A description of tides near the Chesapeake Bay entrance using in situ data with an adjoint model

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. C3, 3075, doi: /2001jc000820, 2003 A description of tides near the Chesapeake Bay entrance using in situ data with an adjoint model Z. R. Hallock, P. Pistek, J. W. Book, and J. L. Miller Naval Research Laboratory, Stennis Space Center, Mississippi, USA L. K. Shay Rosenstiel School of Marine and Atmospheric Science, Miami, Florida, USA H. T. Perkins Naval Research Laboratory, Stennis Space Center, Mississippi, USA Received 30 January 2001; revised 16 September 2002; accepted 31 October 2002; published 13 March [1] Time series of surface elevation and current velocity in the vicinity of the Chesapeake Bay entrance are assimilated into a linear, barotropic model using an iterative adjoint method. Tide surface elevations, NOAA-NOS tidal data, moored acoustic Doppler current profiler (ADCP), and ocean surface current radar (OSCR) data are included in the assimilation. Data are analyzed for three tidal constituents (M 2, S 2, O 1 ); tidal series are constructed from these constituents and then used in the model assimilations. Statistics of predicted currents compare favorably with those calculated from data not used in the assimilation. An error analysis of the data distribution is performed for the M 2 tidal component, showing lower errors near assimilation data series but large errors, particularly for velocity, at some locations. Addition of a hypothetical current mooring dramatically reduces errors over most of the domain. We believe these results give the most demonstrably accurate description now available of tidal currents for this region of strong currents and heavy shipping. INDEX TERMS: 4219 Oceanography: General: Continental shelf processes; 4560 Oceanography: Physical: Surface waves and tides (1255); 4263 Oceanography: General: Ocean prediction; 4255 Oceanography: General: Numerical modeling; KEYWORDS: tides, data assimilation, Chesapeake Bay Citation: Hallock, Z. R., P. Pistek, J. W. Book, J. L. Miller, L. K. Shay, and H. T. Perkins, A description of tides near the Chesapeake Bay entrance using in situ data with an adjoint model, J. Geophys. Res., 108(C3), 3075, doi: /2001jc000820, Introduction [2] The objective of the present study is to use a dataassimilative numerical model to describe barotropic tidal variability in the vicinity of the entrance to the Chesapeake Bay (CBE) in the region shown in Figure 1. The large area of the Chesapeake Bay and its narrow, shallow entrance give rise to strong tidal currents, making the entrance of hydrodynamic interest. It is also, of course, a region of heavy shipping, both commercial and naval. In this and similar regions the barotropic tide is an ever-present phenomenon, affecting other coastal and estuarine processes (such as coastal fronts and river plumes), as well as human activities (e.g., shipping). The method described here allows the prediction of barotropic tidal elevations and currents to accuracies of a few centimeters and centimeters per second, respectively. Our approach is to use an adjoint technique of data assimilation (described in section 3). Data assimilation techniques have been used for decades in meteorology and, more recently, in oceanography. A comprehensive review is Copyright 2003 by the American Geophysical Union /03/2001JC provided by Ghil and Malanotte-Rizzoli [1991]. A variant of this technique has been developed by Thacker and Long [1988] and subsequently advanced by Griffin and Thompson [1996] by applying it to observations on the shelf east of Nova Scotia. We have successfully applied a similar technique to data collected in various areas: on the California coast, in the Yellow sea, and in the Korea-Tsushima Strait [Pistek et al., 1997; Perkins and Pistek, 1998; Book et al., 2001]. With this approach, a solution to the linear, shallow water equations is found that best agrees with in situ measurements over a limited spatial and temporal domain. To find a solution that agrees with available measurements, a related system of equations, the adjoint of the original equations, is used to iteratively calculate the gradient of model prediction-data mismatch conditions with respect to the boundary conditions. This gradient estimate is iteratively used to adjust the boundary conditions until an optimum fit over the entire space-time domain is achieved. Hence boundary conditions are derived along with the solution, using in situ data within the domain of the model. [3] Currents in the region immediately outside the CBE are dominated by tides, density-driven flows, and wind forcing. Barotropic variability is due primarily to the M

