorted by distance along the coast increasing from the south, for the tide gauges, or latitude increasing from the south, for bottom pressure

  • View in gallery

    The amplitudes of the inferred elevation b.c. from the full and partial assimilation cases. The shaded zones are the first estimates of the 95% confidence intervals. The rms difference (rd) between each partial case and the full assimilation case, and the average value (hw) of the half-width of the confidence interval are indicated on each panel

  • View in gallery

    Comparisons between modeled (solid curves) and observed (dashed curves) middepth tidal ellipses at station 26 on the northern Grand Banks for various model solutions: (a) benchmark case, (b) elevation case, (c) decimated case, and (d) full interior assimilation case. The small square indicates a common phase. The site’s water depth is 198 m and the current meter was at 100 m above the seabed

  • View in gallery

    Sensitivity of (a) the rms elevation and (b) velocity misfits to scaling of the base frictional fields. Data for sensitivity cases are marked by circles and those for the base fields are marked by plus in addition

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Application of a Direct Inverse Data Assimilation Method to the M2 Tide on the Newfoundland and Southern Labrador Shelves

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  • 1 Ocean Sciences Division, Fisheries and Oceans Canada, Bedford Institute of Oceanography, Dartmouth, Nova Scotia, Canada
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Abstract

The barotropic M2 tide over the Newfoundland and southern Labrador Shelves and adjacent deep ocean is studied using a linear harmonic finite-element model and a newly developed direct inverse method for data assimilation. The dataset includes harmonic tidal constituents from TOPEX/Poseidon altimetry, coastal tide gauges, bottom pressure gauges, and moored current meters. Three modeling approaches are taken: a conventional modeling approach with boundary conditions specified from along-boundary observations; a full interior data assimilative approach, which provides an optimal domain-wide solution; and a sensitivity study in which the roles of various data subsets and the frictional parameters are investigated.

The optimal solution from the full assimilative approach has rms misfits of 3.5 cm and 1.3 cm s−1 for elevation and current, respectively (in terms of distances on the complex plane), compared to overall rms amplitudes of 30 cm and 6 cm s−1. These misfits are reduced by more than 40% and 70% from those in the conventional solution. Formal confidence limits for the optimal solution can be estimated but depend on assumptions about the spatial covariance of the observational residuals. The sensitivity study provides quantitative indications of the importance of the quantity and location of the observational data and indicates little sensitivity to the specified frictional fields within a reasonable range. In particular, the inclusion of a fraction of the velocity data in the assimilation results in a significant improvement in the model fit to the velocity observations.

Corresponding author address: Dr. Zhigang Xu, Bedford Institute of Oceanography, 1 Challenger Drive, P.O. Box 1006, Dartmouth, NS B2Y 4A2, Canada.

Email: xuz@mar.dfo-mpo.gc.ca

Abstract

The barotropic M2 tide over the Newfoundland and southern Labrador Shelves and adjacent deep ocean is studied using a linear harmonic finite-element model and a newly developed direct inverse method for data assimilation. The dataset includes harmonic tidal constituents from TOPEX/Poseidon altimetry, coastal tide gauges, bottom pressure gauges, and moored current meters. Three modeling approaches are taken: a conventional modeling approach with boundary conditions specified from along-boundary observations; a full interior data assimilative approach, which provides an optimal domain-wide solution; and a sensitivity study in which the roles of various data subsets and the frictional parameters are investigated.

The optimal solution from the full assimilative approach has rms misfits of 3.5 cm and 1.3 cm s−1 for elevation and current, respectively (in terms of distances on the complex plane), compared to overall rms amplitudes of 30 cm and 6 cm s−1. These misfits are reduced by more than 40% and 70% from those in the conventional solution. Formal confidence limits for the optimal solution can be estimated but depend on assumptions about the spatial covariance of the observational residuals. The sensitivity study provides quantitative indications of the importance of the quantity and location of the observational data and indicates little sensitivity to the specified frictional fields within a reasonable range. In particular, the inclusion of a fraction of the velocity data in the assimilation results in a significant improvement in the model fit to the velocity observations.

Corresponding author address: Dr. Zhigang Xu, Bedford Institute of Oceanography, 1 Challenger Drive, P.O. Box 1006, Dartmouth, NS B2Y 4A2, Canada.

Email: xuz@mar.dfo-mpo.gc.ca

1. Introduction

Regional modeling studies with a fine spatial resolution and assimilation of local elevation and current data are often needed to address tidal problems in coastal and shelf regions. Here we report on a study of shelf-scale regional modeling of the M2 tide with high spatial resolution and assimilation of a large and diverse observational dataset including currents. The study region encompasses the Newfoundland and southern Labrador Shelves and the adjacent deep ocean (Fig. 1). The high resolution is achieved through a triangular finite-element mesh of variable size. The dataset comprises harmonic constituent estimates of M2 elevations from in situ and TOPEX/Poseidon (hereafter referred to as T/P) altimetry measurements at over 1400 locations, and of M2 currents from moored measurements at 33 sites.

To represent the basic tidal dynamics, we use the 3D harmonic finite-element model FUNDY5 (Lynch et al. 1992; Naimie and Lynch 1993), which is a discretized version of the linear shallow water equations in the frequency domain. The data assimilation method is the newly developed “direct inverse method” (Xu 1998) for inverting a frequency-dependent boundary value problem. Based on the “stiffness” matrix from the FUNDY5 model, the method first establishes an explicit linear relationship between the interior response (in elevation and velocity) and the boundary forcing, and then casts the assimilation problem as a regression problem. One inversion of the matrix leads directly to an optimal domainwide solution (instead of requiring an iterative procedure as in the commonly used adjoint method), hence the name of the method. It has been shown by Xu (1998) that, for a tidal data assimilation problem, the method has advantages in computational economy and accuracy compared to an iterative adjoint method. Further, since the method uses a regression-model formulation, statistical uncertainty of the assimilative solutions can be estimated if knowledge of the source errors is available or assumed.

The primary goals of the study are (i) to obtain a high-resolution and observationally constrained representation of the elevations and 3D currents associated with the barotropic M2 tide in the region, (ii) to examine the importance of different data types and locations to an assimilative dynamical model of this tide, and (iii) to test the performance of the direct inverse method with a real dataset. The underlying motivations are realistic tidal currents for use in applications such as drilling waste dispersion estimates and iceberg drift prediction at hydrocarbon development sites on the Newfoundland Shelf, and improved tidal elevation estimates for use in detiding altimetry data from the region.

Three modeling approaches are used in the study: a conventional modeling approach, a full interior data assimilative approach, and a partial data assimilative approach. In the conventional approach, the open boundary condition (hereafter b.c.) is specified based on the T/P data and a few in situ observations along or near the boundary, but the interior (including coastal) data are reserved for observational comparisons. This approach provides a benchmark solution for evaluating the improvements provided by the data assimilative approaches. The full interior assimilative approach uses all the interior data to infer an optimal b.c., which is then used to drive a domainwide solution. In the partial assimilative approach, only subsets of the dataset are assimilated to investigate the role that particular data subsets play in constraining the optimal solution.

