tag:blogger.com,1999:blog-7729093380675162051.post8110736011388087005..comments2021-03-22T09:11:17.119+11:00Comments on moyhu: Methods of integrating temperature anomalies on the sphereNick Stokeshttp://www.blogger.com/profile/06377413236983002873noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-7729093380675162051.post-23147447849792072472019-10-27T06:23:48.300+11:002019-10-27T06:23:48.300+11:00Clive,
"You seem to be using what I call a le...Clive,<br /><i>"You seem to be using what I call a level 4 Icosahedral grid with 2562 nodes"</i><br />You tend to work in terms of iterated bisecting; I do fractional divisions. Each triangle is divided into n² elements. So h2 has 80 elements in total (42 nodes); h3 has 180 (92), h4 has 320 (162). So a lot fewer elements.<br /><br /><i>"Do you also interpolate into empty triangles?"</i><br />With higher order polynomials, it's harder, but there is a good solution. You need enough nodes (not just 1) to get a unique fit. The solution is using the Laplace operator. Here is how I do it for the simpler grid infill method. You have a whole lot of grid equations<br />NᵢTᵢ=yᵢ<br />where N is the number of nodes in a tri i, T is the average anomaly, and y is the sum of anomalies. It's a system of equations but may be singular if some Nᵢ=0.<br /><br />Embed it in a matrix system<br />AᵢₖTₖ=yᵢ<br />(summing over k - summation convention) where A is a diagobal matrix with N's on the diag.<br /><br />Let B be a Laplacian matrix. Good enough is for BT=0 to say that each cell value is the average if its neighbors - ie 3 on the diagonal (triangle grid), 0 elsewhere except, for each row, a value of -1 at each neighbor col.<br /><br />Then solve (A+εB)T = y<br />ε needs to be large enough to make A+εB not too ill-conditioned, but small enough not to do too much smoothing of the known data. There is a lot of room there.<br /><br />My new method uses sparse matrix structures and conjugate gradient methods for solving that last equation. With the FEM version, A is like a mass matrix. It's the matrix for the local regression.<br /><br />Nick Stokeshttps://www.blogger.com/profile/06377413236983002873noreply@blogger.comtag:blogger.com,1999:blog-7729093380675162051.post-53703736750694363572019-10-27T04:26:15.438+11:002019-10-27T04:26:15.438+11:00I think I understand your new method . It has got ...I think I understand your new method . It has got me interested again in the use of icosahedral grids (FEM) for the earth's surface. You seem to be using what I call a level 4 Icosahedral grid with 2562 nodes, then using LOESS to fit the station values to a polynomial for each triangle. The result are nice smooth anomaly distributions. Do you also interpolate into empty triangles?<br /><br />I have instead been using a brute force method by stepping up to a level 5 icosahedral grid with 10224 nodes and then averaging all temperatures within the 4 times smaller triangles. Perhaps I should be using R !Clive Besthttps://www.blogger.com/profile/10486120708699060846noreply@blogger.comtag:blogger.com,1999:blog-7729093380675162051.post-77581283399407287152019-10-21T04:25:01.157+11:002019-10-21T04:25:01.157+11:00The issue with the 1940's is that the correcti...The issue with the 1940's is that the corrections are abrupt, drawn along the war-year endpoints when measurement techniques changed dramatically. Differing filtering techniques will either exaggerate or suppress the adjustments made at the boundary years. A high-pass filter will enhance, while a low-pass will suppress. BTW, the strongest multiple El Nino peak also occurred in the early 1940's making it eve more difficult to adjust.<br />pphttps://www.blogger.com/profile/15737287219806254245noreply@blogger.comtag:blogger.com,1999:blog-7729093380675162051.post-42646647984961657842019-10-21T00:45:02.365+11:002019-10-21T00:45:02.365+11:00Hum, it seems that the discrepancy might be relate...Hum, it seems that the discrepancy might be related to some artifact in the 1940's. Maybe you have discovered a test to pinpoint anomalous data here.Yvan Dutilhttps://www.blogger.com/profile/16380549602801539075noreply@blogger.com