## 1. Introduction

Data assimilation can be an expensive business in terms of both computational time and storage (Talagrand 1997). Given the present state of the art, coastal dynamical models are complicated to such a degree that simplified data assimilation techniques are generally required to make computations feasible. Hence, it is important to understand how different simplifications affect the quality of the solution in the context of coastal ocean data assimilation.

The problem of finding the best estimate of an entire time series, given a prior estimate and a series of observations of that time series, is known as the smoothing problem (Bennett 1992, p. 112). The generalized inverse method (GIM), a variational data assimilation scheme, can be used to find a solution to a smoothing problem that is optimal in a least square sense. The GIM yields a solution minimizing a cost functional that is a sum of penalties on errors in the inputs such as the model, initial and boundary conditions, and the data. From a Bayesian statistical perspective the method seeks the most probable state in the space of possible solutions. In the linear Gaussian case this is equivalent to minimizing the error variance of the solution at each point, and at each time. Estimates of the error covariances of the inputs are required for the definition of the penalty terms, and are prerequisites for obtaining the optimal solution. The GIM uses the model as a dynamically based interpolator between the data, providing correction to the model solution globally in space and time. If the measurements contain valuable information about the flows in the past or the future, perhaps somewhere distant from the measurement locations, this information will be acknowledged and “transmitted” by the model. In this way estimates of initial and open boundary values may be improved substantially.

Another class of data assimilation methods are solutions to the “filtering problem,” that is, the problem of finding the best estimate of a time series at a given time, which we may call the present, given only previous and contemporaneous observations, but no subsequent or “future” observations (see Bennett 1992, p. 90). The Kalman filter (KF) gives the solution to the optimal filtering problem (Gelfand and Fomin 1963; Miller 1996). It is a sequential algorithm: the model is run forward, and when the measurements are available, the forecast is corrected by extrapolating forecast–data discrepancies to all state variables. Thus, data assimilation is done in repeated cycles: forecast (model run) followed by correction (analysis). Extrapolation from the data to the full state space is accomplished by means of a gain matrix that evolves with time in such a way that the analysis field has minimum error variance conditional on all past data. Note that at each stage only present data are used for the correction. So, the KF does not require retention of past data. All of the influence of past data is embedded in the evolution of the forecast error covariance. The KF, like the GIM, requires specification of the input error covariances. In the linear case, the KF yields the same result as the GIM after the last data in a series of measurements are assimilated, but remains in general suboptimal compared to the GIM at earlier steps. Filtering methods are especially useful in such applications as numerical weather prediction, in which going back and correcting past estimates is of limited interest.

In oceanographic research, the task is often to reconstruct past flows with the goal of understanding underlying dynamical processes, rather than producing forecasts. In this case, to recover the true flow at any particular time, it would be sensible to use all available data. However, computational costs limit use of GIM with realistic models. In particular, a literal application of the theory would require storage of the model error term at every spatial node and at every time step of the integration period. Computational costs are also a big obstacle for the use of KF. Although the scheme is sequential and does not require storing model errors for all times, the equations for the forecast error covariance matrix, of size *M* × *M,* where *M* is the length of the state vector, must be integrated in time. Exact implementation of such a scheme for a realistic oceanographic model would obviously be impractical.

As an approximation to the KF, optimal interpolation (OI; cf. Daley 1991, p. 98) has been developed. For example, Oke et al. (2001) used an OI scheme to assimilate high-frequency (HF) radar surface current data into a primitive equation model of circulation on the mid-Oregon shelf. The gain matrix was not evolved from one assimilation cycle to the next, so the cost of computation was not significantly higher than for a simple forward model run. However, for OI, the gain matrix still must be chosen somehow. From a statistical perspective, the optimal stationary gain matrix would be the limiting (as *t* → ∞) KF forecast error covariance, which typically becomes stationary after a long period of integration. Instead of deriving this stationary forecast error covariance (a challenging task for a primitive equation model), Oke et al. (2001) used a scaled model solution error covariance. This was computed from averaging over an ensemble of model simulations for 18 different summers forced with observed winds. Thus, it was assumed that data assimilation reduced the error variance of the forecast in comparison with that of the prior model, but left the correlation structure of the errors unchanged. Although ad hoc, this scheme has the virtue of yielding an anisotropic inhomogeneous forecast error covariance with dynamically sensible length scales consistent with local topography.

In oceanography, the amount of empirical information is never sufficient to specify the statistics of the model errors completely. Accuracy of the data is sometimes also difficult to assess, particularly as we must allow for the limited ability of our models to represent the actual data. For example, our ability to represent surface currents from the HF radars will depend critically on the accuracy of our model representation of the surface mixed layer. So, the choice of the input error statistics is always a hypothesis. Possible misspecifications of these statistics will affect the performance of any data assimilation (DA) system.

Here we analyze GIM, KF, and OI in a unified framework to gain insights into several important questions in coastal data assimilation. We address the following issues: how useful future data are for prediction of the flow at a particular time; what the optimal gain matrix for OI should be, and how sensitive the performance of OI is to this choice; and how misspecified input error covariances affect the quality of the solution. A thorough answer to these questions would be difficult to obtain in the context of a model based on the full primitive equations, so our analysis will be based on the idealized coastal dynamical model described in a companion paper (Kurapov et al. 2001, manuscript submitted to *Mon. Wea. Rev.*, hereafter KAME). As application of this simplified model to real data would probably not be instructive, we pursue a theoretical analysis. In KAME, we perform experiments with synthetic data to illuminate some fundamental issues in coastal data assimilation such as restoration of the open boundary conditions and sensitivity of the solution to the choice of weights in the cost functional. The quality of the solution in such experiments depends on the choice of the “true” wind forcing and on initial and boundary conditions. Here, a more general approach, based on the methodology of estimation theory (Cohn 1997), is used. We compute the error statistics of ensembles of solutions to compare the performance of data assimilation methods. For our simple linear model, results may be computed exactly in a straightforward way given the error statistics of the inputs. The advantage of this approach is that the ensemble of actual solutions does not have to be generated, so results are independent of the choice of mean and sampling strategy. The representer functions, obtained analytically in KAME as components of the generalized inverse solution, yield the error covariance of the model solution, given input error statistics. Using these analytic expressions, posterior error analysis requires only simple matrix manipulations. The computations for the GIM- and KF-derived solutions are standard (see Bennett 1992, section 5.6). We will need to devise some new tools to compute the posterior error statistics for OI, and to assess the effect of incorrect input error statistics for all the methods.

In section 2, we give a comparative description of GIM, KF, and OI with the emphasis on how the solution error covariance can be computed. In section 3, we describe briefly the coastal inverse model. In section 4, results of variance computations are presented, including a discussion of possible choices for the gain matrix in OI. The effect of incorrect input error statistics on the performance of the inverse solution is considered in section 5. Finally, section 6 contains a summary and discussion.

