Land masks, mesh and global temperature

I have been writing articles about land masks, leading up to using them to check and maybe improve my triangular mesh based TempLS. As I have tried to emphasise, the core of estimating global average temperature anomaly (or average anything) is numerical spatial integration. The temperature is known at a finite number of points. It has to be inferred for all the rest (interpolation) and the result complete data integrated. To do this successfully, the data has to be fairly homoigeneous, so anomalies are formed to take out variance in long term mean values. Then in the triangle method, linear interpolation is done within triangles.

But another kind of inhomogeneity is between land and sea, and indices often use a land mask to try to pin that down. In the mesh context, and in general, the idea is to ensure that values on land are only interpolated from land data; sea likewise.

The method corresponding to what is done with grids would be to count the mask elements within each triangle, and to divide coast-crossing triangle into a land and sea part. Since all that matters in the end is the weighting of each node, it's only necessary to get the area right. Assigning maybe a million grid elements to triangles is a rather heavy computation. So I tried something more flexible. Here is a snapshot from the WebGL graphic below. It shows a problem section in East Africa. Light blue triangles are those that have two sea nodes, one land, and orange are those with two land, one sea. The Horn of Africa is counted as sea, and there is a good deal of encroachment of sea on land. That is about as bad as it gets, and of course there is some cancelling where land encroaches on sea.

So I refine the mesh. On the longest 20% of lines in such triangles, with land at one end and sea at the other, I make an extra node, and test whether it is sea/land with the mask. Then I give it the value of its matching end type. With the new nodes, I then re-mesh. This process I repeat several times. After respectively four and seven steps I get:

As you see, the situation improves greatly. new nodes cluster around the coast. There is however, still two rather large triangles at sea with a land node. These can show up when everything else seems converged; it is because of the convex hull re-meshing which may make different decisions about some of the large triangles bordering the coast. It slows convergence. As to placement of that new node on the line, that is where the mask with a metric comes in. I know the approx distance of each node from the coast, and can place the new node where I estimate the cost to cross. I don't want it to be too exact, just to minimise the interior nodes created.

What I really want to know is what this does to the integral. So I tried first integrating the mask itself. That is a severe test; the result should show land/sea proportions as in a count of the mask. Then I tried integrating anomalies for February 2017. I'll show those below, but first, here is the WebGL showing of the seven stages of refinement (radio buttons).

Integration results

The table below shows the results of the progression. The left column is the area of the mixed triangles (part land, part sea), as a proportion of total surface. The next shows the result of integrating the mask itself, which should converge to 0.314. The third are the successive integrals of the anomalies for February 2017.

	Mixed area (fraction of sphere)	Integral of mask	Integral of anomaly °C, Feb 2017
Step 0	0.1766	0.3118	0.8728
Step 1	0.1268	0.3228	0.8583
Step 2	0.1097	0.3192	0.8645
Step 3	0.0845	0.3205	0.8655
Step 4	0.0682	0.3212	0.8646
Step 5	0.0578	0.3203	0.8663
Step 6	0.0489	0.3208	0.8624
Step 7	0.0429	0.3199	0.8611

Conclusion

I think it was a coincidence that the mask integration turned out near its target value of 0.314 at step 0 (no mesh change). As I said above, this is the most demanding case, maximising inhomogeneity. It doesn't improve because of the occasional flipping of triangles which leads to the occasional exceptions that show in the WebGL, but also because it started so close. For anomalies, the difference it makes to February 2017 is small at around 0.01°C.

So, while I am glad to have checked on the coast issue, I don't think it is worth incorporating this method in TempLS. It means extra convex hull calculation for each month, which is slow.