Sequential Data Assimilation for Nonlinear Dynamics: The Ensemble Kalman Filter
109
~=8/3. The terms l(t), qY(t) and qZ(t) are assumed to represent the unknown model
errors. Initial conditions for the model are given as
x(O)
z(O)
x
= X o + a ,
z
= Zo + a ,
(30)
where xo, Yo' and Zo are the first-guess values ofthe initial conditions and the terms
if, a Y and a Z represent the errors in the first-guess initial conditions. If all the error
terms were known or equal to zero, these equations would formulate a well-posed
problem having a unique solution in a mathematical sense.
Now a set ofmeasurements, d E 9\M, ofthe true solution are assumed given and
linearly related to the model variables by the measurement equation
d = L[x,y,Z]+E,
(31)
where L E 9\M is a linear measurement functional, E E 9\M is a vector of measurement errors and M is the number of measurements.
6.5.2 Discussion of cases
The initial condition for the reference case is given by
(xo, Yo' zo)= (1.508870, -1.531271, 25.46091)
and the time interval is tE [0,40]. The observations and initial conditions are simulated by adding normal distributed noise with zero mean and variance equal to 2.0,
to the reference solution. The initial conditions used are also assumed to have the
same variance as the observations. These are the same values that were used in
Miller et al. (1994) and Evensen and Fario (1997).
The following examples are discussed.
Experiment A: In this first experiment the distance between the measurements is
Atobs= 0.25, which is the same as was used in Miller et al. (1994). Thus, it is
possible to compare the results presented here with those presented in Miller
et al. (1994) using the extended Kalman filter and a strong constraint variational method.
Experiment B: In order to examine the sensitivity with respect to measurement
density, an additional experiment is now performed where the distance between the measurements is Atobs = 0.5.
The data assimilation estimate from the ensemble Kalman filter is given in Figs.
6.3 and 6.4 for the two cases. The ensemble size is 1000 members. The ensemble
Kalman filter seems to do a reasonably good job in tracking the state transitions
and also in reproducing the correct amplitudes for the peaks of the solution.
109
~=8/3. The terms l(t), qY(t) and qZ(t) are assumed to represent the unknown model
errors. Initial conditions for the model are given as
x(O)
z(O)
x
= X o + a ,
z
= Zo + a ,
(30)
where xo, Yo' and Zo are the first-guess values ofthe initial conditions and the terms
if, a Y and a Z represent the errors in the first-guess initial conditions. If all the error
terms were known or equal to zero, these equations would formulate a well-posed
problem having a unique solution in a mathematical sense.
Now a set ofmeasurements, d E 9\M, ofthe true solution are assumed given and
linearly related to the model variables by the measurement equation
d = L[x,y,Z]+E,
(31)
where L E 9\M is a linear measurement functional, E E 9\M is a vector of measurement errors and M is the number of measurements.
6.5.2 Discussion of cases
The initial condition for the reference case is given by
(xo, Yo' zo)= (1.508870, -1.531271, 25.46091)
and the time interval is tE [0,40]. The observations and initial conditions are simulated by adding normal distributed noise with zero mean and variance equal to 2.0,
to the reference solution. The initial conditions used are also assumed to have the
same variance as the observations. These are the same values that were used in
Miller et al. (1994) and Evensen and Fario (1997).
The following examples are discussed.
Experiment A: In this first experiment the distance between the measurements is
Atobs= 0.25, which is the same as was used in Miller et al. (1994). Thus, it is
possible to compare the results presented here with those presented in Miller
et al. (1994) using the extended Kalman filter and a strong constraint variational method.
Experiment B: In order to examine the sensitivity with respect to measurement
density, an additional experiment is now performed where the distance between the measurements is Atobs = 0.5.
The data assimilation estimate from the ensemble Kalman filter is given in Figs.
6.3 and 6.4 for the two cases. The ensemble size is 1000 members. The ensemble
Kalman filter seems to do a reasonably good job in tracking the state transitions
and also in reproducing the correct amplitudes for the peaks of the solution.