## 2. Posterior statistics

**u**

_{t+1}

**u**

_{t}

*ϵ*_{t+1}

**u**

_{t}is the true solution at time

*t,*𝗔 is a model operator that propagates the solution forward from time

*t*to time

*t*+ 1, and

*ϵ*_{t+1}denotes model error. Here, 𝗔 includes necessary boundary conditions. Measurements are taken regularly at the same

*N*locations at times

*t*= 1, 2, … ,

*T*:

**d**

_{t}

**u**

_{t}

*η*_{t}

**d**

_{t}and

*η*_{t}(of length

*N*) represent data values and errors, correspondingly. In the present study, both 𝗔and 𝗛are linear and time independent.

To simplify the presentation, we will consider the model (1) to be discretized on a computational grid such that **u**_{t} and *ϵ*_{t} are vectors of length *M* while 𝗔 and 𝗛 are matrices of size *M* × *M* and *N* × *M,* respectively. The same analysis holds for continuous **u**, provided the matrix transpose 𝗔′ is interpreted as the adjoint.

*σ*

^{2}; 𝗜

_{N}is the diagonal identity matrix of size

*N*×

*N.*A more general, nondiagonal and time-dependent data error covariance would not alter the derivation, but would make notation more cumbersome.

*posterior covariance,*would not depend on the forcing and prior guesses for initial/boundary conditions, a homogeneous problem is considered. In particular,

**u**

_{0}= 0, and the prior model solution

**u**

^{model}= 0. The initial condition is satisfied exactly. The error covariance of the model solution, or

*prior covariance,*is

*m*≥

*n,*(5) yields

_{m,n}

_{m+1,n}

Note, that throughout this presentation the term prior covariance is used for the covariance of the error in the prior model *solution,* rather than the covariance of errors in the prior model *equations* (**Π**_{i}). Before proceeding with our treatment of OI, it is instructive and useful to review GIM and KF. Our analytical solution for the representers needed for GIM (KAME) provides expressions for all elements of 𝗣_{s,t}, which in turn can be used in computation of the posterior error statistics for KF and OI. Also, by considering the KF, we will see how to construct the optimal gain matrix for OI.

### a. Generalized inverse

*ϵ*_{t}and

*η*_{t}are the estimates of the model and data errors. The inverse solution can be written in terms of the data and representers as

**Ũ**

_{T}

*σ*

^{2}

_{NT}

^{−1}

**D**

_{T}

_{m,n}= 𝗣

_{m,n}𝗛′ are each matrices, of size

*M*×

*N,*so 𝗤 is of size

*MT*×

*NT.*Each column of (11) is the representer corresponding to a particular datum; together 𝗤

_{1,t}, … , 𝗤

_{T,t}give

*N*representers for all the data measured at time

*t.*For point measurements, 𝗣

_{m,n}𝗛′ is just selected columns of 𝗣

_{m,n}.

_{T}is of size

*NT*×

*NT,*composed of blocks 𝗥

_{m,n}= 𝗛𝗣

_{m,n}𝗛′. In the following, we will use special notation for a row of blocks of 𝗤to the left of the diagonal blocks 𝗤

_{t,t}:

_{t}

_{t,1}

_{t,2}

_{t,t−1}

*t*with the solution at all earlier measurement times. For lagged covariances, (6) yields

_{t}

_{t,t}

_{t+1}

_{m,n}. In particular, at time

*t*=

*T,*the inverse solution is

**ũ**

_{T}

_{T}

_{T,T}

_{T}

*σ*

^{2}

_{NT}

^{−1}

**D**

_{T}

*T,*is

### b. Kalman filter

*t,*is usually expressed as (see Miller 1996)

**u**

^{a}

_{t}

**u**

^{a}

_{t−1}

_{t}

**d**

_{t}

**u**

^{a}

_{t−1}

_{t}is

_{t}

_{t}

_{t}

*σ*

^{2}

_{N}

^{−1}

_{t}is the forecast error covariance:

_{t}is the covariance of the model solution conditioned upon the past data.

^{a}

_{t}

_{t}

_{t}

_{t}

*t*− 1 to

*t,*the analysis field

**u**

_{t−1}and the analysis error covariance 𝗦

^{a}

_{t−1}are necessary. One can show that at time

*T,*when the last measurements are available,

**u**

^{a}

_{T}

**ũ**

_{T}, and 𝗣

^{a}

_{T}= 𝗣

^{GIM}

_{T,T}. This is intuitively obvious, since the KF makes optimal use of all prior data, and the GIM of all data.

_{t}can be computed as

_{t}

_{t,t}

_{t}

_{t−1}

*σ*

^{2}

_{N(t−1)}

^{−1}

^{′}

_{t}

^{a}

_{t−1}is equal to the r.h.s. of (16), where

*T*is replaced by

*t*− 1; then, apply (5) and (13). All the entries in (21) are given by the representer solution. The upper-left corner of the representer matrix is 𝗥

_{t−1}, of size

*N*(

*t*− 1) ×

*N*(

*t*− 1). Note that while both 𝗖

_{t}and 𝗣

_{t,t}are matrices of large dimension (

*M*×

*M*), to find the solution we need to operate only with 𝗖

_{t}𝗛′ and 𝗣

_{t,t}𝗛′, which are much smaller since generally

*N*≪

*M.*

### c. Optimal interpolation (OI)

By OI we mean a simplified sequential data assimilation method such that the gain matrix is not changed from one assimilation cycle to the next. As time goes on, the dimension of 𝗦_{t} in (21) increases. Here, 𝗖_{t} will become stationary for large *t* if 𝗣_{t,t} does, and if the lagged covariances 𝗤 _{n,t} are negligible for large *t* − *n.* In this case 𝗚_{t} → 𝗚 _{∞}, a constant gain matrix. In our numerical examples we perform data assimilation over a finite period of time. We check that 𝗖_{t} stabilizes by the end of the assimilation period (*T*), and take the limiting gain matrix 𝗚_{T} ≡ 𝗚 as optimal for OI.

Oke et al. (2001) essentially approximated 𝗖_{T}𝗛′ by *a*𝗣_{T,T}𝗛′, where *a* = const < 1. The estimate of 𝗣_{T,T}𝗛′ was obtained by utilizing time-averaged correlations of model solutions computed with observed winds for 18 different summers. Thus, the assumption was made that the correlation structure of the prior error covariance conditioned upon the past data, 𝗖_{T}, was the same as that of the unconditional prior error covariance, 𝗣_{T,T}. To explore the possible consequences of this assumption, we need to compute the posterior covariance of the OI solution for gain matrices based on 𝗖_{T} and *a*𝗣_{T,T}. We refer to results for these two as cases OI(C) and OI(aP), respectively.

