Marine Ecosystem Model Calibration through Enhanced Surrogate-Based Optimization - Simulation and Modeling Methodologies, Technologies and Application - page 187

κ ∂y ( l )
∂z

∂y ( l )
∂t =

∂
∂z

+ Q ( l ) ( y,u 2 ,...,u n ) , l = N,P,Z,

(1)

κ ∂y ( D )
∂z

∂y ( D )
∂t =

∂y ( D )
∂z

∂
∂z

+ Q ( D ) ( y,u 2 ,...,u n ) − u 1 , l = D,

in ( −H, 0) × (0 ,T ) , with additional appropriate initial values. Here, z denotes the only
remaining, vertical spatial coordinate, and H the depth of the water column. The terms
Q ( l )

are the biogeochemical coupling (or source-minus-sink ) terms for the four tracers

and u =( u 1 ,...,u n ) is the vector of unknown physical and biological parameters,

with n =12 for this specific model. The sinking term (with the sinking velocity u 1 )is
only apparent in the equation for detritus. In the one-dimensional model no advection
term is used, since a reduction to vertical advection would make no sense. Thus, the
circulation data (taken from an ocean model) are the turbulent mixing coefficient κ =
κ ( z,t ) and the temperature Θ = Θ ( z,t ) , which goes into the nonlinear coupling terms
Q ( l ) but is omitted in the notation.
The parameters to be optimized are, for example, growth and dying rates of the

tracers and thus appear in the nonlinear coupling terms Q ( l )

u

l = N,P,Z,D in (1). For the sake
of brevity and for the purpose of this paper we omit the explicit formulation of the
coupling terms as well as the explicit physical meaning of the involved parameter. For
details we refer the reader to [13,16].

2.2 Numerical Solution

The continuous model (1) is discretized and solved using an operator splitting method
[11], an explicit Euler time stepping scheme for the nonlinear coupling terms Q and the
sinking term while using an implicit scheme for the diffusion term. For further details
we refer the reader to [13,14].
More explicitly, in every discrete time step, at first the nonlinear coupling operators
Q j (that depend on t j directly and/or via the temperature field Θ ) are computed at every
spatial grid point and integrated by four explicit Euler steps with step size τ/ 4 . Then,
an explicit Euler step with full step size τ is performed for the sinking term. Finally, an
implicit Euler step for the diffusion operator, again with full step size τ , is applied.
In the original model, the time step τ is chosen as one hour. By choosing this time
step, all relevant processes are captured and further decrease of the time step does not
improve the accuracy of the model. The model with this particular time step will be
referred to as the high-fidelity or fine one in the following.
We furthermore denote by

y j ≈ y ( ·,t j ) the discrete fine model solution of the con-
tinuos model (1) in time step j (containing all tracers N,P,Z,D ) given as

M f I , I = n z n t ,

y j =( y ji ) i =1 ,...,I , j =1 ,...,M f , y ∈ R (2)

where I denotes the number of spatial discrete points n z times the number of tracers
n t , which is four for the considered model, and where M f denotes the total number of
discrete time steps, given the discrete time step τ f . More specifically, the model consists
of n z =66 vertical layers and is integrated over totally M f = 8760 time steps/year ×