The in situ data subset reflects the numerous observational estimates of local M2 tidal elevations and currents in the study region, but shelf-scale syntheses have been limited to Han et al.’s (1996) evaluation of T/P data and a few modeling investigations. Petrie et al. (1987; also see de Margerie and Lank 1986) used a depth-averaged nonlinear barotropic model to show the spatial structure of the elevations and currents for the five largest tidal constituents in the Grand Banks region, and found that the M2 constituent is predominant. Recently, Han et al. (2000) used a 3D nonlinear primitive equation model and a nudging method to assimilate T/P altimetry data for the M2 tide in the same region. The present study differs from these investigations in the least-squares optimality of its assimilation method, an expanded model domain that includes the northeast Newfoundland and southern Labrador Shelves, the use of expanded altimetry and in situ datasets, and an increased resolution of coastlines and topography.

Section 2 briefly describes the study domain and model mesh, the dataset, and the FUNDY5 dynamical model. Section 3 then reviews the direct inverse method and discusses the b.c. treatment in the present application. The results for an optimal representation of the M2 tide are presented in section 4 with a description of the solution from the full interior data assimilative approach and a comparison with the benchmark solution. The importance of various data subsets and sensitivity to the frictional parameters are addressed in section 5. A summary is presented in section 6.

2. Data and model background

a. Study domain and model mesh

The study domain is the northwestern North Atlantic continental shelf, slope, and adjacent deep ocean, extending from the southern Newfoundland Shelf to southern Labrador Shelf (Fig. 1). This includes the Grand Banks, Flemish Cap, and the northeast Newfoundland Shelf including the outer part of the Strait of Belle Isle (connecting to the Gulf of St. Lawrence). Offshore water depths range from 50 to 100 m on the Grand Banks to over 3000 m in the deep-ocean part.

The offshore boundaries were chosen to lie along T/P tracks, except for the southern boundary at 38°N. The horizontal finite-element mesh (Fig. 1) has 10 927 nodes, and triangular elements that vary in size to provide increased resolution over the shelf in general, and in areas of steep topography in particular. The spatial resolution (defined as the square root of the triangular area) varies with the local topographic gradient parameter h/h (where h is the water depth; Hannah and Wright 1995) and ranges from as fine as 0.7 km along the shelf edge to about 22 km in the deep ocean with a domain average of 8 km. With the present focus on the continental shelf, the 5–10-km near-shore resolution used in the study resolves the major bays and promontories, but not the numerous small bays and islands along the Newfoundland and Labrador coasts. Vertical space is transformed into a σ space (σ = z/h where z is the vertical coordinate and h is the water depth), with 21 variably spaced vertical nodes providing increased resolution at the surface and seafloor. There are two separate segments of open boundary on which elevation is required as a b.c. The major offshore segment runs from the southern Newfoundland coast (point A; Fig. 1) counterclockwise to the Labrador coast (point G) with a total length of about 4500 km. A small 25-km segment runs across the Strait of Belle Isle (from point H to I).

b. Observational dataset

The observational dataset of M2 elevations and currents is distributed widely over the study region (Fig. 2). The in situ elevations are taken from coastal tide gauge constituents at 82 locations provided by the Canadian Hydrographic Service, and harmonic analyses of bottom pressure gauge data from 37 offshore sites reported by Lively (1984; Strait of Belle Isle), Petrie et al. (1987; the Grand Banks), Petrie (1991; Avalon Channel), and Wright et al. (1988, 1991; Labrador Shelf and Sea). Coastal elevation constituents from sites (e.g., inlets) not resolved by the model mesh are not included. The bottom pressure constituents have been converted to elevation. The tidal currents are taken from harmonic analyses reported by Lively (1984; Strait of Belle Isle), Lively and Petrie (1990; the NE Grand Banks), Narayanan (1994; NE Newfoundland Shelf), Petrie (1991; Avalon Channel), Petrie et al. (1987; the Grand Banks), Ross et al. (1988; Southeast Shoal; with corrections from J. Loder), and Wright et al. (1988, 1991; Labrador Shelf and Sea). These analyses are from current meter records from 57 different (3D) positions at 33 sites. The various in situ tidal analyses are typically based on the entire length of record, and do not include uncertainty estimates. The in situ datasets generally provide good coverage of coastal elevation, and sparse coverage of elevations and currents on the shelf (Fig. 2). Data on the vertical current structure are particularly limited, but those available show little suggestion of baroclinic M2 tides (internal tides are expected along the shelf edge however).

The T/P tidal coefficients were derived from a local database (G. Han 1998, personal communication) of sea level time series obtained from version A geophysical data records (Benada 1993) and gridded at 0.1° latitude intervals along the T/P tracks. The data came from 123 repeat cycles during the 3.3-yr period from December 1992 to April 1996. A response method (Cartwright and Ray 1990) was used to extract estimates of the geocentric M2 tide from 1327 locations in the study region where a minimum of 60 points in the time series were available for analysis (Fig. 2). The M2 tidal variations are aliased to a period of approximately 62 days in the 10-day T/P sampling, and sea level variability with periods near 62 days (e.g., Gulf Stream eddies and meanders) can also influence the estimates. At 144 locations on the southern Labrador and northeast Newfoundland Shelf, there were insufficient data for the analysis because of seasonal ice cover.

Standard environmental and load tide corrections were applied [see Han et al. (1996) and Han et al. (2000) for further details]. The load tide correction was taken as 0.05 times the geocentric tide to convert estimates of the latter to land-referenced tides, as discussed in Cartwright and Ray (1991). A direct calculation of the M2 load tide at the T/P sampling sites using the methods described by Agnew (1996), the global CSR3.0 model fields (Eanes and Bettadpur 1995), and the Guttenberg-Bullen model A average Earth gave an overall rms difference of only 0.3 cm between the two approaches.

The overall rms amplitude of the observed elevation data in the study domain is 30 cm, and that for current amplitude is 6 cm s−1. Empirical error estimates for the M2 altimetric coefficients were calculated by comparing results at the 55 cross-over points of ascending and descending passes in the study region. Assuming that the errors in tidal coefficients derived from different passes have equal variance and are uncorrelated, the variance of the differences will be twice the variance of the coefficient errors. The resulting estimates of individual M2 coefficient standard errors are approximately 2 cm for amplitude, 8° for phase, and 3 cm for the amplitude of the complex difference.1 Standard errors were also estimated using the jackknife method (Hogg 1979), with essentially identical results. The estimated errors from both methods showed coherent spatial patterns, with higher values in deeper regions consistent with the increased background variability of sea level in the offshore Gulf Stream and North Atlantic Current (e.g., Wunsch and Stammer 1995; Shum et al. 1997).

c. Spatial structure from elevation data

The broad coverage of the observational elevation data allows a crude display of the spatial pattern of M2 tidal elevation without application of a dynamical model. Figure 3 is a contour map of the elevation amplitudes, obtained using triangular linear interpolation, that shows the major features in the region. The largest amplitudes occur in Placentia Bay on the southern Newfoundland Shelf, consistent with previous descriptions of the regional M2 tide (e.g., Petrie et al. 1987). The smallest amplitudes occur at an amphidromic point near the easternmost apex (point E) of the offshore boundary. The present extended domain shows a broad minimum in M2 amplitudes on the northeast Newfoundland Shelf, and increasing amplitudes progressing northward on the Labrador Shelf, consistent with some global tidal model solutions (e.g., Egbert et al. 1994; Kantha 1995a,b; Le Provost et al. 1995; Le Provost 1998).