**u**

^{a}

_{t}

**D**

_{t}

**D**

_{t}is the vector of all data available to time

*t*[see (10)];

^{t−1}

^{t−2}

*M*×

*Nt,*andThe posterior covariance is then

^{s}𝗚. Further derivation depends on the choice of 𝗚. In the case of OI(aP), 𝗚 = 𝗣

_{T,T}𝗛′

**Ω**, where

**Ω**

_{T,T}

*a*

^{−1}

*σ*

^{2}

_{N}

^{−1}

_{T}𝗛′

**Ω̂**

**Ω̂**

_{T}𝗛′ +

*σ*

^{2}𝗜

_{N})

^{−1}. In (27) and (28), we use

**Ω̂**

_{T+i,T}, and

_{T+i,T}instead of

**Ω**, 𝗤

_{T+i},

_{T}, and 𝗥

_{T+i,T}, where

_{n}are blocks of size

*N*×

*N*that form the matrix (𝗭

_{1}| 𝗭

_{2}|· · ·| 𝗭

_{T−1}) = 𝗛𝗦

_{T}(𝗥

_{T−1}+

*σ*

^{2}𝗜

_{N(T−1)})

^{−1}. Using the analytical representer solution and (26)–(30) we can explicitly construct 𝗞 (23) and hence posterior covariances (25) for both the OI(C) and OI(aP) cases.

## 3. Inverse coastal model

Here we give a summary of the coastal model described in detail in KAME. A number of simplifications are made to make analytical progress possible. We consider subinertial linear motions near a straight coast in a basin of constant depth *H,* with linear ambient stratification. A long-wave boundary layer approximation is made. Effects of the surface and bottom Ekman layers are parameterized. Friction is parameterized in Rayleigh form. The problem is considered in terms of the perturbation pressure *p,* and only the baroclinic part is retained.

All variables are made nondimensional: time by the inverse of the Coriolis parameter *f*^{−1}, and horizontal distance by the Rossby radius of deformation *NH*/*f*, where *N* is the buoyancy frequency. Our computational domain represents a part of the North American western coast, with the *x* axis of the coordinate system directed offshore (0 ≤ *x* < ∞), the *y* axis toward the south along the coast (0 ≤ *y* ≤ *L*), and the *z* axis in the vertical (0 ≤ *z* ≤ 1) (Fig. 1).

*x,*

*y,*

*z,*and

*t*denote partial differentiation;

*ϵ*

^{( · )}on the right-hand sides represent errors in the governing equation, and initial and boundary conditions;

*τ̂*

*α*

_{m}and

*α*

_{c}are dissipation parameters. Condition (34) is for the initial potential vorticity (IPV). Expressions (35) and (36) provide initial and boundary conditions for coastal trapped baroclinic waves that travel from the south (

*y*=

*L*) to the north (

*y*= 0). In the analysis of the second-order posterior statistics below we assume homogeneous initial and boundary conditions, and no forcing in (32). Error-free boundary conditions at the surface and bottom (33) are assumed so that

*p*and

*ϵ*

^{( · )}can be obtained as a series of cosine baroclinic modes. For instance,

*p*(

*x*

_{k},

*y*

_{k},

*z*

_{k},

*t*

_{k}) =

*d*

_{k}are available, where

*k*= 1, … ,

*K.*The cost functional is

*ϵ*

^{(m)},

*ϵ*

^{(c)},

*ϵ*

^{(IPV)}, and

*ϵ*

^{(I)}are correlated in the alongshore direction with Gaussian covariances

*C*(

*y*−

*y*

_{1}) = exp[−(

*y*−

*y*

_{1})

^{2}/

*L*

^{2}

_{y}

*L*

_{y}is the error decorrelation length scale. Errors in the south boundary condition,

*ϵ*

^{(B)}, are correlated in time. As described by the term IB of (39), the errors in (35) are correlated with the errors in (36) mode by mode, in a manner consistent with the dynamics of the long alongshore waves. In this term, integration is performed over the parametric variable

*s*such that

*y*first varies from 0 to

*L*while

*t*= 0, and then

*y*=

*L*while

*t*varies from 0 to

*T*;

*L*

_{n}=

*L*+ (

*nπ*)

^{−1}

*T.*Higher weights are assigned to higher modes of

*ϵ*

^{(I)}and

*ϵ*

^{(B)}at the rate

*w*

_{n}= (

*nπ*)

^{β}to inhibit singular behavior of the solution at the surface. This is equivalent to assigning a specific covariance in the vertical:

*β*= 2 in the term IB. In section 4, we will consider a similar covariance in the vertical for

*ϵ*

^{(m)}and

*ϵ*

^{(IPV)}, with

*β*= 0 or

*β*= 2. Terms

*m*and IB in (39) are written for

*β*= 0, which corresponds to the assumption that the errors are not correlated in the vertical. Terms

*m*and IPV of (39) contain exponential factors to eliminate the prior variance far offshore, as discussed in KAME.

The representers [see (8)] are computed as a sum of the baroclinic modes, and for each baroclinic mode, an analytical expression was obtained in KAME. Each representer has stationary and propagating components, the latter moving as long coastal trapped baroclinic waves.

**u**(

*t*), which is here continuous in time and still discrete in space, and

**e**(

*t*) is the error in the continuous equations, which is assumed to be temporally uncorrelated, with spatial covariance

**Ψ**

_{t}=

**e**(

*t*)

**e**(

*t*)′

^{τ}is defined as

^{τ}

**v**

**u**

*τ*

**u**(

*t*) is the solution to

_{s,t}will coincide with that for the continuous problem

**u**(

*t*)

**u**(

*s*)′

## 4. Posterior variance computations

Our coastal inverse model includes a number of parameters that will be chosen with relevance to the mid-Oregon coast where an HF radar system has been operating for the past several years (Kosro et al. 1997). In this area, a nondimensional time interval of 8 would correspond to approximately 1 day. In computations, *α*_{m} = *α*_{c} = 1/24 except for one special case discussed in section 4a. The unit nondimensional horizontal distance will correspond to about 25 km. The HF radar system covers a region approximately 50 × 50 km^{2}, and observations are available on an hourly basis. Our zone of interest would thus be of nondimensional length *L* = 2 alongshore. We will assimilate synthetic pressure data measured from time 0 to *T* = 80 (∼10 days) every *δt* = 2 at 21 points on the surface (*x*_{k} = 0.1, 0.3, 0.5; *y*_{k} from 0.1 to 1.9 spaced 0.3; see Fig. 1). The posterior covariance does not depend on the actual data values, but only on the data error statistics (*σ*^{2}) as well as model error statistics. In the computations, the error decorrelation length scale *L*_{y} is the same for the model, and initial and boundary conditions. The performance of the methods will be assessed by comparison of the prior and posterior error variances, *E*_{prior} and *E*_{post}, as functions of spatial coordinates and time.

### a. Performance of GIM in the vertical and time

The first question asked with regard to sea surface data assimilation is how much information such data contain about the ocean state at depth. In KAME, we tried to answer this question by performing twin experiments, with the governing equation (31) assumed to be a strong constraint. Here, we consider a weak constraint case (*w*_{m} < ∞) and compare the prior and posterior variances as functions of the vertical coordinate at a particular location (*x* = 0.2, *y* = 1). Two specific questions are addressed. The first is the effect of the model error (input) covariance in the vertical [see (40)] on the performance of the inverse model at depth. The second question is the restriction of the growing error in the prior model solution with time, both at the surface and the bottom. In KAME it was shown that without dissipation in the governing equation [*α*_{m} = 0 in (31)] the prior variance grows linearly with time. We are interested whether data assimilation helps to bound the growth in uncertainty. In Fig. 2, prior and posterior error variances are plotted as functions of depth for three selected times: *t* = 10, 40, and 70. Plots in Figs. 2a and 2b correspond to the case of *α*_{m} = 1/24, and Figs. 2c and 2d to the case *α*_{m} = 0. For Figs. 2a and 2c the model errors were not correlated in the vertical, while for Figs. 2b and 2d the errors in the model were vertically correlated with *β* = 2 in the covariance of (40). In all of these cases, the weight for the first baroclinic mode was the same.

