The basic process for that integration is to match the data to a locally smooth function which can be integrated analytically. That function S(x) is usually expressed as a linear combination of basis functions Bᵢ(x), where x are surface coordinates
| s(x) = bᵢBᵢ(x) | 1 |
The fitting process is usually least squares; minimise over coefficients b:
| (yj-bᵢBᵢ(xj)) Kjk (yₖ-bₘBₘ(xₖ)) | 2 |
where the yj are the data points, say monthly temperature anomalies, at points xj) and K is some kernel matrix (positive definite, usually diagonal).
This all sounds a lot more complicated than just calculating grid averages, which is the most common technique. I'd like to fit those into this framework; there will be simplifications there.
Minimising (2) by differentiating wrt b gives
| (yj-bᵢBᵢ(xj))KjkBₘ(xₖ) | 3 |
| or in more conventional matrix notation | |
| HwH⊗b = Hw⊗y | 4 |
where H is the matrix Bᵢ(xj)), Hw is the matrix Bᵢ(xj))Kjk and HwH = Hw⊗H* (* is transpose and ⊗ is matrix multiplication). So Eq (4) is a conventional regression equation for the fitting coefficients b. The final step in calculating the integral as
A = bᵢaᵢ
where aᵢ are the integrals of the functions Bᵢ(x), or
| A=a*⊗(HwH⁻¹)⊗Hw⊗y | 5 |
The potentially big operations here are the multiplication of the data values y by the sparse matrix Hw, and then by the inverse of HwH. However, there are some simplifications. The matrix K is the identity or a diagonal matrix. For the grid method and mesh method (the early staple methods of TempLS), HwH turns out to be a diagonal matrix, so inversion is trivial. Otherwise it is positive definite, so a conjugate gradient iterative method can be used, which is in effect the direct minimisation of the quadratic in Eq (2). That requires only a series of multiplication s of vectors by HwH, so this product need not be calculated, but left as its factors.
I'll now look at the specific methods and how they appear in this framework.
Grid method
The sphere is divided into cells, perhaps by lat/lon, or as I prefer, using a more regular grid based on platonic solids. The basis functions Bᵢ are equal to 1 on the ith element, 0 elsewhere. K is the identity, and can be dropped.The matrix H has a row for each cell and a column for each data point. It is zero except for a 1 where data point j is in cell i. The product HwH is just a diagonal matrix with a row for each cell, and the diagonal entry is the total number of data points in the cell (which may be 0). So that reduces to saying that the coefficient b for each cell is just the average of data in the cell. It is indeterminate if no data, but if such cells are omitted, the value would be unchanged if the cells were assigned the value of the average of the others.
Multiplying H by vector y is a sparse operation, which may need some care to perform efficiently.
Mesh method
Here the irregular mesh ensures there is one basis function per data point, central value 1. Again K is the identity, and so is H (and HwH). The net result is just that the integral is just the scalar product of the data values y with the basis function integrals, which area just 1/3 of the area of the triangles on which they sit.More advanced methods with sparse matrix inversion
For more advanced methods, the matrix HwH is symmetric positive definite, but not trivial. It is, however, usually well-conditioned, so the conjugate gradient iterative method converges well, using the diagonal as a preconditioner. To multiply by HwH, you can successively multiply by H*, then by diagonal K, then by H, since these matrix vector operations, even iterated, are likely to be more efficient than a matrix-matrix multiplication.Spherical Harmonics basis functions
My first method of this kind used Spherical Harmonic (SH)functions as bases. These are a 2D sphere equivalent of trig functions, and are orthogonal with exact integration. They have the advantage of not needing a grid or mesh, but the disadvantage that they all interact, so do not yield sparse matrices. Initially I followed the scheme of Eq (5) with K the identity, and H the values of the SH at the data points. HwH would be diagonal if the products of the SH's where truly orthogonal, but with summation over irregular points that is not so. The matrix HwH becomes more ill-conditioned as higher frequency SH's are included. However the condition is improved if a kernel K is used which makes the products in HwH better approximations of the corresponding integral products. A kernel from the simple grid method works quite well. Because of this effect, I converted the SH method from a stand alone method to an enhancement of other methods, which provide the kernel K. This process is described here.For a long time I used SH fits, calculating the b coefficients as above, for graphic shade plots of the continuum field. But some other advanced methods provide their own fitted functions.