Figure 4 shows M2 amplitudes and phase lags along the open boundaries. These were obtained by triangular interpolation of T/P measurements on or near the bounding tracks, supplemented with coastal tide gauge data from the southern Newfoundland and Labrador Shelves and the Strait of Belle Isle. Smoothed versions of the boundary amplitudes and phases, shown as dashed curves in Fig. 4, were created by a smoothing spline fit for the specified b.c. case discussed below. The boundary amplitudes feature coastal maxima and an offshore minimum associated with the amphidrome. The phase lags show an abrupt change near the offshore amphidrome. The northern half of the domain leads the southern half, reflecting the counterclockwise propagation of the M2 tide around the North Atlantic. The amplitude estimates show prominent along-track variations from midsegment BC to just beyond point E, with peak-to-peak changes of more than 10 cm and a dominant wavelength of approximately 300 km. This part of the open boundary coincides with the high-variability Gulf Stream and North Atlantic Current, so these variations are likely of nontidal origin. However, a contribution from baroclinic tides (e.g., Ray and Mitchum 1997) cannot be ruled out.

d. Governing equations and FUNDY5 model

The frequency-dependent linear 3D shallow water equations used in this barotropic tidal application of the FUNDY5 model (Lynch et al. 1992; Naimie and Lynch 1993) are
i1520-0426-18-4-665-e1
with surface and and bottom b.c.’s
i1520-0426-18-4-665-e4
and lateral b.c.’s from either a priori specification or data assimilation. The meaning of the notation is as follows:
  • i   the imaginary unit, (−1)1/2;

  • ω   the angular frequency of motion with solutions assumed to be ∼eiωt;

  • xyz   the coordinates in a right-hand Cartesian system (eastward-northward-upward with z = 0 at the sea surface in the absence of motion);

  • uυw   the complex amplitudes of depth-dependent velocity components in the xyz directions, respectively;

  • ξ   the complex amplitude of sea surface elevation;

  • g   the gravitational acceleration;

  • f   the Coriolis parameter taken here as a constant;

  • h   the water depth (specifically, the depth at which the bottom b.c.’s are applied);

  • ν   the vertical eddy viscosity, which can be a function of (x, y, z), but is taken here as vertically uniform; and

  • κ   the bottom friction parameter, which can be a function of (x, y).

Factors neglected in these equations include baroclinic dynamics, stratification influences, sea-ice influences, latitudinal variations in f, the earth’s curvature, and forces associated with tidal potential and yielding seabed. The inaccuracies associated with these approximations are expected to be small in the focal shelf region, but need further investigation. For example, Foreman et al. (1993) found that inclusion of the tidal potential and loading tide in a shelf tidal model off British Columbia changed the largest M2 amplitudes by only 2.5% and the largest K1 amplitudes by less than 1%. We have chosen an oblique Mercator map projection (Evenden 1990) to minimize the distortion from mapping the spherical coordinate system to our Cartesian coordinate system by taking the line of no distortion along the Newfoundland-Labrador shelf break.2

Similar to Greenberg et al. (1997), we specify the two frictional parameters
i1520-0426-18-4-665-e6
where the overbars and the subscripts b indicate depth-averaged and bottom velocity amplitudes, respectively, νmin = 0.002 m2 s−1, β = 0.2s, κmin = 3.5 × 10−4 m s−1, and Cd = 5 × 10−3. The minimum values are included to represent contributions from background (unmodeled) flow components (of order of 0.1 m s−1) while, following Davies and Furnes (1980) and Lynch and Naimie (1993), the velocity-dependent terms provide approximations to spatial variability using a mixing-length representation of viscosity and a quadratic bottom stress law. With this frictional parameterization, the dynamical equations are no longer linear so we adopt an iterative approach. We begin with a model run with the specified b.c. (Fig. 4) and spatially uniform values of ν and κ, and then use the computed depth-averaged and bottom velocity amplitudes to update ν and κ via (6) and (7). With the updated spatially varying ν and κ, the model is rerun with the same b.c. to obtain refined velocity amplitudes and frictional parameters, and the procedure is repeated until stable values of ν and κ are obtained (requiring five to six iterations). These values are then taken as the specified spatial-varying frictional parameters for all further modeling cases. The domain-averaged values of ν and κ are 3.4 × 10−3 m2 s−1 and 5.0 × 10−4 m s−1, and their peak values are 7.4 × 10−2 m2 s−1 and 2.5 × 10−3 m s−1. The sensitivity of our model solution to the frictional values will be examined in section 5f.
The FUNDY5 model provides a mathematical formalism for reducing (1)–(5) to a single second-order equation for surface elevation, and a finite-element method for solving the equation and constructing the depth-dependent velocities. With this method, the governing equations are reduced to the single matrix equation
Aξ
where the sea surface elevation ξ (at both boundary and interior nodes) is an N × 1 vector and N is the total number of nodes. The coefficient matrix A, of size N × N, is the so-called “stiffness” matrix, which contains the dynamical constraints imposed by the governing equations (Lynch et al. 1992). The absence of u and υ from (8) reflects the method’s decoupling of the calculations for elevation and depth-dependent velocity. This does not compromise the 3D sense of the solution but helps in substantially reducing the dimensionality of the solution space. This method provides a computationally efficient procedure for high-resolution circulation models used in several recent applications to shelf circulation (Lynch and Naimie 1993; Naimie et al. 1994; Han et al. 1999).

3. The direct inverse method and b.c. spatial structure assumptions

In this section, we briefly describe the direct inverse method and our assumptions regarding the along-boundary spatial structure of the b.c. For more details on the inverse method, readers are referred to Xu (1998).

a. A dynamically consistent regression model

Splitting the elevation vector ξ into a part ζ defined on the boundary nodes and a part η defined on the interior nodes, and similarly splitting the stiffness matrix A into boundary and interior parts (Abc and Ain), the matrix equation (8) can be written as
AinηAbcζ.
If a weight matrix W is introduced as a solution to the equation
AinWAbc
then
ηWζ
gives all the interior elevations as responses to an arbitrary boundary forcing ζ.

The weight matrix is very large, specifically (Nm) × m where m is the number of boundary nodes where the elevation must be specified. However, in data assimilation applications, it can be dramatically reduced to n × m, where n is the number of observation positions, by extraction of the rows appropriate to these positions (a linear interpolation of rows of W will be necessary if the positions are not coincident with the mesh nodes, as is usually the case). So, in regression analysis of (11), only a small subset of W is used, making computations feasible and economical. In the following discussion, the notation W may stand for the whole set or a subset, depending on the context.