2 20-2 HALLOCK ET AL.: TIDES NEAR THE CHESAPEAKE BAY ENTRANCE when assimilated into the model, yields an accurate description of the tides. That is, where independent (not assimilated) time series exist, statistics of model velocities compare favorably with those based on observations. However, an error analysis shows that a broader data distribution is needed to establish demonstrably correct fields in some parts of the model domain. The approach used here does not require a priori initial and boundary values. It is an efficient method which can describe the barotropic tidal environment in the vicinity of and just outside the CBE and to provide such a description with an assessment of its validity. [5] The following section describes the observations, consisting of elevation and current time series, that are assimilated into the model to provide a short-term prediction, as well as independent data sets used for comparisons with model results. Section 3 describes the adjoint model formulation. In section 4 we discuss results of the model calculations, compare statistics with those based on historical data sets, and assess the uniqueness/reliability of the solution for the M 2 constituent. Figure 1. Map of the COPE-1 area showing locations of data sets used in the assimilations. Elevation series, h 1, h 2, (triangles), ADCP (diamond), and OSCR gridpoints (circles). A hypothetical mooring location (square) is used in the error analysis. tide [Shay et al., 2001; Clarke, 1991], while the Chesapeake outflow plume, modulated by wind forcing, constitutes most of the baroclinic flow. Farther offshore (>40 km), tidal currents become less important and the flow is increasingly dominated by slope currents and shelf waves [Biscaye et al., 1994]. Shay et al. [2001] report on the strength of the M 2 tidal current constituent from the immediate CBE area to about 25 km south of Cape Henry, based on ocean surface current radar (OSCR) and moored acoustic Doppler current profiler (ADCP) records (these same records are utilized in the present study but are assimilated into the adjoint model). Their results show dominance of the M 2 tide at the CBE (about 60% of the signal) with decreasing amplitudes southward. Biscaye et al. [1994], who describe observations just northeast of the region shown in Figure 1, find that subtidal fluctuations are aligned generally alongshore but that except very close to the coast, tidal currents are polarized across isobaths (70% 90% of variance). phase change in surface height for the dominant M 2 tide along the eastern seaboard is quite small, generally less than an hour between New York and the CBE (see, for example, Doodson [1958]). [4] In this work we assimilate time series of vertically averaged, in situ tidal current velocity, surface elevation, and surface current velocities obtained from remotely sensed OSCR measurements. To estimate the effectiveness of the available data in describing the M 2 constituent over the model domain, an error map is developed based on the data distributions in space and time. We find that this sparse collection of current velocity and surface elevation data, 2. Observations 2.1. Data Used in the Assimilations [6] Concurrent observations of time series of current velocity and surface elevation were used as input to the adjoint model assimilation. Observation locations are shown in Figure 1. [7] During the first Chesapeake Outflow Plume Experiment (COPE-1; September 1996), two upward looking acoustic Doppler current profilers (ADCPs) were deployed within 1 km of each other near N, W from 9 to 27 September 1996, in water depths of about 17 m. The data records from these instruments are virtually the same for the purposes of this study, so only one is used. Current velocity samples were acquired in 1 m vertical bins and 1 min ensemble averages of 60 pings each, giving an uncertainty in speed of about 1.5 cm s 1. The data were temporally smoothed to remove energy at periods shorter than 40 minutes and subsampled at 20 min to be compatible with other data sets. The location of this instrument is denoted by the diamond in Figure 1. [8] Surface elevation data from two locations (triangles, Figure 1) during COPE-1 are used. These include a NOAA- NOS prediction, based on harmonic analysis of historical data, for a location near Cape Henry (site 1) and a nearshore tide (pressure) gauge south of Cape Henry deployed as part of COPE-1 (site 2) (D. Johnson personal communication, 2001). [9] An ocean surface current radar (OSCR) system in its HF mode (25.4 MHz) was deployed from 15 September to 6 October 1996 to include the intensive observation period of COPE-1. Details of the OSCR land-based radar installations as well as preliminary data processing are described by Shay et al. [2001] and Haus et al. [1998]. Briefly, the OSCR system provides a two-dimensional record of surface current velocity every 20 min, with spatial resolution of about 1 km. The OSCR coverage during COPE-1 was somewhat larger than shown by the subsampled set of 12 gridpoints (filled circles) in Figure 1. [10] Since our objective is to describe barotropic tidal variability, all input data were analyzed for tidal content.

