Level Coordinate System
For the simple level coordinate system, we allow various options for the representing the bathymetry.Hybrid Coordinate System
For the hybrid sigmaz coordinate option, we assume that the mixed layer and topography do not intersect. This is not so great, for example, in the Arctic Ocean, but doesn't unduly upset us in our current work. However, we did not want to abandon Dake's vertical mixing parameterization which requires a variable depth mixed layer. So a hybrid vertical structure was implemented  a variable depth mixed layer, patched to a zcoordinate lower ocean by a predetermined division of the intermediate water into sigma layers.Hybrid coordinate system  

patched to z levels using intermediate sigma levels z levels which can intersect topography 
In this example,

In order to preserve the condition
we must satisfy the compatibility condition
The remaining degree of freedom was used to place the first value of z in the center of the first layer, so that z(1) = h(1)/2. This allows easy coordinate transformations between h and z.