Eknes and Evensen (1997) solved the weak-constraint variational problem for a linear Ekman-flow model using the representer method. They computed the weak constraint solution for a long time series of velocity measurements. Additionally, they considered a parameter-estimation problem which rendered the problem nonlinear. Here we will focus on the representer method’s properties for a linear problem. The model is simple and allows for a straightforward interpretation and demonstration of the method. For more details, we refer to Eknes and Evensen (1997) and Evensen  (2009b).

1 Ekman-Flow Model

The Ekman-flow model describes the horizontal velocity field as a function of depth in the ocean’s upper surface layer when subject to a wind forcing. The model provides the so-called Ekman spiral due to the Coriolis force and we can write its equations in a nondimensional form as

$$\begin{aligned} \frac{\partial {\mathbf {u}}}{\partial t} + {{\mathbf {k}}} \times {{\mathbf {u}}} = \frac{\partial }{\partial z}\left( A \frac{\partial {\mathbf {u}}}{\partial z}\right) , \end{aligned}$$

where \({\mathbf {u}}(z,t)\) is the horizontal velocity vector, \({\mathbf {k}}\) is a vertically pointing unit vector, and \(A = A(z)\) is the diffusion coefficient. The initial conditions are

$$\begin{aligned} {\mathbf {u}}(z,0) = {\mathbf {u}}_0, \end{aligned}$$

and we specify boundary conditions as a wind-drag at the surface \(z=0\) and zero drag or no friction at the lower boundary \(z=-H\) as

$$\begin{aligned} \left. A \frac{\partial {\mathbf {u}}}{\partial z}\right| _{z=0}&= \left( c_d\sqrt{u_a^{2} + v_a^{2}}\right) {\mathbf {u}}_a , \end{aligned}$$
$$\begin{aligned} \left. A \frac{\partial {\mathbf {u}}}{\partial z}\right| _{z=-H}&= {\mathbf {0}}, \end{aligned}$$

where \(c_d\) is the wind drag coefficient, and \({\mathbf {u}}_a\) is the atmospheric wind speed. Following the procedure outlined in Sect. 5.5, we can derive the Euler–Lagrange equations and the representer solution. We refer to the original work by Eknes and Evensen  (1997) for the detailed derivation.

2 Example Experiment

We now discuss a simple example to illustrate the representer method. We use a constant wind with \({\mathbf {u}}_a = (10\,\mathrm {m}\,\mathrm {s}^{-1}, 10\,\mathrm {m}\,\mathrm {s}^{-1})\) to spin up the velocity structure in the first-guess solution, which we initialize with \({\mathbf {u}}(z,0)={\mathbf {0}}\) and then perform 50 h of integration. We construct the reference case and extract velocity data by continuing the integration for another 50 h.

By measuring the reference case and adding Gaussian noise, we generate nine simulated measurements of \({\mathbf {u}}\); i.e., we have a total of 18 measurements of the u and v components of the velocity at three different depths. Figure 17.1 illustrates the measurement locations, together with the first-guess, the reference solution, and the posterior mode from the representer method. The reference solution is regenerated quite well, even though the first-guess solution is out of phase with the reference case, and the measurements do not resolve the period of the oscillation. A single measurement may suffice for reconstructing the correct phase since the corresponding representer will carry the information both forward and backward in time. However, the errors will be more significant with fewer measurements.

To illustrate the solution procedure using the representer method in more detail, Fig. 17.2 presents the u-components of the variables \({\mathbf {s}}_5\), \({\mathbf {r}}_5\), \(\boldsymbol{\lambda }\), and the convolutions \({\mathbf {C}}_\textit{qq}\bullet {\mathbf {s}}_5\) and \({\mathbf {C}}_\textit{qq}\bullet \boldsymbol{\lambda }\). The \(\bullet \) denote convolution in space and time, but we set the model errors’ time correlation to zero in this example. The symbols follow the notation of Sect. 5.5. The subscript five denotes the measurement number. Measurement number five corresponds to the u component at the location \((z,t)=(-20.0,25.0)\). These plots demonstrate how the information from the measurements influences the solution.

Fig. 17.1
figure 1

Reproduced from Eknes and Evensen (1997)

The u components of (from top to bottom) the first-guess estimate \({\mathbf {u}}_F\), the reference case \({\mathbf {u}}\) and the posterior mode solution for \({\mathbf {u}}\). The contour intervals are 0.05 m s\(^{-1}\) for all the velocity plots. The measurement locations are marked with a bullet. The v components are similar in structure and not shown.

Fig. 17.2
figure 2

Reproduced from Eknes and Evensen  (1997)

The u component of (top to bottom) \({\mathbf {s}}_5\), \({\mathbf {C}}_\textit{qq}\bullet {\mathbf {s}}_5\), \({\mathbf {r}}_5\), the adjoint \(\boldsymbol{\lambda }\), and \({\mathbf {C}}_\textit{qq}\bullet \boldsymbol{\lambda }\). The measurement locations are marked with a bullet. The v components are similar in structure and not shown.

The upper plot shows the u-component of adjoint representer \({\mathbf {s}}_5\) forced by the impulse function at the measurement location, see Eq. (5.41). This information is then propagated backward in time while the u and v adjoint representer velocity components interact during the integration.

