Plotting triangulation with variables defined at the barycenter

I have created a triangulation of a 2D space, here reported you can see it:

On such triangulation I have perfomed some calculations to see the evolution in time of a certain variable (the water depth, if you need to picture it) defined at the barycenter of each triangle. I wish to plot such triangulation together with the value of the variable at the barycenter on a 3d surface in order to see such surface moving in time. I searched on the documentation and the closest thing that I have found was this procedure reported at https://www.mathworks.com/help/matlab/math/delaunay-triangulation.html

Now the plot that I want is something like this, but the only issue is that in the documentation the values of the z axis (Depth in Feet) are defined on the vertices of the triangles while in my code the values of the variable are defined only at the barycenter of the triangles. Is there a way to realise a plot similar to the picture reported where the values of the variable are defined at the barycenter of the triangles instead? Assume I have all the coordinates of both vertices and barycenters.

Thanks in advance for your suggestions and help.