the third category of optimal design based on spatial sampling error study and high resolution dataset.
The optimal design for a one-category-instrument network is a simplified case of OOSD. This kind of optimal network design with a prescribed mean sampling error F0 can be defined as follows:
where r presents a spatial point, L and M location dependent zonal and meridional sampling distances, F an error function which is selected as mean sampling error and S the area covered by the observing system. Since the solution is obtained for the whole system, the optimal design is called a global design.
3.2 Simplification of the global optimal network design
With the Lagrangian multiplier λ, the network design (I) can be written as follows:
The control equations of the optimal solution are:
Noting that the integration in the objective function J is for the whole system, this gives rise to two difficulties in solving the optimization problem with optimization methods (such as optimal descending techniques). One is that the objective function here is usually not quadratic and positive definite and the final numerical solution may not be convergent owing to unknown features of objective functions in the parameter space. The second difficulty is that there exists a large number of freedoms in a comprehensive global design problem and this leads to a heavy computation.
To make the problem to be tractable, a notion of the local homogeneous condition is introduced. This means that the whole tropical Pacific can be divided into a finite number of homogeneous sub-regions. Considering that the error function and sampling parameters are spatially uniform in each sub-regions 1, control equations (1)-(3) can be simplified as the following forms by substituting Equations in (I') into Eqs. (1)-(3):
