In this section we present 4D optimal interpolation technique, in both space and
time, allowing retrieving the entire evolution of tidal currents from the survey data.
The technique employs tight space-time correlations in the tidal flow field that can
be accurately simulated by the existing state-of-the-art numerical models. A brief
description of the method and its application to assessing tidal circulation in a
limited size area (Boulogne Harbour) are given below. The harbour is located in the
French sector of the Dover Strait, in the vicinity of the area covered by the HFR
observations (Fig. 2).
A straightforward method of statistically consistent spatial interpolation of a
vector field is the well-known optimal interpolation (OI) pioneered by Gandin [7].
Since then, the approach was widely adopted in geosciences (e.g., [2, 30]). The OI
technique can be easily extended to include time dimension by using the space-time
correlation functions. In this approach, the optimal correction to the evolution of a
background vector field u
m (x, t) defined on a regular (model) grid is represented by
a linear combination of the weighted differences between the background trajectory
and the observed velocities. The weights a i are chosen so as to minimise the mean
square difference between observations u i
* and the background field values u
m ,
interpolated into the space-time locations of the observations by the (linear) operator H
i , projecting gridded velocity values onto the i-th observation point from the
apexes of the enveloping grid cell:
J u = ⟨ u
m + ∑
i
a i ðH
i u
m
− u
*
i Þ
2
⟩ → minða i Þ
ð 1Þ
Here, angular brackets denote the statistical (ensemble) average, and summation
is made over all (distributed in space and time) velocity values measured during the
survey period. Given the space-time covariance matrices of the model B = ⟨u
m (x,
t) u
m (x′, t′)⟩ and observations R ij = ⟨u i
* u j
*
⟩, and assuming that observation errors
are not correlated with the model (background) errors, the OI interpolation formula
takes the form:
u opt = u
m + ∑
ij
B H
T
j ðH i B H
T
j + R ij Þ
− 1 ðH
i u
m
− u
*
i Þ
ð 2Þ
In order to apply Eq. (2) for interpolating the surveyed velocities, u
m (x, t) and
B are specified using the output statistics from the regional model MARS-3D [13]
configured for high resolution simulations in the surveyed area [25].
The result of interpolation allows accurate tracking of the motion of the
anti-cyclonic eddy derived from underway velocity measurements by towed ADCP
within the Boulogne harbor on March 27, 2012 (Fig. 7a). Figure 7 also shows two
snapshots of the velocity field: 20 min after the beginning of the survey, at
HW − 1.2 h, and at the end of the survey, at HW + 0.5 h. As it can be seen, the
optimal interpolation enables relatively accurate assessment of the circulation pattern: at the beginning of the period (Fig. 7b), the eddy center was located just south
Monitoring Strong Tidal Currents in Straits and Nearshore Regions
531
time, allowing retrieving the entire evolution of tidal currents from the survey data.
The technique employs tight space-time correlations in the tidal flow field that can
be accurately simulated by the existing state-of-the-art numerical models. A brief
description of the method and its application to assessing tidal circulation in a
limited size area (Boulogne Harbour) are given below. The harbour is located in the
French sector of the Dover Strait, in the vicinity of the area covered by the HFR
observations (Fig. 2).
A straightforward method of statistically consistent spatial interpolation of a
vector field is the well-known optimal interpolation (OI) pioneered by Gandin [7].
Since then, the approach was widely adopted in geosciences (e.g., [2, 30]). The OI
technique can be easily extended to include time dimension by using the space-time
correlation functions. In this approach, the optimal correction to the evolution of a
background vector field u
m (x, t) defined on a regular (model) grid is represented by
a linear combination of the weighted differences between the background trajectory
and the observed velocities. The weights a i are chosen so as to minimise the mean
square difference between observations u i
* and the background field values u
m ,
interpolated into the space-time locations of the observations by the (linear) operator H
i , projecting gridded velocity values onto the i-th observation point from the
apexes of the enveloping grid cell:
J u = ⟨ u
m + ∑
i
a i ðH
i u
m
− u
*
i Þ
2
⟩ → minða i Þ
ð 1Þ
Here, angular brackets denote the statistical (ensemble) average, and summation
is made over all (distributed in space and time) velocity values measured during the
survey period. Given the space-time covariance matrices of the model B = ⟨u
m (x,
t) u
m (x′, t′)⟩ and observations R ij = ⟨u i
* u j
*
⟩, and assuming that observation errors
are not correlated with the model (background) errors, the OI interpolation formula
takes the form:
u opt = u
m + ∑
ij
B H
T
j ðH i B H
T
j + R ij Þ
− 1 ðH
i u
m
− u
*
i Þ
ð 2Þ
In order to apply Eq. (2) for interpolating the surveyed velocities, u
m (x, t) and
B are specified using the output statistics from the regional model MARS-3D [13]
configured for high resolution simulations in the surveyed area [25].
The result of interpolation allows accurate tracking of the motion of the
anti-cyclonic eddy derived from underway velocity measurements by towed ADCP
within the Boulogne harbor on March 27, 2012 (Fig. 7a). Figure 7 also shows two
snapshots of the velocity field: 20 min after the beginning of the survey, at
HW − 1.2 h, and at the end of the survey, at HW + 0.5 h. As it can be seen, the
optimal interpolation enables relatively accurate assessment of the circulation pattern: at the beginning of the period (Fig. 7b), the eddy center was located just south
Monitoring Strong Tidal Currents in Straits and Nearshore Regions
531