Grid Infill and the Laplacian
As mentioned above, the weakness of grid methods is the existence of cells with no data. Simple omission is equivalent to assigning those cells the average of the cells with data. As Cowtan and Way noted with HADCRUT's use of that method, it has a bias when the empty cell regions are behaving differently to that average; specifically, when the Arctic is warming faster than average. Better estimates of such cells are needed, and this should be obtained from nearby cells. Some people are absolutist about such infilling, saying that if a cell has no data, that is the end of it. But the whole principle of global averaging is that region values are estimated from a finite number of known values. That happens within grid cells, and the same principle obtains outside. The locally based estimate is the best estimate, even if it isn't very good. There is no reason for using anything else.I described here a rather ad hoc method for iteratively estimating empty cells from neighbors with data, which worked quote well. But the matrix formulation above gives a more scientific way of doing this. You can write a matrix which expresses a requirement that each cell should have the average values of its neighbors. For example, a rectangular grid might have, for each cell, the requirement that 4 times its value - the values of the four neighbors is zero. You can make a (sparse symmetric) matrix L out of these relations, which will have the same dimension as HwH. It has a rank deficiency of 1, since any constant will satisfy it.
Now the grid method gave an HwH which was diagonal, with rows of zeroes corresponding to empty cells, preventing inversion. If you form HwH + εL, that fixes the singularity, and on solving does enforce that the empty cells have the value of the average of the neighbors, even if they are also empty. In effect, it fills with Laplace equation using known cell averages as boundaries. You can use a more exact implementation of the Laplacian than the simple average, but it won't make much difference.
There is a little more to it. I said L would be sparse symmetric, but that involves adding mirror numbers into rows where the cell averages are known. This will have the effect of smoothing, depending on how large ε is. You can make ε very small, but then that leaves the combination ill-conditioned, which slows the iterative solution process. For the purposes of integration, it doesn't hurt to allow a degree of smoothing. But you can counter it by going back and overwriting known cells with the original values.
I said above that, because HwH inversion is done by conjugate gradients, you don't need to actually create the matrix; you can just multiply vectors by the factors as needed. But because HwH is diagonal, you don't need a matrix-matrix multiplication to create HwH + εL, so it may as well be used directly.
I have gone into this in some detail because it has application beyond enhancing the grid method. But it works well in that capacity. My most recent comparisons are here.
FEM/LOESS method
This is the most sophisticated version of the method so far, described in some detail here. The basis functions are now those of finite elements on a triangular (icosahedron-based) mesh. They can be of higher polynomial order. The matrix H can be created by an element assembly process. The kernel K can be the identity.The question of empty cells is replaced by a more complicated issue of whether each element has enough data to uniquely specify a polynomial of the nominated order. However, that does not have to be resolved, because the same remedy of adding a fraction of a Laplacian works in the same way of regularising with the Laplace smooth. It is no longer easy to just overwrite with known values to avoid smoothing where you might not have wanted it, but where you can identify coefficients added to the matrix which are not wanted, you can add the product of those with the latest version of the solution to the right hand side and iterate, removing spurious smoothing with rapid convergence. But I have found that there is usually a range of ε which gives acceptably small smoothing with adequate improvement in conditioning.
LOESS method
This is the new method I described in April. It does not fit well into this framework. It uses an icosahedral mesh; the node points are each interpolated by local regression fitting of a plane using the 20 closest points with an exponentially decaying weight function. Those values are then used to create an approximating surface using the linear basis functions of the mesh. So the approximating surface is not itself fitted by least squares. The method is computationally intensive but gives good results.Modes of use of the method - integrals, surfaces and weights.
I use the basic algebra of Eq 4 in three ways:| HwH⊗b = Hw⊗y | 4 |
| A = b•a | 4a |
- The straightforward way is to take a set of y values, calculate b and then A, the integral which, when divided by the sphere area, gives the average.
- Stopping at the derivation of b then gives the function s(x) (Eq 1) which can be used for graphics.
- There is the use made in TempLS, which iterates values of y to get anomalies that can be averaged. This required operating the multiplication sequence in reverse:
A = ((a•HwH⁻¹)⊗Hw)•y 6
By evaluating the multiplier of y, a set of weights is generated, which can be re-used in the iteration. Again the inversion of HwH on a is done by the conjugate gradient method.
Summary
Global temperature averaging methods, from simple to advanced, can mostly be expressed as a regression fitting of a surface (basis functions), which is then integrated. My basic methods, grid and mesh, reduce to one that does not require matrix inversion. Spherical harmonics give dense matrices and require matrix solution. Grid infill and FEM/LOESS yield sparse equations, for which conjugate gradient solution works well. The LOESS method does not fit this scheme.Appendix - code and libraries
I described the code and structure of TempLS V4 here. It uses a set of library functions which are best embedded as a package; I have described that here. It has all been somewhat extended and updated, so I'll call it now TempLS V4.1. I have put the new files on a zip here. It also includes a zipreadme.txt file, which says:This zip contains:
LS_wt.r master file for TempLS V4.1
LS_fns.r library file for TempLS V4.1
Latest_all.rda a set of general library functions used in the code
All.rda same as a package that you can attach
Latest_all.html Documentation for library
templs.html Documentation for TempLS V4.1
The main functions used to implement the above mathematics are in Latest_all
spmul() for sparse matrix/vector multiplication and
pcg() for preconditioned conjugate gradient iteration
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