In every computation, *E*_{prior}(*z*) is symmetric with respect to *z* = 0.5, minimum at middepth, and maximum near the surface and bottom where the largest variations of the pressure are allowed by the model, based on our assumptions about the statistics of the errors in the equations. In the case of Fig. 2a we see that the flow at the bottom is not recovered nearly as well as that at the surface [*E*_{post}(*z* = 0)/*E*_{post}(*z* = 1) ≈ 3.4 at *t* = 40 and 70]. If the errors in the governing equation are correlated in the vertical (see Fig. 2b), the ratio of the posterior variance at the bottom and the surface becomes smaller [*E*_{post}(*z* = 0)/*E*_{post}(*z* = 1) ≈ 1.9 at *t* = 40 and 70]. If there is no dissipation in the governing equation, data assimilation may to some degree restrict the growth of the error at the surface, but not at the bottom (see Fig. 2c). Correlation of the model errors in the vertical does not help to restrict the growth of the uncertainty at the bottom (see Fig. 2d). Thus, to have a model suitable for the study of the sequential algorithms, it is essential to have dissipation inside the domain, not only at the coast. Otherwise, the gain matrix will not become stationary.

### b. Comparison of GIM and KF

For our coastal inverse model, the analytical representer solution can be used to compute the KF solution at any time *t*_{k} when measurements are available, by applying GIM in the time window 0 ≤ *t* ≤ *t*_{k} using only data available up to time *t*_{k}. This GIM solution at *t* = *t*_{k} will be identical to what the KF would yield. By applying GIM repeatedly over larger and larger time intervals, involving more and more data, we can obtain the output of the KF as a function of time, and compare this with the solution of the GIM solution using data from the whole time interval 0 ≤ *t* ≤ *T.*

It should be noted that in our coastal model the errors in the south boundary condition should be correlated in time (both physically, and for technical reasons; see KAME). At the same time the analysis of section 2b is given for temporally uncorrelated model errors. The KF can be extended to allow for temporally correlated errors (Gelb 1974), although this would introduce some complications into the equations of section 2b. However, our computation of the KF solution, based on equivalence of GIM and KF at *t* = *T,* easily accommodates temporally correlated errors.

Prior and posterior variances are computed as functions of time at six points in the computational domain: one point each at the surface and the bottom at the north, middle, and south cross sections (*y* = 0, 1, and 2 correspondingly), all at an offshore distance *x* = 0.2. For these computations, weights are chosen so that the prior variance *E*_{prior} is on average close to 1 over the full assimilation time interval, and the decorrelation length scale is *L*_{y} = 2. The data weight *w*_{d} = 100 corresponds to *σ*^{2} = 0.01.

We first consider the case when the error in the model solution is due to uncertainty in the wind forcing, and initial and boundary conditions for the coastal trapped waves (Fig. 3a). In this case, which we refer to as c + IB, there are no errors in the governing equation or the IPV condition, so the representers have only a propagating component that moves from the south to the north, passing through the data locations at the time of measurement. At the south boundary, the GIM yields a posterior variance 1/3 that of the KF-derived solution. The advantage of GIM remains profound up to *t* = 70, when its error variance starts increasing to match the performance of the sequential method at *t* = *T* = 80 as it must. It is quite clear why the GIM works better at the south boundary. Future data contain much information about the present flow south of all measurement points, and this information is not utilized by the KF. Past data provide maximum information somewhere to the north of the measurement points, and essentially no information at the south boundary. At the middle cross section, the advantage of the GIM is still appreciable (about 20% of the variance), while at the northern boundary the future data are almost useless, and both KF and GIM perform with the same accuracy.

Next, consider the case m + IPV where we assume that the coastal boundary condition as well as the initial and boundary conditions for the traveling waves are perfect, and errors are due only to uncertainty in the governing equation and the IPV condition (Fig. 3b). In this case, the representer has a substantial stationary part and the advantage of the GIM can be subtle. In this example, data of relatively high quality totally control the solution at the time of the measurement, so that data obtained in the future or past bring very little new information. However, the advantage of GIM is greater if the data errors become larger (Fig. 3c; *w*_{d} = 1 instead of 100). Since we have assumed that the model errors are not correlated with depth, performance of both methods is relatively poor at the bottom, and the difference between GIM and KF is seen only at the surface.

Finally, we compute the variance as a function of *x* in the middle cross section, at the surface and bottom, for time *t* = 40. The most distant data from the coast are at *x* = 0.5, and we would like to see how well the solution is restored farther offshore. In the case c + IB the data provide maximum correction to the solution at *x* = 0, and both the prior and posterior variances decay offshore exponentially (Fig. 4a). However, for the case m + IPV, posterior variance increases beyond *x* = 0.5 (Fig. 4b). To diminish the effect of the error in the potential vorticity field at *x* > 0.5, measurements farther offshore would be necessary. In this case errors in the equations are not correlated in the vertical. At the surface, the posterior variance is reduced locally at the data locations indicating that the pressure field is undersampled for the assumed covariance structure. If the errors in the governing equation and the IPV condition are correlated with depth [*β* = 2 in (40)], the performance at the bottom is much more like that at the surface. Then GIM is apparently better than KF at the bottom, not just at the surface, and the local depression of the posterior variance near the data sites that would be associated with smaller scales introduced by the higher modes is smoothed (Fig. 4c).

### c. OI: The gain matrix and posterior variance

The results of section 2c for OI were derived under the assumption that the model errors are uncorrelated in time. To apply these results to our coastal inverse model, we put *w*_{IB} = ∞, corresponding to zero variance in the south boundary condition, and move the boundary so far to the south that the exact boundary condition does not affect the solution in the region of interest (0 ≤ *y* ≤ 2). Due to dissipation, the prior covariance associated with the error in the open boundary condition decays alongshore as *e*^{αcnπ(y−L)}*n* is the number of the baroclinic mode. Thus, for *α*_{c} = 1/24, *L* = 20 is a safe choice for the south boundary. The extended computational domain is one of the prices we pay to use the simpler OI data assimilation routine.

For OI the forecast error covariance 𝗖_{T} must be chosen a priori. Recall that 𝗖_{T} can be interpreted as the covariance of the model solution error conditional upon the past data. In theory one should know the lagged prior covariances 𝗦_{T} [see (21)] to construct an optimal conditional covariance. A simpler approach was used by Oke et al. (2001) who essentially assumed that the spatial correlation structure of 𝗖_{T} is the same as that of the (unconditional) prior covariance 𝗣_{T,T}. This avoids computation of lagged covariances, but is this assumption reasonable? For our simplified model we can explicitly compute the full covariances 𝗖_{T}𝗛′ and 𝗣_{T,T}𝗛′ and assess the validity and impact of this approximation. To compare the structure of the two covariances, we sample them on a uniform discrete grid: *x* is from 0.05 to 0.95 each 0.1, *y* is from 0 to 2 each 0.1, and *z* is from 0 to 1 each 0.1, a total of *M* = 2310 mesh points.