3 HALLOCK ET AL.: TIDES NEAR THE CHESAPEAKE BAY ENTRANCE 20-3 The three constituents, M 2, S 2, and O 1, the most energetic constituents in the model domain, were fitted to all data series by least squares, and tidal time series of the sum of these constituents were calculated. Although the N 2 constituent is of the same order as S 2 in this region, it cannot be separated from M 2 with the length of the available data records; hence we assume that N 2 energy (the ratio of N 2 to M 2 amplitude is about 0.2) is folded into the M 2 estimates. These tidal time series are used in all subsequent calculations. The tidal fitting is straightforward for the elevation sites and for the ADCP record after calculation of a depthaveraged time series. However, since the OSCR data represent only surface current, we must ask how representative they are of the depth-averaged flow. Figure 2 shows vertical profiles of ADCP tidal constituent ellipse parameters. Clearly, M 2 is dominant and all amplitudes are largely depth-independent. S 2 and O 1 show more change of ellipse orientation than M 2, particularly for O 1 near the bottom. Hence the use of a barotropic tide model is quite appropriate. The subset of OSCR points (Figure 1) was therefore selected to avoid the highly stratified outflow plume [e.g., Hallock and Marmorino, 2002], which flows southward along the coast, extending out about 10 km except during strong upwelling wind events. The selected OSCR records are thus considered to be representative of depth-averaged flow Historical Data Used for Validation [11] Current meter data were acquired during four deployments between May 1985 and March 1986, from the CBE area (W. Boicourt, personal communication, 1998). These records were depth-averaged (where more than one instrument was on a mooring) and fit with the tidal constituents. Principal standard deviations (PSDs) [Preisendorfer, 1988, Figure 2.1] were computed with reconstituted tidal series for comparison with model results. Similarly, PSDs were computed with tides based on records from five ADCP moorings deployed from early October to mid November 1997 and from one ADCP mooring deployed for about 10 days in summer 2000 in the same region as COPE-1. These data were collected as part of NRL s COPE-3 [Hallock and Marmorino, 2002] and COJET (D. Johnson, personal communication, 2001) experiments. 3. Model Description 3.1. Model Equations [12] The method is described in detail by Griffin and Thompson [1998]; here only the essential steps are shown. We assume that the study area (the model domain is the area depicted in Figure 1) can be adequately represented by the linearized, vertically integrated (shallow water) equations, u t fv þ gh x ¼ ku H v t þ fu þ gh y ¼ kv H h t þ ðuhþ x þðvhþ y ¼ 0; ð1þ Figure 2. Tidal ellipse parameters for COPE-1 ADCP record. (a) Amplitude (negative semiminor axis value indicates clockwise rotation of the current vector). (b) Orientation. where u and v are vertically integrated velocity components in the x (eastward) and y (northward) directions, respectively, and h is free surface elevation. H is water depth, f is the Coriolis parameter, and k is a linear bottom drag coefficient. Since only tides are considered, there is no direct external forcing term present. Instead, the model is effectively forced from the open boundaries through the assimilation of observational data in the interior of the model domain. [13] Boundary conditions are needed to control these equations where the ocean meets the land or an edge of the model domain. For land boundaries a condition of zero normal flow is used. For open ocean boundaries the condition is, n ¼ t pffiffiffiffiffiffi H gh h; ð2þ H where n is the normal component of velocity at the boundary and t is a specified normal flux at the boundary. The term proportional to elevation in equation (2) is a radiation condition which minimizes reflections of outgoing waves (positive sign for upper or right boundary) from the boundary [Thompson et al., 2000] Discrete Formulation [14] The equations are solved on a rectangular grid in x and y over uniformly spaced times. An Arakawa C grid with