The weight matrices for the velocity components u and υ can be calculated as functions of W (Xu 1998). Thus, both horizontal components of the interior depth-dependent velocity can also be calculated as weighted sums of the boundary elevations:
i1520-0426-18-4-665-e12
One may use (11), (12), and (13) to assimilate elevation and velocity data simultaneously, through the augmented regression model
i1520-0426-18-4-665-e14
where the ɛ’s have been introduced to account for observational noise and any missing dynamics, and λ is a scaling factor to adjust the velocity data to numerical values that have comparable magnitudes to those of the elevation data, so that the two data types have comparable weight (per position) in the regression. In the present study, the averaged velocity amplitudes (in cm s−1) are about one-fifth of the averaged elevation amplitudes (in cm, averaged over all the data), so that λ is taken as 5 s.3

b. Interior response to unit boundary forcing

The weight matrices (W, Wu, and Wυ) have important physical meanings. Each column is the Green’s function for the elevation or velocity response to a unit elevation forcing at a boundary node. Each row is the distribution of the weights at an interior point for all the boundary nodes. Using W as an example, Fig. 5 shows the Green’s functions for elevation with unit forcing at six different boundary nodes: three “influential” nodes from which information penetrates well into the model domain, and three boundary nodes with limited information penetration. Those boundary nodes with little influence fall into the so-called “null space” from which forcing is mapped to zero (or near zero) in the interior.4

From the data assimilation point of view, the existence of the null space means that a complete b.c. cannot be inferred from a given set of interior observations. If the b.c. is considered to consist of two parts, then only the part that is responsible for the observed interior variances is inferable, while the other is noninferable. Mathematically, the difficulty is that the weight matrices in (14) are singular. One approach using the singular value decomposition (SVD) technique yields a particular solution for the inferable part of the b.c. Xu (1998) discussed the need to smooth such an SVD solution and proposed a null-space smoother. A second approach is to constrain the spatial structure of some variables so the dynamical system becomes nonsingular. In this study, we explore the latter approach by assuming an along-boundary structure for the elevations.

c. Assumed spatial structure of the b.c

The assumed along-boundary structure consists of piecewise linear and trigonometric (spatial Fourier expansion) functions, as illustrated in Fig. 6. As suggested by the along-boundary observations on the major (A–G) segment (Fig. 4), we choose point A, F, G, and a point slightly north of E as the controlling points for its piecewise linear functions. The additional trigonometric functions should be helpful in resolving nonlinear structure in the true b.c. Lynch et al. (1998) have used similar trigonometric expansion along their open boundary on Georges Bank. For the short Strait of Belle Isle segment, a linear structure using H and I as the controlling points should be sufficient. The spatial structure along the open boundaries is constrained by these functions, but their (complex) amplitudes are to be determined by the interior data.

Formally, if we introduce a matrix L for the linear interpolation, and matrices Hcos and Hsin for the cosine and sine basis functions, then the boundary elevation vector ζ can be expressed as
i1520-0426-18-4-665-e15
where ζc is the elevation at the six controlling nodes, and vectors a and b contain the coefficients of the cosine and sine base functions. Appendix A gives detailed expressions for matrices L, Hcos and Hsin. The regression model (14) now becomes
i1520-0426-18-4-665-e16
The number of unknown elements in the vector [ζcab] will be much smaller than that in the vector ζ.

4. Full data assimilative and benchmark solutions

In this section we discuss the assimilation of the full interior dataset and compare the results with the benchmark solution obtained from the specified b.c. shown in Fig. 4. The full interior dataset includes M2 tidal coefficients for the elevations and current from a total of 1266 records (elevation data from the bounding T/P tracks are excluded). We first report how the number of Fourier modes for the spatial structure of the b.c. is determined and describe the optimally inferred b.c. We then present FUNDY5 model solutions driven by the specified and inferred b.c.’s. The two model solutions are evaluated against the overall and various subsets of interior dataset. Since both the observed tidal constituents and the model solutions are complex numbers, the overall misfit is the distance between the observed and modeled vectors in the complex plane. In the discussion that follows, we also quote misfits of amplitude and phase.

a. Number of Fourier modes and the optimally inferred b.c

As discussed in section 3c, the spatial structure of the inferred elevation b.c. is taken to comprise a linear function on small segment HI, and piecewise linear and trigonometric (Fourier) functions on major segment AG. The number of Fourier modes on the AG segment was determined by a set of trial regressions with an increasing number of modes. Figure 7 shows the rms elevation and current distance misfits to the interior data for the different trials. For both datasets, the misfits decrease rapidly as the number of modes changes from zero to three, with little further change with the addition of more modes. Therefore, the Fourier expansion was truncated at three modes. The regression determines the six complex controlling values for the piecewise linear part of the b.c. and the six complex coefficients for the sine and cosine parts of the Fourier modes. The 12 complex parameters for the basis function are much fewer than the number of the open boundary nodes (184).

With the form of the boundary structure fixed, the solution of (16) with its left-hand side specified by all the interior current and elevation data gives an optimal b.c. that minimizes the weighted sum of the squared differences between the observations and the model. The results are shown in Fig. 8 for amplitude (top) and phase (bottom). Also shown are the unsmoothed along-boundary observations (circles) and the smoothed version of the observations used for the specified b.c. control run (dashed line). Note that the inferred b.c. is independent of the along-boundary data.

While the inferred b.c. resembles the overall features of the observed data (or the specified b.c.), there are noticeable differences between them, especially for the amplitudes. To the left of point E, the inferred b.c. exhibits some wavelike features around the specified b.c. and to the right of point E, the inferred b.c. is systematically lower than the specified b.c. The wavelike feature perhaps reflects the projections of similar features in the interior data near that boundary section onto the three trigonometric modes of the assumed boundary structure. (Although they are not assimilated into the inferred b.c, the wiggles seen in the along-boundary data to the left of point E reflect more explicitly the same features existing in the nearby interior data shown by Fig. 3.) To the right of point E, the inferred b.c. is closer to a straight line, but systematically lower than the specified b.c. For the phase lags (Fig. 8 bottom), the agreement between the inferred b.c. and the data is better. The inferred b.c., being a continuous solution, has smoothed the phase discontinuity at the amphidromic point slightly. In short, the interior data has demanded its own version of the b.c., and we shall see its advantages over the specified version when the misfits between the data and the model solutions are compared.

b. Solutions and statistics of their misfits

Figure 9 shows corange and cophase maps for the benchmark and full assimilative solutions on shaded co-range contours based on the data as in Fig. 3. Both model solutions capture the observed large-scale features of an amphidromic system. The tide enters the model domain at northern boundary segment EF and sweeps through the domain in an anticlockwise sense around the amphidromic point near vertex E. A comparison with basin-scale tidal solutions (e.g., Schwiderski 1980;Egbert et al. 1994; Kantha 1995a,b; Le Provost et al. 1995; Le Provost 1998) shows that this regional solution captures about one-quarter of the North Atlantic M2 amphidromic system. The greatest tidal range occurs on the southern Newfoundland Shelf, with peak amplitudes exceeding 60 cm in Placentia Bay in southern Newfoundland. Amplitudes of approximately 50 cm, the second highest in the study region, are found near boundary segment FG on the southern Labrador Shelf. Intermediate amplitudes characterize the connecting coastal zone, except the Strait of Belle Isle where amplitudes are low. The equatorward phase propagation and shoreward amplification of tidal amplitude resemble a Kelvin-like coastal wave.

A visual comparison of the modeled amplitudes with the raw data in Fig. 9 suggests that the assimilative solution provides more realistic results in the interior of the study region than the benchmark solution. As shown in Fig. 8, the improvement in the interior for the assimilative case has been at the expense of greater disagreement with the boundary observations, especially near boundary vertex F. Both solutions underestimate the observed amplitude near this vertex where the maximum open-boundary amplitude is found. Figure 10 gives a more quantitative view by showing where the amplitude misfits to observed values exceed a 3-cm threshold. The benchmark solution has more interior misfits exceeding this threshold than the assimilative solution. The region of greatest improvement is the eastern and northeastern Newfoundland Shelf north of about 46° latitude.