_{T}𝗛′ and 𝗣

_{T,T}𝗛′ can be compared in several ways. First of all we would like to find, for a range of covariance parameters, the scale factor

*a*≤ 1, that makes

*a*𝗣

_{T,T}𝗛′ the closest to 𝗖

_{T}𝗛′. Specifically, we vary

*L*

_{y}and

*w*

_{d}, and find

*a*=

*a*

_{min}(

*L*

_{y},

*w*

_{d}) that minimizes the norm

*a*) = 𝗖

_{T}𝗛′ −

*a*𝗣

_{T,T}𝗛′. Then, we plot the norm of the difference (46) computed for

*a*=

*a*

_{min}as a function of

*L*

_{y}and

*w*

_{d}(Fig. 5). The difference is apparent when

*w*

_{d}is large enough such that the use of the data makes sense. In the present example, if

*w*

_{d}> 1, the norm of the difference becomes larger with decreasing

*L*

_{y}. We take this result to indicate that accounting for lagged correlations in computing the gain matrix is probably more important if the model errors vary on shorter spatial scales. In a wind-driven circulation model, the dynamical errors on the surface will probably have large decorrelation length scales associated with uncertainty of the wind stress. However, topography will introduce smaller scales. In our case, the difference is largest when

*L*

_{y}= 0.5, and it is smaller for

*L*

_{y}= 0.2. A possible explanation of this is that the covariance fields represented by the columns of both 𝗖

_{T}𝗛′ and 𝗣

_{T,T}𝗛′ become almost singular (

*δ*functions) when

*L*

_{y}is small, and the norm (46) is poor at distinguishing between such matrices.

We next look at the correlation structure of the two covariances. Each column of 𝗖_{T}𝗛′ and 𝗣_{T,T}𝗛′ represents a three-dimensional field giving the covariance between a particular observation location and all other points in the domain; for 𝗣_{T,T}𝗛′ this is the representer. To illuminate differences in spatial structure, we normalize the covariances by their values at the corresponding observation location, and plot several alongshore sections (Fig. 6). If *L*_{y} = 0.5 (left panels in Fig. 6), the conditional covariance field is distorted in comparison with the unconditional (representer) field, with some decrease of the alongshore correlation scale. If *L*_{y} = 2 (right panels in Fig. 6), the maximum of the conditional covariance is shifted to the south of the observation point (for an intermediate weight *w*_{d}, see Fig. 6b). With the larger data weight, when the present data alone provide good control of the solution and the past data bring almost nothing new, the maximum of the covariance shifts back to the data location at the surface, but is still displaced at the bottom. Accordingly, at *L*_{y} = 2 the norm of (46) is small for large *w*_{d} (see Fig. 5).

Another way to compare 𝗖_{T}𝗛′ and 𝗣_{T,T}𝗛′ is to look at their singular vectors. By doing so we operate with all the columns of the matrices. In Fig. 7, we show alongshore sections of a few of the singular vectors of 𝗣_{T,T}𝗛′ (on the left) and 𝗖_{T}𝗛′ (on the right), and *L*_{y} = 0.5. The dominant modes of 𝗖_{T}𝗛′ that describe large-scale alongshore variability are distorted and systematically shifted to the south.

To understand how much the structural difference between the conditional and unconditional prior covariances affects the solution, we compute the posterior variance of both the OI(C)- and OI(aP)-derived solutions at the same six points used in section 4b (Fig. 8). As expected, the posterior variance curve for OI(C) merges with the curve for KF some time after the beginning of data assimilation, when the initial condition has been forgotten and the gain matrix for the KF has stabilized. The posterior variance for OI(aP) is larger than that for OI(C). If the approximate conditional covariance *a*𝗣_{T,T} is used with optimal *a* = *a*_{min}, the difference is no more than 10% in our example (see Figs. 8a,b). However, if *a* is misspecified, the difference will be larger. The limit *a* → 0 corresponds to the case 𝗛 → 0, and *E*_{post} → *E*_{prior}. The limit *a* → 1 (see Figs. 8c,d) means that the effects of past data were disregarded during the construction of the gain matrix. As a consequence, the difference between posterior variance plots for KF and OI(aP) is larger in the area where the past data would have been significant, that is, in the middle and at the north.

## 5. How bad is a wrong hypothesis about error covariances?

_{t,t}, 𝗦

_{t}, 𝗤

_{t,t}, and 𝗥

_{t}are computed with the representers based on the true error covariances, while 𝗞 is based on the hypothesized covariances. Similarly, for GIM (and thus also for KF), the expression for the posterior covariance that replaces (14) is

*σ*

^{2}𝗜

_{K}and quantities without the hat, including the first term on the r.h.s. (prior covariance), are computed with the true error covariances, while those marked by the hat are computed with the hypothesized covariances.

A number of statistical parameters such as weights and decorrelation length scales need to be prescribed to define the covariances. As we have an apparatus for the assessment of the effect of their misspecification, a large number of computations could be produced. We choose two examples illustrating the effect of misspecification of the correlation structure. In each example, we compare the performance of the methods at time *t* = 70, after the initial condition has already been forgotten, but when there are still future data containing potentially valuable information. Plots show the variance as a function of depth. The posterior variance is computed with both correct and incorrect hypotheses about the input error covariances. Results for OI are obtained using the conditional covariance 𝗖_{T} in the gain matrix. At time *t* = 70, the variance for OI(C) coincides with that for KF. The south boundary is again at *L* = 20, far from the region of interest.

*Example 1*: The uncertainty in the model solution is due to the error in the governing equation (31) while the coastal boundary condition (32) is exact. For true error statistics the model errors *ϵ*^{(m)} are correlated with depth (*β* = 2), while the assumed hypothesis is that the errors are not correlated (*β* = 0), or vice versa (Fig. 9). First, if in fact *β* = 2 (Fig. 9a), GIM with the misspecified error covariance (*β* = 0) is significantly less efficient than the GIM with the correct covariance, both at the surface and the bottom. In contrast, when *β* = 0 (Fig. 9b), misspecification has little effect on the performance. In this example, we are safe if we assume errors are correlated whether they are or not. When the coastal boundary condition is exact the representers are dominated by the stationary part, and the GIM has little advantage over the sequential algorithms. Thus, analysis for OI gives posterior variances close to those achieved by GIM (Figs. 9c,d). In case the errors are indeed correlated in the vertical, specification of the right covariance is more important than the use of future data (compare the dashed line, Fig. 9a, and the solid line, Fig. 9c).