4 20-4 HALLOCK ET AL.: TIDES NEAR THE CHESAPEAKE BAY ENTRANCE a size of M y by M x is used. The grid resolution is approximately 1.4 km for the bathymetry and 2.9 km for u, v, and h. Let S y =(M y 1)/2 and let S x =(M x 1)/2. The sizes of u, v, and h are then S y by S x +1,S y +1byS x, and S y by S x, respectively. Each of these gridded variables can be identified by a single index. Time steps are denoted by the index n = 1,..., N. All dependent variables, which include boundary velocities, are represented at time t n by the vector, Xn T ¼ ½ u 1n; u 2n ;...; v 1n ; v 2n ;...; h 1n ; h 2n ;... Š; ð3þ where the column vector X n, here represented by its transpose (indicated by the superscript T ), is of length M = (3M y M x M y M x 1)/4. The dynamical equations together with their boundary conditions, are then expressed as a matrix operator, D 1, that advances X n by one time step and a matrix operator, D 2, that incorporates the boundary forcing vectors, t n =[t 1n,t 2,n...]: X n ¼ D 1 X n 1 þ D 2 t n ; n ¼ 1;...; N: ð4þ [15] Observational data are represented as a sequence of vectors d n of length L. Each value of one of these vectors represents an observation at a particular location at time n. A measure of the mismatch between model solution and data over all measurement points is evaluated as a cost function, J ¼ 1 X N 2s 2 0 n¼0 ðax n d n Þ T WðAX n d n Þ; ð5þ where the diagonal elements of matrix W are weights used to nondimensionalize and adjust the relative influence of various data. Hence J is implicitly a function of all t n.in this study the weights set the influences so that a 0.5 cm s 1 prediction error in ADCP velocity has equal effect on J as a 1.0 cm/s error in OSCR data and a 3.0 cm error in surface elevation. Only the relative values of the weights affect the solution. The different weights used for the ADCP and OSCR velocities is based on the expectation that the ADCP measurements more accurately represent the barotropic tide than do the OSCR measurements. Different weights for velocity and surface elevation are chosen to enhance the effect of velocity measurements. A is an L by M matrix that performs bilinear interpolation between the grid points and the data locations. s 0 is an overall weighting factor to scale J. [16] In presenting results below, J is represented as a sum of contributions from different data types, thus giving a measure of the success in assimilating a particular type. For example, we can take J = J c + J h where J c and J h, each of the form of equation (5), are respective contributions from current and elevation data. [17] We select the normal fluxes, t, in equation (4) as control variables. The solution we seek minimizes the cost function, J, with respect to these control variables. Fluxes (rather than elevations) are chosen because they are easier to specify as forcing (K. Thompson, personal communication, 2000). The bottom friction parameter, k, is not explicitly included as a control variable but a series of model runs using k between ms 1 and ms 1 shows a relative minimum in the cost function, where J 5% of its initial value, for k ms 1. [18] In particular, transports across each of the open boundaries of the model domain are expressed as the sum of nonoverlapping boxcar functions with unknown amplitudes, giving 14 unknown time series to be determined by the assimilation. The boxcar functions group adjacent boundary fluxes together thus reducing model resolution with regard to the fluxes (radiation condition resolution is not reduced). The distribution is deliberately coarse since a finer resolution would not adequately constrain the model with the distribution of observations available for this study. For this same reason the unknown amplitudes are defined in 1.5 hour time steps rather than at the fine time steps needed to satisfy the CFL limit of the model. Amplitudes for the functions for times between these coarse time steps are determined by a temporal interpolation function (see, for example, equation 7 of Griffin and Thompson [1996]). The use of this interpolation function with 1.5 hourly data can adequately reproduce fluctuations at the S 2 frequency, the highest frequency considered in this study. There is a tradeoff between independent data input and control variable resolution. This issue is discussed in section Solution Equations [19] A solution for the assimilated fields that satisfies the dynamical equations and minimizes the mismatch with data can be obtained by finding a saddle point of the Lagrange function, L¼J þ XN l T n n¼1 ð X n D 1 X n 1 D 2 t n Þ: ð6þ Setting the derivatives of L with respect to the Lagrange multipliers (l) and model variables (X ) to zero leads to X n D 1 X n 1 D 2 t n ¼ 0; n ¼ 1;...; N ð7þ l n D* 1 l nþ1 þ 1 s 2 A T W T ðax n d n Þ ¼ 0; n ¼ N;...; 0; ð8þ 0 where equation (7) is just the dynamical equation (4), which the solution must satisfy exactly. Equation (8) is the adjoint equation (for l), where the matrix D* 1 is the adjoint of D 1, but for linear dynamics it is the transpose of D 1. This equation steps backward in time, calculating the Lagrange multiplier vectors, l n. The otherwise undefined vectors are assigned the values l 0 ¼ 0; l nþ1 ¼ 0; X 0 ¼ 0: ð9þ [20] Given a first guess for the control variables, t, the above system of equations can be solved, yielding values for the Lagrange multipliers. Setting the derivative of L with respect to the control variables to zero n D* 2 l n ¼ 0; n ¼ N...0: ð10þ Equation (10) allows the calculation of the derivatives of J with respect to t for this particular guess of t. A conjugate