The statistics of the elevation misfits for the benchmark and assimilative solutions are summarized in Fig. 11. The rms distance misfit for the assimilative solution is 3.5 cm, which is not much different from the overall noise estimate of 3 cm for individual T/P tidal constituents (section 2b). This misfit is about 60% of the corresponding value (5.9 cm) for the benchmark solution (bottom). The reduction is more than 50% for the rms amplitude misfit (top), and about 15% for the rms phase misfit (middle). The amplitude misfits for the assimilative solution are distributed more narrowly and symmetrically around smaller mean values.

Surface tidal current ellipses in the Grand Banks region from the assimilative solution are displayed in Fig. 12, together with the available near-surface observed ellipses. A pronounced spatial variation is apparent with the strongest currents found in the shallower shelf areas. The currents are relatively weak in the deep ocean, and become substantially stronger on the Grand Banks with a clockwise rotation and peak amplitudes of about 20 cm s−1 on the Southeast Shoal. The strongest currents in the model domain occur in the Strait of Belle Isle (not shown) where they are rectilinear with peak amplitudes about 50 cm s−1. Elsewhere along the coasts of Newfoundland and southern Labrador, the currents are generally rectilinear with amplitudes of 5–10 cm s−1 (but recall that the coastline is not fully resolved in the model, so that there may be additional local current amplifications in the ocean). Overall, the spatial variation in the model current ellipses is in good agreement with the observations.

In Fig. 13, 3D displays are used to compare the observed tidal ellipses with the modeled ones from both the benchmark and assimilative solutions, for three sites with multilevel observations—stations 1 and 3 on Southeast Shoal (Ross et al. 1988), and station 7 near Hibernia (Fig. 12) on the northeastern the Grand Banks (Petrie et al. 1987). The assimilative solution shows substantially improved agreement with the observed ellipses, in both size and phase at all three sites, and also in ellipse orientation at Hibernia. At the shallower Southeast Shoal site (station 1) with stronger currents, the model solutions also show the observed reduction in ellipse size with depth, with the lowest current meter showing the effects of the bottom Ekman layer. In contrast, it appears that the bottom Ekman layer thickness may be underestimated in the model solution at station 3.

The statistics of the current misfits for all stations are summarized in Fig. 14 for both the benchmark and assimilative solutions. Significant reductions in the misfits for the data assimilation solution are seen, both in the distance and ellipse-parameter misfits. For example, the rms distance misfit of the benchmark solution is 4.5 cm s−1, while that of the assimilative solution is only 1.3 cm s−1—a reduction of more than 70%. This indicates that a data assimilative model provides the potential for even greater improvements in the realistic representation of tidal currents than of tidal elevations.

c. Track-by-track comparisons

A track-by-track comparison of the data and model amplitudes along the 14 interior T/P tracks plotted as a function of along-track distance is presented in Fig. 15. Results are given for both the benchmark and full assimilative solutions. Two additional panels in Fig. 15 show similar results for the two in situ data assemblages—coastal tide gauge data and bottom pressure data.

Figure 15 amplifies the results of Figs. 9 and 10 by showing a systematic tendency for the benchmark solution to overestimate the amplitudes from interior T/P, coastal tide gauge, and bottom pressure data. This is especially notable in the northeastern part of the study area, as seen on T/P tracks 4–6 and 11–14 and in the tide gauge and bottom pressure comparisons. These biases contribute significantly to the overall rms data-benchmark solution misfit of 5.9 cm, compared with the rms amplitude value of 30 cm for the 1033 T/P interior points. On the other hand, the least squares assimilative solution gives better agreement with the interior data.

Figure 15 also reveals some spatially organized differences between the assimilative model and the interior data. The amplitude of the assimilative solution is consistently lower than the T/P data along track 6 next to the outer boundary segment EF. This mirrors the discrepancy between the inferred b.c. and the T/P data along this boundary segment shown in Fig. 8. In contrast, the coastal tide gauge comparison shows consistently higher amplitudes for the assimilative solution in the northern part of the study area. The comparison based on bottom pressure data shows an opposing bias in this region, with assimilative solution amplitudes generally less than suggested by the data. Overall, it seems that there is a competition between the coastal and offshore data in the northern part of the domain, since the specified b.c. results in an overestimate of the coastal elevation whereas assimilation of the interior data reduces the coastal elevation overestimate substantially but results in an underestimate of the boundary elevation as Fig. 8 shows. It is not known if the apparent competition reflects data sparsity and quality or model deficiencies such as inadequate near-shore spatial resolution and the assumption of constant f.

Figure 15 shows wavelike variations along the tracks for the observed T/P amplitudes (circles) in the southern part of the model domain. Tracks 2, 3, and 7 provide particularly large examples of such features. Similar to the wavelike variations along the boundary (Fig. 4), they may arise from low-frequency sea level variations associated with the Gulf Stream and North Atlantic Current, but contributions from possible baroclinic tides in that area cannot be ruled out.

Formal error estimates at each grid point are available from the regression analysis for the assimilative model. An associated parameter hw gives the half-width of a 95% confidence interval for amplitude. Diagnostic track-averaged values of hw, computed on the assumption that the residuals have a spatially uniform variance and are uncorrelated, are included in Fig. 15. The misfit between the assimilative model and the data, including the normalized current misfit variance, was used to estimate the overall error variance. The formal errors reflect the spatial distribution of the data. Relatively small values are noted on central tracks 2–5 and 8–9; relatively large values on near-boundary tracks 1, 6, 7, and 11; and relatively large values on short tracks 12–14 in the northern part of the model domain. The average hw values for the coastal tide gauge and bottom pressure data are notably higher than for the central T/P tracks.

It is recognized that the error model associated with these hw values is unrealistic. Regional variations in residual amplitude are apparent and the residuals are spatially correlated. The construction of a quantitative error model allowing for these considerations is beyond the scope of the present study. As a first step, estimates of elevation residual autocovariance as a function of spatial lag were calculated for each of the 14 interior T/P tracks. The number of degrees of freedom for residual variance estimates (e.g., Jenkins and Watts 1968) averaged over all 14 tracks was approximately one-fifth of the corresponding average number of data points. This suggests that increasing the formal model elevation errors by a factor of (5)1/2 (∼2.2) would give more realistic error estimates. This does not account for cross-track correlations, which would further reduce the number of effective degrees of freedom and increase the size of the error bars.

5. Sensitivity studies

In the preceding section, we presented an optimal solution from assimilating the full interior dataset (which we refer to as the “full” assimilative case) and a “benchmark” solution from a specified b.c. We now examine how the assimilative model solutions are affected when some of the data are excluded. For this purpose, we have carried out four partial data assimilative cases:

  • an “in situ” case in which only the data from coastal tide gauges, bottom pressure gauges, and current meters are assimilated;

  • a “T/P” case in which all the interior T/P data plus in situ data from two Strait of Belle Isle sites are assimilated;

  • an “elevation” case in which all the elevation data (from tide and pressure gauges and altimetry) are included, but the current data are excluded; and

  • a “decimated” case in which only about one-third of the full dataset (elevations and currents) is assimilated.