*Example 2*: Let us turn to the case where GIM has a significant advantage over the sequential schemes. That is, let the error in the model solution be due to uncertainty in the wind forcing while the equation for the variation of the potential vorticity (31) is exact; consider the variance at *y* = 2, that is, south of all the data. The true alongshore error decorrelation length scale is *L*_{y} = 0.5, and the misspecified value is *L*_{y} = 2, or vice versa (Fig. 10). Overestimating the decorrelation length scale when it is small (Fig. 10a) is apparently more dangerous than underestimating the decorrelation length scale when it is large (Fig. 10b). Refer to Table 1 where we show the ratio of the posterior variances computed with correct and incorrect input statistics: the larger the value in the table, the more sensitive the data assimilation is to the choice of the prior error hypothesis. Use of the correct decorrelation length scale *L*_{y} is more important for GIM than OI. However, GIM yields a smaller variance than OI.

In reality, the general situation is that wind forcing varies on large scales compared to the size of our domain. In this case GIM with the misspecified error decorrelation length scale yields the same performance as OI with the correct prior error covariance (cf. the dashed line, Fig. 10b, and the solid line, Fig. 10d).

It may be noticed that assuming the errors have too large a scale in the vertical does not cause problems (example 1; see Fig. 9b), but overspecification of the horizontal scale does indeed cause problems (example 2; see Fig. 10a). Equation (31) provides sufficient smoothing in the vertical to the singularities in the representer solution, so that the first mode is dominant in this solution even if the covariance (40) is singular (*β* = 0), as in example 1. In this case, modal amplitudes of the representer decrease as *n*^{−2}, where *n* is the mode number (see KAME). Excessive smoothing in the vertical means dampening higher, relatively insignificant, modes, which is harmless. In contrast, in the alongshore direction, the dynamics does not provide an intrinsic dominant length scale. The alongshore length scale is determined by the scale of the wind forcing error, which for our model translates into the error in the coastal boundary condition (32), as discussed in example 2. Now excessive smoothing in the alongshore direction prevents the model from interpreting small-scale variations presented in data as signal, so the data assimilation performance is degraded.

## 6. Summary

We have compared performance of different data assimilation schemes with a simplified coastal baroclinic model. The analytical representer solution allows us to compute the posterior error variance of the GIM, KF, and OI solutions. For KF and OI, we do not need to integrate the equations explicitly from one assimilation cycle to another, but rather can use results obtained in the framework of GIM. The tools of statistical analysis derived here can also be applied to assess the impact of misspecified input error covariances. These tools are independent of a particular dynamical model, and can be used with other applications.

Although our model is highly idealized, it sheds light on a number of issues that would be relevant in a more realistic environment. In comparison with the previous study of Scott et al. (2000), retaining alongshore and temporal variability allowed us to analyze the use of distant data. GIM has an advantage over the sequential algorithms since it uses all the data, including future data, in an optimal way to obtain an estimate of the flow. In a practically relevant case, GIM with misspecified input error statistics produces a result of the same quality as the sequential methods with correctly specified input error covariances.

In the present study, the variational method had the greatest advantage over sequential schemes when the coastal boundary condition was assumed imperfect, while the governing equation describing evolution of the potential vorticity was a strong constraint. However, we do not want the reader to get the impression that GIM generally works better in the strong constraint case than in the weak constraint case. Cases where either the coastal boundary condition or the governing equation are imperfect have been separated in an attempt to systematize the analysis. In the former case, the representer has only a propagating component, while in the latter case a substantial stationary component is present (KAME). In our dynamical model the wind stress forces the coastal boundary condition that describes the balance of the surface Ekman and interior cross-shore flows. In realistic coastal circulation models, the wind stress is probably a significant source of error in model solutions (see Oke et al. 2001). We have tried to emphasize the role of data assimilation in reducing this error, rather than to derive any conclusions about the relative value of strong and weak constraint model formulations. Even in the case where the governing equation is a weak constraint, the advantage of GIM over KF is significant if the data error becomes large.

To provide a concise description, we confined our study to a particular observation system that would be relevant to the present HF radar system currently in operation along the mid-Oregon coast. Based on our results, we may presume that deployment of more radars to the north or south of our test region (which was about 50 km alongshore) would help to improve prediction in the test region. Since information propagates with the long coastal trapped waves from south to north, the new data would help to eliminate the error associated with misspecified wind stress or boundary conditions at the southern edge of the domain. However, data obtained far up or down the coast (farther than ∼400 km) would probably not contribute much because of dissipation. Information about second and higher baroclinic modes becomes insignificant on even shorter length scales.

Analysis of the OI sequential scheme shows how lagged covariances of the model solution error can be used to optimize the form of the gain matrix [see (18) and (21)]. Even with a complex primitive equation model, 𝗣_{T,T}𝗛′ can be estimated based on the ensemble of the model runs (e.g., Oke et al. 2001). The elements 𝗤_{T,T−t} = 𝗣_{T,T−t}𝗛′, necessary to build 𝗦_{T} in (21), can be estimated in a similar way from the same ensemble. In practice, a restricted number of 𝗤_{T,T−t} need be computed since lagged covariances corresponding to large differences *T* − *t,* for example, more than 3 days in our example, are negligible because of dissipation. Note that in (26) the scaling factor *a* for the approximate forecast error covariance gets mixed up with the data variance *σ*^{2}, which complicates specification of this statistical parameter. Computation of the optimized gain matrix would make specification of *a* unnecessary.

In our computational example, the difference in the performance of OI(aP) was nearly as good as OI(C) or KF. However, we have observed that the difference in structure between the conditioned and unconditioned prior covariances, important for OI, can be relatively large when the error decorrelation length scale is small. Such short scales may be introduced by the variable bathymetry and curved coastline. Topography will also influence the way propagating waves carry information. These topics will be subjects of future study.

## Acknowledgments

The research was supported by the Office of Naval Research Ocean Modeling and Prediction Program under Grant N00014-98-1-0043.

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Prior (dashed lines) and posterior (solid lines) variance as a function of *z,* at *x* = 0.2, *y* = 1; *w*_{m} = 1/*π*^{β}, *w*_{d} = 5, and the other weights are ∞. Each column corresponds to a specific time (10, 40, or 70): (a) *α*_{m} = 1/24, *β* = 0; (b) *α*_{m} = 1/24, *β* = 2; (c) *α*_{m} = 0, *β* = 0; and (d) *α*_{m} = 0, *β* = 2

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Prior (dashed lines) and posterior (solid lines) variance as a function of *z,* at *x* = 0.2, *y* = 1; *w*_{m} = 1/*π*^{β}, *w*_{d} = 5, and the other weights are ∞. Each column corresponds to a specific time (10, 40, or 70): (a) *α*_{m} = 1/24, *β* = 0; (b) *α*_{m} = 1/24, *β* = 2; (c) *α*_{m} = 0, *β* = 0; and (d) *α*_{m} = 0, *β* = 2

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Prior (dashed lines) and posterior (solid lines) variance as a function of *z,* at *x* = 0.2, *y* = 1; *w*_{m} = 1/*π*^{β}, *w*_{d} = 5, and the other weights are ∞. Each column corresponds to a specific time (10, 40, or 70): (a) *α*_{m} = 1/24, *β* = 0; (b) *α*_{m} = 1/24, *β* = 2; (c) *α*_{m} = 0, *β* = 0; and (d) *α*_{m} = 0, *β* = 2