5 HALLOCK ET AL.: TIDES NEAR THE CHESAPEAKE BAY ENTRANCE 20-5 gradient technique is used to adjust the values of t from their first guess values using this derivative information. This procedure is repeated iteratively until convergence on the minimum of J has been obtained. For this study, convergence was obtained within 10 iterations. 4. Results and Comparisons [21] The adjoint model generates a predictive data set consisting of horizontal velocity and surface elevation at each gridpoint in space and time. We emphasize here that since we are only considering tides, mean currents do not develop and solutions consist of oscillations about zero. The solution fits the dynamics of the model exactly and with this strong constraint fits the various input data sets in a leastsquared sense. Hence one test of model performance is how well each of the assimilated data sets are reproduced by the model. Figure 3 shows a comparison of modeled and observed elevation at the site near Cape Henry and velocity at the ADCP Site. Good agreement is apparent in both amplitude and phase for this example, particularly for elevation. For velocity (Figures 3b and 3c), U is slightly underestimated and V is overestimated, implying a difference in direction as well as magnitude. A small phase difference is also evident in the velocity components. Comparisons at other sites (not shown) for both elevation and velocity exhibit about the same level of agreement Correlation and Error [22] To provide a concise, quantitative comparison of observations and model results correlation (r), root-meansquared difference (E ) and regression slope (S ) are calculated and listed in Table 1a. For the elevation h for example, r ¼ h oh m ; s ho s hm h E ¼ ðh o h m Þ 2 i1 2 ; S ¼ h oh m s h 2 o ð11þ where the subscripts o and m denote observation and model, respectively. s is standard deviation, and the overbar denotes a time average. Statistics for velocities are calculated similarly but use the complex correlation coefficient [Kundu, 1976] and associated regression coefficient which yields correlation and regression magnitudes (r, S) and a veering angle (f); this angle is essentially an average direction difference between the observed and modeled velocity. For the 12 assimilated OSCR velocities (Figure 1), minimum and maximum values are listed in Table 1a (the 87 intermediate, nonassimilated points yield consistent error statistics). In Table 1a the annotation SW next to a statistic indicates that the value is at or near the southwest corner of the OSCR box shown in Figure 1, similarly for SE and NW. Statistics calculated at the OSCR points west of the box (not shown) indicate large errors; this is expected because of stratification effects of the outflow plume (discussed further below). If the model were in perfect agreement with the observations, r and S would be equal to unity and E and f would be zero. The correlations (Table 1a) are everywhere quite high, all RMS Figure 3. Examples of observed (solid) and modeled (dashed) series of tidal (a) elevation near Cape Henry and (b, c) velocity components at the ADCP site. differences for elevation are about 7 cm near at the Bay entrance and 3 cm south of Cape Henry, but slopes are near unity, suggesting small phase differences. Velocity errors range from about 1 to 5 cm s 1. The model versus ADCP velocity error (E ADCP = 4.1 cm s 1 ) is near the maximum model versus OSCR error (E OSCR ) within the rectangle where data were subsampled for assimilation. The regression slope for the ADCP velocity is slightly less than one and the veering angle is essentially zero. E OSCR values are greater near the west side of the assimilation rectangle. In particular, E OSCR is maximum at the northwest corner and f OSCR is greatest to the southwest. S OSCR differs significantly from unity at both these locations. Furthermore, E OSCR (not listed) increases rapidly to the west of this region, most likely because of stratification associated with the outflow plume. This increase in errors closer to the shore results even when these additional OSCR data are assimilated (not shown). The large errors west of the the OSCR points shown in Figure 1 do not imply that the OSCR-measured surface currents are inaccurate, nor do they imply that the model prediction for the barotropic tide is necessarily wrong. More likely, it shows that the measured surface current is not representative of the depth-averaged flow associated with the tide. This view is supported by results in the next subsection.