We will focus on two aspects to summarize the results: the inferred b.c. for each case, and the rms misfits to the interior observational data. Figure 16 shows the inferred b.c.’s together with 95% confidence intervals estimated under the assumption of uncorrelated residuals with spatially uniform variance. As discussed in the preceding subsection, this assumption results in an underestimation of the uncertainty, but it gives an indication of how the confidence levels are affected when the number and location of the data change. Table 1 summarizes the rms distance misfits for each of the cases, for both the assimilated and reserved data. We discuss the results for each of the partial assimilation cases in the following subsections.

We conclude this section with an evaluation of the sensitivity of the model solutions to the velocity data and to the frictional parameters.

a. In situ case

This case is used to examine whether the in situ data alone can yield a reasonable domainwide solution. The inferred b.c. has the same general form as in the full assimilative case (Fig. 16), implying a well-defined solution. However, the 95% confidence zone has widened considerably, to a few cm near the two coasts (points A and G) and to 10–15 cm on the southern part of the deep-ocean boundary (near points C and D). This is because there are no in situ data in the latter area, with most of these data being along the coast. Nevertheless, the rms elevation misfits for the in situ case are reduced from those in the benchmark solution by about 30% for both the assimilated and reserved (T/P) data, compared to a reduction by 41% in the full case (Table 1). The current misfits are correspondingly reduced by 73%, a value comparable to that in the full case. These substantial reductions in the misfits indicate that the in situ data alone can constrain a realistic domain-wide M2 solution, but the confidence level is lower than with the T/P data included.

b. T/P case

This case was intended to examine whether the T/P data alone can provide a reasonable assimilative solution, and in particular how well this solution does in predicting the in situ data. However, it was found that the Strait of Belle Isle is a particularly sensitive area, with an open boundary segment (HI) and no nearby T/P data. Inclusion of in situ data from two sites in the strait was required to obtain a solution that is realistic locally. This is understandable since forcing in this area is not influential on the rest of model domain (Fig. 5), and hence will not be represented in the T/P data. In contrast, a reasonable b.c. for most of the major segment AG can be inferred from the interior T/P data (Fig. 16). The confidence zone for the inferred b.c. is comparable to that in the full case for most of the segment, but the uncertainty increases at the two coastal ends (A and G) where there are no T/P data.

As seen in Table 1, the rms elevation misfit to the (assimilated) T/P data is reduced by 46% in the T/P case (relative to that in the benchmark case), but the misfit to the reserved (in situ) data is reduced by only 15%. This indicates that inclusion of the interior T/P data results in only a small improvement in the prediction of the in situ data over the benchmark solution. This is consistent with the discussion in section 4c about the opposing influences of the T/P and coastal data in the northern part of the model domain. For the velocity, there is a substantial (45%) reduction in the rms misfit compared to the benchmark case, but the improvement is significantly less than the 71% provided by the full assimilation solution.

As noted in the introduction, Han et. al. (2000) recently conducted a data assimilation study for a subdomain of the present study area centered over the Grand Banks with a different method and a different model. They only assimilated T/P data, similar to our T/P case here, and then compared their solution with their withheld in situ data. It is therefore interesting to compare results of the two studies for the same area. They reported the rms misfits to the elevation and current data to be 2.7 and 7.7 cm s−1, respectively. In our T/P case, the rms misfit to the same in situ elevation data is 2.0 cm and the rms misfit to the current data in the same region is 2.9 cm s−1. In our full assimilative case the misfits are 2.4 and 1.5 cm s−1. While the differences in the elevation misfits between the two studies may not be of statistical significance, our study results in a significant improvement in the current fields.

c. Elevation case

This case was intended to evaluate the model’s skill in predicting tidal currents with only elevation data as input. Relative to the full assimilation case, the T/P case showed a doubling of the rms velocity misfit when only T/P data were assimilated. The elevation case uses all the elevation data, including both in situ and remote observations, so that the differences between the results of this case and the full case are due solely to the velocity data. The confidence zone on the inferred b.c. is comparable to that in the full case (Fig. 16). The rms misfits (Table 1) indicate that this case does about as well as the full case in fitting the elevation data, but not much better than the T/P case in fitting the velocity data. This suggests that the in situ elevation data contribute only limited additional information on the velocity field. Collectively, the above cases indicate that: 1) assimilation of the elevation data alone can reduce the current misfits, in the present cases by about 45% (relative to that in the specified b.c. solution); and 2) a further significant reduction of the current misfits can be obtained from also assimilating the current data, in this study by about 25%.

d. Decimated case

This case assimilates only about one-third of the full dataset, in order to investigate the sensitivity of the misfits to the amount of assimilated data. In situ elevation data from every second site were reserved (excluded), most of the T/P data were reserved except at or near the cross-over points of the ascending and descending passes, and most of the current data were reserved except for seven sites (stations 2, 6, 16, 17, 22, 27, and 32; Fig. 2). The inferred b.c. (Fig. 16) still shows a substantially narrower confidence zone than in the in situ case, but slightly wider than in the full and elevation cases. These differences are related to the amount and location of the assimilated data in the various cases. The rms misfits in the decimated case are not significantly different from those in the full case (Table 1), for both elevations and currents. This suggests that there is little gain in including the present elevation dataset at its highest spatial resolution (e.g., the 0.1° latitude resolution of the T/P data), presumably since the spatial scales of the tidal elevation are much larger (e.g., Fig. 3).

e. Tidal ellipse sensitivity

We have already noted the differences in the rms current misfits for assimilation cases with and without velocity data (section 5c). The decimated case provides additional information regarding the importance of spatial resolution of the velocity data. Table 1 suggests a progressive improvement in the velocity fitting with the inclusion of more current data. Comparison of the modeled and observed tidal ellipses at station 26 on the northern edge of the Grand Banks (Fig. 17) further illustrates the importance of assimilating current data. The observed data at this location were not included in the decimated case, with the nearest assimilated station being about 100 km away. The benchmark solution shows poor agreement with the observed ellipse, with substantial differences in ellipse orientation. The ellipse in the elevation assimilation case is close to that observed in size and orientation, but rotates counterclockwise (instead of clockwise as observed) and has a significant difference in phase. In contrast, the ellipses in both the full and decimated assimilation cases show much improved overall agreement with the observed ellipse, with approximate agreement in size, orientation, phase, and in rotation sense. This reinforces the importance of assimilating current data, but suggests that the observational current data may not need high spatial resolution in areas of gently sloping topography.

f. Sensitivity to frictional parameters

We now investigate the sensitivity of the model solutions to the frictional parameters. Our approach is to run the model under a fixed b.c. with different frictional parameter fields, and then examine how the elevation and current misfits vary. We choose the b.c. that was inferred from the full data assimilation case as the fixed b.c., and take the fields ν and κ used in all the proceeding assimilations as the base fields. We then use different values of the scaling factor M (chosen as 1/100, 1/10, 1/4, 1/2, 1, 2, 4, 10, 100) to produce new fields, that is, [νnew(x, y), κnew(x, y)] = M[νbase(x, y), κbase(x, y)].

Figure 18 summarizes the results, showing the rms elevation misfits (upper panel) and rms velocity misfits (lower panel) versus the scaling factor. We can see that there is a substantial range of the scaling factor over which the misfits change little, particularly for elevation. The range is smaller for the velocity misfits, and the minimum misfit occurs for friction values that are half of the base values. This limited investigation of the sensitivity to friction suggests that our base case is a reasonable approximation for the purposes of examining how to infer optimal boundary forcing from a given set of interior observations, but it is recognized that improved velocity solutions can be expected with more advanced turbulence closures.