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Prior and posterior variance of the GIM- and KF-derived solutions vs time at six different points, all at the offshore distance *x* = 0.2. Columns: (I) south (*y* = *L* = 2), (II) middle (*y* = 1), and (III) north (*y* = 0). Weights (a) *w*_{m} = *w*_{IPV} = ∞, *w*_{c} = 10, *w*_{IB} = 0.084, *w*_{d} = 100; (b) *w*_{m} = 0.2, *w*_{IPV} = 0.017, *w*_{c} = *w*_{IB} = ∞, *w*_{d} = 100, *β* = 0; (c) the same as in (b) but with the lower data weight *w*_{d} = 1

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Prior and posterior variance of the GIM- and KF-derived solutions vs time at six different points, all at the offshore distance *x* = 0.2. Columns: (I) south (*y* = *L* = 2), (II) middle (*y* = 1), and (III) north (*y* = 0). Weights (a) *w*_{m} = *w*_{IPV} = ∞, *w*_{c} = 10, *w*_{IB} = 0.084, *w*_{d} = 100; (b) *w*_{m} = 0.2, *w*_{IPV} = 0.017, *w*_{c} = *w*_{IB} = ∞, *w*_{d} = 100, *β* = 0; (c) the same as in (b) but with the lower data weight *w*_{d} = 1

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Prior and posterior variance of the GIM- and KF-derived solutions vs time at six different points, all at the offshore distance *x* = 0.2. Columns: (I) south (*y* = *L* = 2), (II) middle (*y* = 1), and (III) north (*y* = 0). Weights (a) *w*_{m} = *w*_{IPV} = ∞, *w*_{c} = 10, *w*_{IB} = 0.084, *w*_{d} = 100; (b) *w*_{m} = 0.2, *w*_{IPV} = 0.017, *w*_{c} = *w*_{IB} = ∞, *w*_{d} = 100, *β* = 0; (c) the same as in (b) but with the lower data weight *w*_{d} = 1

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Variance of the GIM- and KF-derived solutions vs *x* at the points, *y* = 1, *t* = 40, surface and bottom. Legend is the same as in Fig. 3. (a) Case c + IB, *w*_{m} = *w*_{IPV} = ∞, *w*_{c} = 10, *w*_{IB} = 0.084, *w*_{d} = 100; (b) case m + IPV, *β* = 0, *w*_{m} = 0.2, *w*_{IPV} = 0.017, *w*_{c} = *w*_{IB} = ∞, *w*_{d} = 1; (c) case m + IPV, *β* = 2, *w*_{m} = 0.2/*π*^{β}, *w*_{IPV} = 0.017/*π*^{β}, *w*_{c} = *w*_{IB} = ∞, *w*_{d} = 1

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Variance of the GIM- and KF-derived solutions vs *x* at the points, *y* = 1, *t* = 40, surface and bottom. Legend is the same as in Fig. 3. (a) Case c + IB, *w*_{m} = *w*_{IPV} = ∞, *w*_{c} = 10, *w*_{IB} = 0.084, *w*_{d} = 100; (b) case m + IPV, *β* = 0, *w*_{m} = 0.2, *w*_{IPV} = 0.017, *w*_{c} = *w*_{IB} = ∞, *w*_{d} = 1; (c) case m + IPV, *β* = 2, *w*_{m} = 0.2/*π*^{β}, *w*_{IPV} = 0.017/*π*^{β}, *w*_{c} = *w*_{IB} = ∞, *w*_{d} = 1

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Variance of the GIM- and KF-derived solutions vs *x* at the points, *y* = 1, *t* = 40, surface and bottom. Legend is the same as in Fig. 3. (a) Case c + IB, *w*_{m} = *w*_{IPV} = ∞, *w*_{c} = 10, *w*_{IB} = 0.084, *w*_{d} = 100; (b) case m + IPV, *β* = 0, *w*_{m} = 0.2, *w*_{IPV} = 0.017, *w*_{c} = *w*_{IB} = ∞, *w*_{d} = 1; (c) case m + IPV, *β* = 2, *w*_{m} = 0.2/*π*^{β}, *w*_{IPV} = 0.017/*π*^{β}, *w*_{c} = *w*_{IB} = ∞, *w*_{d} = 1

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Norm (46) of 𝗖_{T}𝗛′ − *a*_{min}𝗣_{T,T}𝗛′ as a function of *L*_{y} and *w*_{d}: (a) *w*_{m} = ∞, *w*_{c} = 10; (b) *w*_{m} = 0.2, *w*_{c} = ∞

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Norm (46) of 𝗖_{T}𝗛′ − *a*_{min}𝗣_{T,T}𝗛′ as a function of *L*_{y} and *w*_{d}: (a) *w*_{m} = ∞, *w*_{c} = 10; (b) *w*_{m} = 0.2, *w*_{c} = ∞

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Norm (46) of 𝗖_{T}𝗛′ − *a*_{min}𝗣_{T,T}𝗛′ as a function of *L*_{y} and *w*_{d}: (a) *w*_{m} = ∞, *w*_{c} = 10; (b) *w*_{m} = 0.2, *w*_{c} = ∞

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Covariance fields corresponding to a column of 𝗣_{T,T}𝗛′ and 𝗖_{T}𝗛′ normalized by the value at the datum site (*x*_{k}, *y*_{k}) = (1, 0.1) (denoted by star), shown in the alongshore section *x* = 0.1; *w*_{m} = 0.2, *w*_{c} = ∞, *β* = 0. (left panels) *L*_{y} = 0.5; (right panels) *L*_{y} = 2. Shaded area is for the positive values, contour offset is 0.2: (a) 𝗣_{T,T}𝗛′, (b) 𝗖_{T}𝗛′, *w*_{d} = 10^{−1}, and (c) 𝗖_{T}𝗛′, *w*_{d} = 10^{3}

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Covariance fields corresponding to a column of 𝗣_{T,T}𝗛′ and 𝗖_{T}𝗛′ normalized by the value at the datum site (*x*_{k}, *y*_{k}) = (1, 0.1) (denoted by star), shown in the alongshore section *x* = 0.1; *w*_{m} = 0.2, *w*_{c} = ∞, *β* = 0. (left panels) *L*_{y} = 0.5; (right panels) *L*_{y} = 2. Shaded area is for the positive values, contour offset is 0.2: (a) 𝗣_{T,T}𝗛′, (b) 𝗖_{T}𝗛′, *w*_{d} = 10^{−1}, and (c) 𝗖_{T}𝗛′, *w*_{d} = 10^{3}

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Covariance fields corresponding to a column of 𝗣_{T,T}𝗛′ and 𝗖_{T}𝗛′ normalized by the value at the datum site (*x*_{k}, *y*_{k}) = (1, 0.1) (denoted by star), shown in the alongshore section *x* = 0.1; *w*_{m} = 0.2, *w*_{c} = ∞, *β* = 0. (left panels) *L*_{y} = 0.5; (right panels) *L*_{y} = 2. Shaded area is for the positive values, contour offset is 0.2: (a) 𝗣_{T,T}𝗛′, (b) 𝗖_{T}𝗛′, *w*_{d} = 10^{−1}, and (c) 𝗖_{T}𝗛′, *w*_{d} = 10^{3}