6 20-6 HALLOCK ET AL.: TIDES NEAR THE CHESAPEAKE BAY ENTRANCE Table 1a. Error Statistics h 1 h 2 ADCP OSCR min OSCR max r E (cgs) (SE) 4.5 (NW) j ( ) (SE) 8.3 (SW) S (NW) 1.07 (NW) [23] In Table 1b we list the same statistics for a model run where no OSCR velocities are assimilated. It is not surprising that E OSCR is greater in this case and that E ADCP is less. What is interesting is that E h2 is nearly twice the value in Table 1a, and both elevation slopes are farther from one Independent Data Comparisons [24] It is useful to compare modeled tidal current statistics to those of independent observations. Figure 4 is a map of principal standard deviation (PSD) crosses (standard deviations of velocity components along principal axes; see Preisendorfer [1988]) at selected gridpoints of the adjoint model together with those calculated with W. Boicourt s (personal communication, 1998) current meter data, the 1997 COPE-3 and 2000 COJET ADCP records. Good agreement between historical data and adjoint model results is evident, both in magnitudes and orientations of the PSD crosses in and near locations where data were assimilated. Even near the eastern boundary of the domain, near N, assimilation results are similar to Boicourt s easternmost mooring (slightly outside the model domain). However, at the COJET (D. Johnson, personal communication, 2001) mooring site, just outside the northern boundary of the model domain, a discrepancy in orientation of the principal axis is evident. Furthermore, the model-predicted orientation in this area disagrees with the historical record [Biscaye et al., 1994; Clarke, 1991]. [25] A map of the predicted M 2 elevation amplitude and phase (Figure 5) shows little variation away from the CBE with a gradual decrease in amplitude and increase in phase on approaching Cape Henry. The amplitude (0.36 m) and phase (15 ) in the CBE are consistent with the results of Brown and Fisher [1988, Figure 18]; details of the shapes of the contours in Figure 5 differ but the basic levels agree Uniqueness of the Solution [26] Figure 4 shows that the model predictions away from the assimilated data locations may be inaccurate in parts of the domain. There are three possible causes for such error. The first is that the model dynamics may be incorrect in such a way to require incorrect solutions away from the data sites in order to generate accurate solutions at the data sites. Second, data errors at assimilation locations could be influencing the optimized boundary conditions and thus inducing error in the entire model domain. The third Figure 4. Map showing tidal principal standard deviation crosses for adjoint model output (thin), Boicourt s current meter data (thick), COPE-3 ADCP data, and COJET ADCP data (dashed). possibility is that particular realistic solutions that are dynamically permissible in the model may not affect the tides at the assimilated data locations. Such solutions produce minimal values of J (equation (5)), and therefore the adjoint modeling technique cannot determine them. Table 1b. Error Statistics (OSCR Data Not Assimilated) h 1 h 2 ADCP OSCR min OSCR max r (NE) 0.91 E (cgs) (SE) 10.2 (NW) j ( ) (SW) S (NW) 1.21 (NE) Figure 5. Map showing M 2 elevation amplitude (dashed, in meters) and phase (solid, in degrees) for adjoint results.