6. Summary

A realistic, high-resolution representation of the barotropic M2 tide on the Newfoundland and southern Labrador Shelves has been obtained through an application of a newly developed direct inverse method for data assimilation. The assimilation model is based on a linear, unstratified version of the finite-element FUNDY5 model and a model geometry with 0.7–22-km horizontal resolution. The model is computationally efficient and can be extended to higher spatial resolution for detailed coastal or regional studies. The method was applied to tidal constituents derived from a large dataset of elevations and current observations. The dataset included sea level measurements from coastal tide gauges and the TOPEX/Poseidon satellite, bottom pressure measurements, and moored current measurements.

The direct inverse method casts the model equations as a least-squares regression problem where the open boundary elevations are the unknowns. This allows the derivation of the set of boundary elevations that best fit interior elevation and current data. In the present application, the boundary conditions are projected onto a simple set of basis functions with a small number of free parameters.

Three modeling approaches have been used to evaluate the importance of assimilating various data. Boundary elevations based on smoothed boundary (or nearby) observations were used in a conventional tidal application of the FUNDY5 model to obtain a benchmark solution. An application of the direct-inverse method to all elevation and current data available in the interior of the model domain provided an inferred boundary condition and an associated full assimilation interior model solution. Sensitivity studies were also made to test the relative importance of data types and spatial coverage, and to justify our choice for the frictional parameter fields.

The assimilation of the full interior dataset provided a solution that agreed with the interior data to within an rms elevation misfit of 3.5 cm (in terms of distance on a complex plane), or approximately 10% of the overall rms M2 tidal amplitude of about 30 cm. The model current field reproduced the main features of the observations, both in horizontal spatial variations and in vertical variations due to frictional effects. The overall rms current misfit was 1.3 cm s−1, approximately 20% of the 6 cm s−1 rms amplitude of the M2 tidal current observations. The full assimilation model successfully simulated tidal currents in regions of relatively strong flows, including the shallow Southeast Shoal of the Grand Banks where tidal currents exceeding 20 cm s−1.

In the full interior assimilative solution, the rms misfits to the interior data were reduced by more than 40% and 70% for elevation and currents, respectively, from those of the benchmark solution. A prominent overestimation of interior elevation amplitude by the benchmark solution was removed by the assimilative approach. The inferred boundary conditions tend to filter out noticeable spatial variations in the tidal amplitudes derived from the altimetry measurements that are prominent in the southern (deep-ocean) part of the study domain. The regression basis of the assimilation model also provides formal estimates of uncertainty in the solution. Estimates based on a simplified statistical model of the residuals suggest uncertainties for the full assimilation solution of order 2 cm.

On the other hand, the rms misfit between the boundary conditions inferred by the direct inverse method and the near-boundary observations interpolated to the model grid increased by nearly 50% compared with the smoothed version of the observations used to drive the benchmark model. In addition to aliased signals in the observational data, an additional contribution to this discrepancy may be that the constraints imposed on the spatial structure of the inferred boundary condition filtered out some real but small-scale features. This may be pertinent to the observed differences near the northern limits of the model domain where the highest boundary elevations were observed. Another factor may be the relative sparsity of interior observations in the northernmost part of the domain.

The sensitivity studies suggest several points. First, there is a strong relationship between the uncertainty of the inferred boundary conditions and the number and location of the assimilated data. Confidence in the inferred boundary conditions generally decreases with the increased distance of the data from the boundary, but also depends on the extent of the null spaces from which the boundary has reduced influence on the interior. Second, the assimilation of the offshore altimetric dataset in the northern (Labrador Sea) part of the study domain was not sufficient to reproduce the coastal sea level observations along the Labrador coast. This may be partly associated with limited altimetric data on the Labrador Shelf associated with interference from the seasonal ice cover, but may also reflect model inadequacies. Inclusion of coastal tide gauge data in the assimilation resulted in an improved local solution. Third, the assimilation of elevation data can help to improve the model’s fit to velocity observations, but with a limit. A further considerable improvement can be achieved if some velocity data are also assimilated. This is consistent with the importance of elevation gradients in the basic tidal dynamics, and with the strong influence of topography on currents. Inclusion of velocity as well as elevation data can be important in obtaining high-resolution representations of continental shelf tidal currents from assimilative models. Finally, our choice for the frictional parameter fields appears to be a reasonable approximation.

A number of areas deserve further investigation. The altimetric dataset used here provided just more than 3 yr of observations and the tidal coefficients derived from this source showed considerable spatial variability in areas of energetic mesoscale variability. Estimates with lower noise levels could be obtained from the extended TOPEX/Poseidon dataset now available. The model framework provides a motivation for obtaining, and a mechanism for making use of more tide gauge, bottom pressure, and (especially) moored current measurements for tidal analysis. More sophisticated error models would allow the calculation of more realistic model error estimates. This study used piecewise linear segments and a truncated Fourier expansion to span the elevation boundary values, but a more complicated set of basis functions and alternative approaches for constraining the inferred boundary conditions are possible. Areas of potential improvements in the model dynamics include the direct tidal generating force, loading tides, coastal resolution, variable Coriolis parameter, density stratification, the treatment of frictional effects, and the use of a spherical-polar coordinate system (e.g., Greenberg et al. 1998). Potential future applications include the extension of the approach to other tidal constituents for both sea level variability and shelf current applications, and to other regions.

Acknowledgments

We extend special thanks to Guoqi Han for making available the altimetry time series and analysis procedure used in this study. We also thank Dave Greenberg for contributions to grid generation and availability of the coastal sea level data, Hong Zhang for assembly of the other in situ datasets, Brian Petrie and Peter Smith for their discussions and internal review of the manuscript, and two anonymous reviewers for constructive comments. We also acknowledge the input and support of Bill Crawford, Mike Foreman, Charles Hannah, Jim Helbig, Dan Lynch, and Chris Naimie. This work was partially supported by the Hydrocarbons Environmental Forecasting Task and the Transportation Task (through the Canadian Hydrographic Service) of the (Canadian) Federal Interdepartmental Program of Energy, Research and Development (PERD).

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APPENDIX

Spatial Structure of b.c

Consider a set of N boundary nodes, n = [1, 2, 3, . . . , N]. Denote sn as a curve length measured counterclockwise along the boundary from the first (n = 1) node to node n. Following the open boundary, choose successively I boundary nodes, {ni} for i = [1, 2, . . . , I] in which the first and the last boundary nodes are always included (i.e., n1 = 1 and nI = N). Assume the following spatial structure for the b.c.:
i1520-0426-18-4-665-ea1
for any node n that falls within the boundary section spanned by ni and ni+1. In the above,
i1520-0426-18-4-665-ea2
that is, S is the total boundary distance plus a mean distance increment, and Ka and Kb are bounded by
i1520-0426-18-4-665-ea3

The first square bracket of (A1) contributes a linear interpolation of the controlling variables and the second bracket contributes a Fourier expansion to the elevation at any boundary node. The controlling variables and the Fourier coefficients, the a’s and b’s, are to be determined from the regression analysis. The assumption of linear spatial structures between the controlling nodes, which is often seen in the data assimilation literature (e.g., Lardner, 1992), yields piecewise straight lines as an approximation to the underlying true b.c. However, the quality of this approximation depends on the choice of the controlling nodes, which can be subjective. The inclusion of the Fourier expansion may relieve this dependence and relax the piecewise linear structure overall. Note that the Fourier modes start at one instead of zero. This is because the zero mode is not independent of the piecewise linear functions in the first bracket.