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

First singular vectors of 𝗣_{T,T}𝗛′ (left panels) and 𝗖_{T}𝗛′ (right panels) shown in the alongshore section *x* = 0.15; shaded area is for the positive values, contour offset is 0.01. *L*_{y} = 0.5, *w*_{m} = 0.2, *w*_{c} = ∞, *w*_{d} = 100. Percent numbers indicate how much variability is explained by the mode

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

First singular vectors of 𝗣_{T,T}𝗛′ (left panels) and 𝗖_{T}𝗛′ (right panels) shown in the alongshore section *x* = 0.15; shaded area is for the positive values, contour offset is 0.01. *L*_{y} = 0.5, *w*_{m} = 0.2, *w*_{c} = ∞, *w*_{d} = 100. Percent numbers indicate how much variability is explained by the mode

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

First singular vectors of 𝗣_{T,T}𝗛′ (left panels) and 𝗖_{T}𝗛′ (right panels) shown in the alongshore section *x* = 0.15; shaded area is for the positive values, contour offset is 0.01. *L*_{y} = 0.5, *w*_{m} = 0.2, *w*_{c} = ∞, *w*_{d} = 100. Percent numbers indicate how much variability is explained by the mode

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Variance of OI- and KF- derived solutions vs time at six different points at *x* = 0.2, *L*_{y} = 0.5. Columns: (I) south (*y* = *L* = 2), (II) middle (*y* = 1), and (III) north (*y* = 0). Rows: (a) *w*_{m} = *w*_{IPV} = ∞, *w*_{c} = 1, *w*_{d} = 1, *a* = *a*_{min} = 0.5; (b) *w*_{m} = 0.2, *w*_{IPV} = 0.017, *w*_{c} = ∞, *w*_{d} = 1, *a* = *a*_{min} = 0.45; (c) and (d) are the same as in (a) and (b), correspondingly, but with *a* = 1

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Variance of OI- and KF- derived solutions vs time at six different points at *x* = 0.2, *L*_{y} = 0.5. Columns: (I) south (*y* = *L* = 2), (II) middle (*y* = 1), and (III) north (*y* = 0). Rows: (a) *w*_{m} = *w*_{IPV} = ∞, *w*_{c} = 1, *w*_{d} = 1, *a* = *a*_{min} = 0.5; (b) *w*_{m} = 0.2, *w*_{IPV} = 0.017, *w*_{c} = ∞, *w*_{d} = 1, *a* = *a*_{min} = 0.45; (c) and (d) are the same as in (a) and (b), correspondingly, but with *a* = 1

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Variance of OI- and KF- derived solutions vs time at six different points at *x* = 0.2, *L*_{y} = 0.5. Columns: (I) south (*y* = *L* = 2), (II) middle (*y* = 1), and (III) north (*y* = 0). Rows: (a) *w*_{m} = *w*_{IPV} = ∞, *w*_{c} = 1, *w*_{d} = 1, *a* = *a*_{min} = 0.5; (b) *w*_{m} = 0.2, *w*_{IPV} = 0.017, *w*_{c} = ∞, *w*_{d} = 1, *a* = *a*_{min} = 0.45; (c) and (d) are the same as in (a) and (b), correspondingly, but with *a* = 1

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Effect of misspecification of the model error covariance in the vertical: prior and posterior variance are shown as functions of *z* at *x* = 0.2, *y* = 1, *t* = 70. *w*_{m} = 0.2/*π*^{β}, *w*_{c} = ∞, *w*_{d} = 1, *L*_{y} = 0.5. A legend for each plot shows the value of the smoothing parameter *β*: the number next to curve prior is the “true” statistical parameter for all the curves on the plot; the numbers next to curves “post” (posterior variance) are our statistical hypothesis, correct or incorrect: (a), (b) GIM; (c), (d) OI

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Effect of misspecification of the model error covariance in the vertical: prior and posterior variance are shown as functions of *z* at *x* = 0.2, *y* = 1, *t* = 70. *w*_{m} = 0.2/*π*^{β}, *w*_{c} = ∞, *w*_{d} = 1, *L*_{y} = 0.5. A legend for each plot shows the value of the smoothing parameter *β*: the number next to curve prior is the “true” statistical parameter for all the curves on the plot; the numbers next to curves “post” (posterior variance) are our statistical hypothesis, correct or incorrect: (a), (b) GIM; (c), (d) OI

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Effect of misspecification of the model error covariance in the vertical: prior and posterior variance are shown as functions of *z* at *x* = 0.2, *y* = 1, *t* = 70. *w*_{m} = 0.2/*π*^{β}, *w*_{c} = ∞, *w*_{d} = 1, *L*_{y} = 0.5. A legend for each plot shows the value of the smoothing parameter *β*: the number next to curve prior is the “true” statistical parameter for all the curves on the plot; the numbers next to curves “post” (posterior variance) are our statistical hypothesis, correct or incorrect: (a), (b) GIM; (c), (d) OI

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Effect of over- or underestimation of the error decorrelation length scale for the wind stress: prior and posterior variance are shown as functions of *z* at *x* = 0.2, *y* = 2, *t* = 70. *w*_{m} = ∞, *w*_{c} = 1, *w*_{d} = 1. A legend for each plot shows the value of the smoothing parameter *L*_{y}: the number next to curve prior is the true statistical parameter for all the curves on the plot; the numbers next to curves post (posterior variance) are our statistical hypothesis, correct or incorrect: (a), (b) GIM; (c), (d) OI

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Effect of over- or underestimation of the error decorrelation length scale for the wind stress: prior and posterior variance are shown as functions of *z* at *x* = 0.2, *y* = 2, *t* = 70. *w*_{m} = ∞, *w*_{c} = 1, *w*_{d} = 1. A legend for each plot shows the value of the smoothing parameter *L*_{y}: the number next to curve prior is the true statistical parameter for all the curves on the plot; the numbers next to curves post (posterior variance) are our statistical hypothesis, correct or incorrect: (a), (b) GIM; (c), (d) OI

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Effect of over- or underestimation of the error decorrelation length scale for the wind stress: prior and posterior variance are shown as functions of *z* at *x* = 0.2, *y* = 2, *t* = 70. *w*_{m} = ∞, *w*_{c} = 1, *w*_{d} = 1. A legend for each plot shows the value of the smoothing parameter *L*_{y}: the number next to curve prior is the true statistical parameter for all the curves on the plot; the numbers next to curves post (posterior variance) are our statistical hypothesis, correct or incorrect: (a), (b) GIM; (c), (d) OI

Citation: Monthly Weather Review 130, 4; 10.1175/1520-0493(2002)130<1009:DAIABC>2.0.CO;2

Ratio of posterior variances computed with correct or incorrect error decorrelation length scale for wind forcing. Values of *L _{y}* in parentheses next to

*E*

_{post}show the hypothesis scale