7 HALLOCK ET AL.: TIDES NEAR THE CHESAPEAKE BAY ENTRANCE 20-7 Model predictions at locations in the domain where these solutions have significant energies are not unique. The model predicts the portion of the tides at these sites contained in solutions affecting the tides at the data locations, but all other portions of the tides at these sites are not determined. Imposing further criteria on the cost function would, in effect, select amplitudes for these undetermined solutions, based on the criteria. By not doing this, the amplitudes of the undetermined solutions stay near their initial values of zero (equation (9)), and thus we do not bias the results to a particular form. Because the undermined solutions are valid solutions of the model, their forms can be found and the errors they may cause (if they are in fact present) can be estimated. In this approach, the results depend only on the data sampling scheme, not the actual data. [27] As shown by Dowd and Thompson [1996] and Thompson et al. [2000], equation (4) can be converted to the frequency domain, thus becoming m g ¼ R g b: ð12þ In this equation, m g is a length 2 M vector containing the sine and cosine amplitudes from a harmonic analysis of all X n, R g is a response matrix, and b is a vector of length 28 containing the sine and cosine amplitudes from a harmonic analysis of all the 14 t time series. R g was found (from the time domain model) using the second method described in Appendix A of Dowd and Thompson [1996]. A second equation is needed to relate the observations to the boundary forcing, m d ¼ R d b: ð13þ m d is a length 2 L vector containing the sine and cosine amplitudes from a harmonic analysis of all d n, and R d is a different response matrix. R d can be derived from R g and A 1. After converting W to its frequency domain equivalent, W f, covariance estimates for m d and m g can be found as was done in Appendix B of Dowd and Thompson [1996]. However, a different approach was used for this study because of several problems encountered in the application of the above method. It requires a matrix inversion of R d T W f R d, which is singular or nearly singular for this application. Also, more realistic estimates can be obtained by prohibiting solutions from exceeding particular velocity and elevation thresholds. Such a requirement is not included in the standard regression calculations of Dowd and Thompson [1996]. [28] By using 14 control variables per time step, we have in effect constrained the model for a particular frequency (here and in all subsequent analysis we will consider only the M 2 constituent) to 28 degrees of freedom (the length of b). By a singular value decomposition of R d, R d = USV T,a unitary matrix, V, is obtained that spans this space and is ordered according to the amount of response at the data sites (given by the non-zero elements of the diagonal matrix S ) that will be produced by using columns of V as b in equation (13). The columns of V are orthonormal, and each one is associated with a particular solution of the model (a column of U for m d and R g times the column of V for m g ). Therefore any permissible model prediction can be represented as a weighted sum of R g times the columns of V, m g ¼ X28 j¼1 a j R g V j : ð14þ The expected errors in m g directly follow from estimates of the expected errors in the weights. For ease of notation we will refer to the columns of V as the basis vectors of the problem. Consider a spurious flow generated by a particular basis vector. The cost function value for such a flow is J ¼ a 2 j V T j R T d W f R d V j : ð15þ With each iteration J decreases toward a minimum, J min.as the number of iterations increases further, J eventually crosses J min and then oscillates about it, yielding an uncertainty of the order of dj. An estimate for the expected error due to undetermined solutions in a j is thus s aj sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ dj Vj T R T d W : f R d V j ð16þ High values of s aj indicate that the solution associated with V j is unconstrained by the model. [29] For estimating s aj for the Chesapeake Bay assimilation we used the difference in J between the ninth and tenth iterations as an estimate of dj. For the data configuration used in this study, solutions associated with the first 10 basis vectors are well determined. Solutions associated with basis vectors 11 through 14 are somewhat determined and solutions associated with basis vectors 15 through 28 are undetermined. Although a 15:28 are not well constrained by the model, if these weights become too high, tides in parts of the domain become unrealistic. An upper bound on s aj can be found by requiring all values in m gj = s aj R g V j to be below 2 m or 1 m/s. This calculation caps the values of s a15:28 below those determined by equation (16). Finally, an estimate of the error due to undetermined solutions at the model grid points can be found by using equation (14) and the calculated or capped values of s aj, s mg vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ux 28 ¼ t 2 s aj R g V j ; ð17þ j¼1 where the square and square root in (17) denote element by element operations. [30] Figure 6 shows the estimate of h tidal amplitude error and an average of the estimates of the u and v tidal amplitude errors, all derived from equation (17). Figure 6a indicates that h predictions are relatively well determined and unique throughout most of the model domain. Only the southeast and northwest corners have large areas of potentially high h error. In contrast, Figure 6b indicates that the velocity is not well determined or unique away from the data sites. Areas along the north and south border, the southeast corner, and the Chesapeake Bay mouth all have potentially high velocity prediction errors. This map qualitatively agrees with the comparisons presented in Figure 4.