After that, we use \({\mathbf {s}}_5\) on the right-hand side of the forward Eqs. (5.38) and (5.39) to evaluate the representer’s initial and boundary conditions and forcing fields. The convolution \({\mathbf {C}}_\textit{qq}\bullet {\mathbf {s}}_5\), is a smoothing of \({\mathbf {s}}_5\) according to the covariance functions contained in \({\mathbf {C}}_\textit{qq}\), as seen from the second plot in Fig. 17.2.

The representer \({\mathbf {r}}_5\) is smooth and is oscillating in time with a period reflecting the inertial oscillations described by the dynamical model. Note that the representers will have a discontinuous time derivative at the measurement location since the right-hand side \({\mathbf {C}}_\textit{qq}\bullet {\mathbf {s}}_5\) is discontinuous there. However, if we had included a time-correlation in \({\mathbf {C}}_\textit{qq}\), then \({\mathbf {C}}_\textit{qq}\bullet {\mathbf {s}}_5\) would be continuous, and the representer \({\mathbf {r}}_5\) would be smooth.

After computing and measuring the representers of all observations to generate the representer matrix \(\boldsymbol{\mathcal {R}}\), we solve for the vector \({\mathbf {b}}\) from Eq. (5.42). We then use \({\mathbf {b}}\) in (5.43) to decouple the Euler–Lagrange equations. The u-component of \(\boldsymbol{\lambda }\) (Fig. 17.2) illustrates how the various measurements have a different impact determined by values of the coefficients in \({\mathbf {b}}\). After solving for \(\boldsymbol{\lambda }\), we construct the right-hand side in the forward model equation through the convolution  \({\mathbf {C}}_\textit{qq}\bullet \boldsymbol{\lambda }\), given at the bottom of Fig. 17.2. The role of this term is to force the solution to smooth the measurements.

3 Assimilation of Real Measurements

We will now apply the representer method with the LOTUS–3 data set  (Bowers et al., 1986) in a similar setup to the one used by Yu and O’Brien (1991, 1992). The LOTUS–3 experiment sampled measurements in the northwestern Sargasso Sea (34\(^\circ \) N, 70\(^\circ \) W) during summer 1982. Current meters at depths of 5, 10, 15, 20, 25, 35, 50, 65, 75, and 100 m measured the currents. A wind recorder mounted on top of the LOTUS–3 tower measured the wind speed. The sampling interval was 15 min, and we are using data from June 30 to July 9, 1982. We have further subsampled the data every five hours at the depths 5, 25, 35, 50, and 75 m. The reason for not using all the measurements is to reduce the size of the representer matrix. The data still resolve the inertial period and the vertical length, and we expect only to ignore small-scale noise by the subsampling.

Fig. 17.3
figure 3

Weak-constraint results from the LOTUS–3 assimilation experiment from Eknes and Evensen  (1997). Inverse estimate for the u component of velocity (red lines), the time series of measurements (blue lines), and the subsampled measurements (bullets), at 5, 25 and 50 m

We initialized the model from the first measurements collected on June 30, 1982. The standard deviation of the small-scale variability of the velocity observations was close to \(0.025\,\mathrm {m}\,\mathrm {s}^{-1}\), and we used this value to determine the error variances for the observations and the initial conditions. We specified the model error variance after a few runs to give a relatively smooth posterior mode estimate. We want a solution that nearly satisfies the model equations and is close to the observations without over-fitting them.

Figure 17.3 shows the results from the estimation as time series of the u component of the velocity at various depths. The figure plots the posterior-mode estimate together with the complete time series of measurements. We denote the measurements used in the estimation as bullets. The estimate’s amplitude and phase agree well with the measurements at all times and depths. Note also that the estimate is smooth and does not precisely interpolate the measurements. By a closer examination of the solution, it is possible to see that the time derivative is discontinuous at measurement locations due to neglecting the time correlation in the model error covariances.

For comparison, we present a strong-constraint solution in Fig. 17.4. It is clear from comparisons that the strong-constraint solution, as determined by the initial conditions, in the upper part of the ocean is reasonably in phase with the measurements. At the same time, the amplitudes are not as good as in the weak-constraint solution. The only way the amplitudes can change when the model is assumed to be perfect is by vertical transfer of momentum from the surface. Thus, we obtain a good fit near the surface, while there is hardly any effect from the wind stress at depth, and the strong-constraint solution is also far from the measurements. The strong-constraint estimate is close to a sine curve representing the model’s inertial oscillations. These results indicate that model deficiencies, such as neglected physics, should be accounted for through a weak-constraint variational formulation to ensure the solution agrees with the measurements.

The representer solution provides the optimal minimizing solution of the linear inverse problem. Furthermore, the M-dimensional space spanned by the representers contains the optimal solution update. Note that the equation for \({\mathbf {b}}\), (5.42) is similar to the one solved in the analysis scheme in the standard Kalman filter. The representers correspond to the measurements of the space-time error covariance of the first-guess solution. Thus, there are strong similarities between the analysis step in the ensemble Kalman smoother and in the representer method. To summarize, the representer method is a highly efficient approach for solving linear inverse problems, and it is also applicable to many nonlinear dynamical models.

Fig. 17.4
figure 4

Strong-constraint results from the LOTUS–3 assimilation experiment from Eknes and Evensen  (1997). Inverse estimate for the u component of velocity (red lines), the time series of measurements (blue lines), and the subsampled measurements (bullets), at 5, 25 and 50 m