For succinctness, we introduce the following matrix notation. Denote ζc as a vector of controlling of variables
ζcζn1ζn2ζniζnIT
Denote L as a n × I Lagrangian linear interpolation matrix whose elements on the nth row are all zero except for two successive elements. Specifically, if ninni+1,
i1520-0426-18-4-665-eq2
where the colon sign in the matrix L stands for all the columns of the matrix. Suggested by the first panel of Fig. 4, boundary nodes n = 1 (A, Fig. 1), n = 112 (a point slightly north of E), n = 155 (F), n = 175 (G), n = 176 (H), and n = 184 (I) are picked up as the controlling nodes. Note that since HI is a separate boundary, there should no linear functions from G to I in calculation of the matrix L.
Use Hcos and Hsin to denote matrices consisting of the Fourier base functions, whose nth rows are
i1520-0426-18-4-665-ea6
respectively. In the above, consideration has been given for not applying the Fourier expansion on the piece of the boundary defined by n > m. This is a useful flexibility since, for example, in this study there is a short segment boundary, HI (n = 176 to 184), for which a linear function should be sufficient. Therefore m = 175 for this particular study.
With these matrix notations, the boundary condition vector ζ can be expressed as
i1520-0426-18-4-665-ea8
where
i1520-0426-18-4-665-ea9

Fig. 1.
Fig. 1.

Model domain, triangular mesh, and topography. The bathymetry is based on the ETOPO5 dataset, complemented by finer-resolution (order 7 km) Canadian hydrographic charts in shallow water (⩽1000 m). The minimum water depth in the model is 10 m

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 2.
Fig. 2.

Distribution of the observational datasets. The T/P tracks are sequentially numbered with letters a and b indicating the southern and northern ends. The current stations referred to in the text are numbered in italic bold face

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 3.
Fig. 3.

Contours of M2 elevation amplitude (cm) from the observational data

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 4.
Fig. 4.

The specified b.c. (dashed lines) with the observations (circles) superimposed. (top) Amplitude and (bottom) phase lags (relative to Greenwich)

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 5.
Fig. 5.

Green’s function maps for a unit (100 cm) set up on a few selected boundary nodes illustrating (a) “influential” nodes and (b) nodes with limited influence on the interior solutions

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 6.
Fig. 6.

Controlling points and basis (piecewise linear and trigonometric) functions used to constrain the spatial structure on the open boundary segments A–G. For clarity, only the first mode of the trigonometric functions is plotted for the segment AG. For the separate HI section, only the piecewise linear function is used for the basis function

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 7.
Fig. 7.

Changes in the rms distance misfit between the full assimilative model solution and observations, as a function of the number of Fourier modes used in the b.c. spatial structure. (a) Rms elevation misfit; (b) rms current misfit

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 8.
Fig. 8.

Comparison between the specified b.c. (dashed curves), the inferred b.c. from full interior data assimilation (solid curves), and the along-boundary observations (circles). (top) Elevation amplitudes and (bottom) elevation phase lags

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 9.
Fig. 9.

Elevation amplitudes (solid dark lines in cm) and phase lags (dashed dark lines in degrees) from (a) the benchmark solution and (b) the full interior data assimilative solution. These are superimposed on the contoured amplitudes (shades with white contours and larger black labels) from the raw data (Fig. 3)

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 10.
Fig. 10.

Geographical distribution of the elevation amplitude misfits for (a) the benchmark and (b) the assimilative solutions, with circles indicating those ≥ 3 cm and dots indicating smaller misfits

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 11.
Fig. 11.

Histograms and statistics of the elevation misfits by (left) the specified b.c. and (right) full interior assimilative solutions. (top) The misfits in amplitude, (middle) the misfits in phase lag, and (bottom) for the distances on a complex plane. The misfit statistics shown are the mean, std dev (std), rms, minimum (min), and maximum (max) values

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 12.
Fig. 12.

Surface tidal ellipses in the Grand Banks region from the full interior assimilative solution (solid ellipses) and from available observations nearest to the surface (dotted ellipses). Station numbers are indicated for observed ellipses

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 13.
Fig. 13.

Comparison between the observed tidal ellipses (dashed curves) and the modeled ellipses (solid curves) from (left) the benchmark and (right) full interior assimilative solutions. The phases are indicated by the radial lines, and the station locations are shown in Fig. 12

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 14.
Fig. 14.

Histograms and statistics of the current misfits by (left) the specified b.c. and (right) full interior assimilative solutions. Proceeding from top to bottom, the panels are for the misfits in the major axis, minor axis, orientation, and distances on a complex plane

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 15.
Fig. 15.

Track-by-track comparisons of the observational data (circles), the benchmark solution (dashed lines), and the data assimilative solution (solid lines). Distances are measured from “a” ends to “b” ends of each track (cf. Fig. 2). The hw value shown for each track is an estimate of the average half-width of the 95% confidence interval. The in situ results are plotted against a sequential data index after being sorted by distance along the coast increasing from the south, for the tide gauges, or latitude increasing from the south, for bottom pressure

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 16.
Fig. 16.

The amplitudes of the inferred elevation b.c. from the full and partial assimilation cases. The shaded zones are the first estimates of the 95% confidence intervals. The rms difference (rd) between each partial case and the full assimilation case, and the average value (hw) of the half-width of the confidence interval are indicated on each panel

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 17.
Fig. 17.

Comparisons between modeled (solid curves) and observed (dashed curves) middepth tidal ellipses at station 26 on the northern Grand Banks for various model solutions: (a) benchmark case, (b) elevation case, (c) decimated case, and (d) full interior assimilation case. The small square indicates a common phase. The site’s water depth is 198 m and the current meter was at 100 m above the seabed

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Fig. 18.
Fig. 18.

Sensitivity of (a) the rms elevation and (b) velocity misfits to scaling of the base frictional fields. Data for sensitivity cases are marked by circles and those for the base fields are marked by plus in addition

Citation: Journal of Atmospheric and Oceanic Technology 18, 4; 10.1175/1520-0426(2001)018<0665:AOADID>2.0.CO;2

Table 1.

Summary of rms distance misfits to the observational elevation and current data for each of the model cases. The misfits are included for both the assimilated and reserved portions of each dataset, as well as for the overall dataset in each. The numbers in parentheses for the assimilation cases are the percentages by which the respective misfit is reduced compared to the misfit for the benchmark case

Table 1.

1

The errors quoted here are for the harmonic data. In the literature, errors are sometimes reported as time-domain rms values, which will be smaller than ours by a factor of 1/(2)1/2.

2

The great circle, to which the mapping cylinder is tangent, goes through the points (54.9, −53.5) and (42, −46) in (latitude, longitude), and is more or less parallel to the Newfoundland Shelf break and divides the domain roughly into two halves. Along this line the distance distortion is zero, and the distortion is tolerably small elsewhere in the model domain.

3

In order for the total source error, ɛ = [ɛηλɛuλɛυ], to have a common unit (taken to be that of elevation), λ has a unit of time.

4

The null space is generally a linear combination of all the boundary nodes, rather than comprising only isolated nodes, in which case it could be easily eliminated.

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