8 20-8 HALLOCK ET AL.: TIDES NEAR THE CHESAPEAKE BAY ENTRANCE Figure 6. second). Map of M 2 prediction error for (a) elevation (centimeters) and (b) velocity (centimeters per For example, the Boicourt current meter site near N and W agrees well with the model s PSD crosses; here the estimate of the average of the velocity component amplitude errors is 5 cm s 1. At the COJET mooring site, just outside the northern boundary of the model, measured velocities are quite different from the model predictions to the south; here the nearest estimate of the average of the velocity component amplitude errors is 36 cm s 1. [31] Since the available velocity data for assimilation are confined to a relatively small portion of the model domain, Figure 7. Map of M 2 prediction error for (a) elevation (centimeters) and (b) velocity (centimeters per second) with added hypothetical velocity mooring data.

9 HALLOCK ET AL.: TIDES NEAR THE CHESAPEAKE BAY ENTRANCE 20-9 it is instructive to evaluate model performance with the addition of a hypothetical mooring close to the northern boundary (see Figure 1). This can be done because the calculations do not depend explicitly on the assimilation data but only on their distribution in space and time. The resulting error maps for this case (Figure 7) show a dramatic reduction in errors for both elevation and velocity. Hence a relatively small number of input data sets, if optimally located, will yield accurate model prediction over most of the model domain. 5. Summary [32] In the region around the CBE, tidal elevations and velocities for the M 2, S 2 and O 1 constituents, can be described well using a relatively small number of in situ data series (several of elevation and several of velocity) and the linear barotropic adjoint method of Griffin and Thompson [1996]. Model predictions, using available observational data acquired during COPE-1 in September of 1996, agree with data used in the assimilation (Figure 3, Table 1). Comparisons with independent observations of depth-averaged velocity are quite good (Figures 4 and 5) over part of the model domain, but discrepancies are apparent near the northern boundary. An error analysis of the method shows good results for elevation, considering the sparse data distribution. Those for velocity are less satisfactory, particularly in the northern part of the model domain. However, the addition of only one or two more velocity data series could dramatically improve the prediction (Figure 7). The error analysis method provides an effective tool for sampling design. The method described allows, using relatively few direct measurements, the prediction of tidal elevations to within a few cm and currents to within a few cm s 1. [33] Acknowledgments. We are indebted to Bill Boicourt for providing his extensive current data for use in our comparisons. We thank Keith Thompson for providing assimilation modeling guidance. Don Johnson provided the COPE-1 tide gauge data at site 5 used in the assimilations and the COJET ADCP data used for comparisons. This work was supported by ONR through PE-61153N (WU 5938 and WU 7815). The Cope-1 OSCR measurements were acquired on a grant from ONR Remote Sensing Division (N ) to the University of Miami. L. K. Shay further acknowledges support from ONR Numerical Modeling Division (N ). References Biscaye, P. E., C. N. Flagg, and P. G. Falkowski, The Shelf Edge Exchange Process experiment, SEEP-II: An introduction to hypotheses, results and conclusions, Deep Sea Res., Part II, 41, , Book, J. W., W. J. Teague, P. Pistek, H. T. Perkins, B. H. Choi, G. A. Jacobs, M. S. Suk, K. I. Chang, and J. C. 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W., Principal Component Analysis, 425 pp., Elsevier Sci., New York, Shay, L. K., T. M. Cook, B. K. Haus, H. C. Graber, J. Martinez, and Z. R. Hallock, The strength of the M 2 tide at the Chesapeake Bay mouth, J. Phys. Oceanogr., 31(2), , Thacker, W. C., and R. B. Long, Fitting dynamics to data, J. Geophys. Res., 93, , Thompson, K. R., M. Dowd, Y. Lu, and B. Smith, Oceanographic data assimilation and regression analysis, Environmetrics, 11, , J. W. Book, Z. R. Hallock, J. L. Miller, H. T. Perkins, and P. Pistek, Naval Research Laboratory, Code 7332, Stennis Space Center, MS 39529, USA. (book@nrlssc.navy.mil; hallock@nrlssc.navy.mil; jmiller@onrifo.navy.mil; hperkins@nrlssc.navy.mil; pistek@ssc.usm.edu) L. K. Shay, Rosentiel School of Marine and Atmospheric Science, 4600 Rickenbacker Causeway, Miami, FL 33149, USA. (nick@rsmas.miami.edu)