Tuesday, September 19, 2017

GISS August up 0.01°C from July.

GISS showed a very small rise, going from 0.84°C in July to 0.85°C in August (GISS report here). TempLS mesh showed a very slight fall, which I posted at 0.013°C, although with further data is is now almost no change at all. I see that GISS is now using ERSST V5, as TempLS does.

The overall pattern was similar to that in TempLS. Warm almost everywhere, with a big band across mid-latitude Eurasia and N Africa. Cool in Eastern US and high Arctic, which may be responsible for the slowdown in ice melting..

As usual here, I will compare the GISS and previous TempLS plots below the jump.

Monday, September 18, 2017

Grids, Platonic solids, and surface temperature (and GCMs)

This follows a series, predecessor here, in which I am looking at ways of dealing with surface variation of Earth's temperature, particularly global averaging. I have written a few posts on the cubed sphere, eg here. I have also shown some examples of using an icosahedron, as here. Clive Best has been looking at similar matters, including use of icosahedron. For the moment, I'd like to write rather generally about grids and ways of mapping the sphere.

Why grids?

For surface temperature, grids are commonly used to form local averages from data, which can then be combined with area weighting to average globally. I have described the general considerations here. All that is really required is any subdivision of reasonable (compact) shapes. They should be small enough that the effect of variation within is minimal, but large enough that there is enough data for a reasonable estimate. So they should be of reasonably equal area.

The other requirement, important later for some, is that any point on the sphere can be associated with the cell that contains it. For a regular grid like lat/lon, this is easy, and just involves conversion to integers. So if each data point is located, and each cell area is known, that is all that is needed. As a practical matter, once the cell locations are known for an inventory of stations, the task of integrating the subset for a given month is just a look-up, whatever the geometry.

I described way back a fairly simple subdivision scheme that works for this. Equal latitude bands are selected. Then each band is divided as nearly as possible into square elements (on the sphere). The formula for this division can then be used to locate arbitrary points within the cells. I think this is all that is required for surface averaging.

However, for anything involving partial differentiation, such as finite element or GCM modelling, more is required. Fluxes between cells need to be measured, so hey have to line up. Nodes have to be related to each cell they abut. This suggests regular grids. In my case, I sometimes want to use a kind of diffusive process to estimate what is happening in empty cells. Again, regular is better.

Platonic solids

Something that looks a bit like a sphere and is easy to fit with a regular grid is a Platonic solid. There are five of them - I'll show the Wiki diagram:



Regular means that each side has the same length, and each face is a congruent regular polygon. The reason why there are only five is seen if you analyse what has to happen at vertices (Wiki):

Friday, September 8, 2017

August global surface temperature down 0.013°C

TempLS mesh anomaly (1961-90 base) was down from 0.69°C in July to 0.677°C in August. This very small drop compares with the small rise of 0.038°C in the NCEP/NCAR index, and a bigger rise (0.12) in the UAH LT satellite index. The August value is less than August 2015 or 2016, but higher than 2014.

There was a moderate fall in Antarctica, which as usual affects TempLS mesh and GISS more than others. I'd expect NOAA and HADCRUT to show increases for August. Regionally, the Old World was mostly warm; US was cold cental and East, but N Canada was warm. S America mostly warm (still awaiting a few countries). :


Thursday, September 7, 2017

August NCEP/NCAR global anomaly up 0.038°C

In the Moyhu NCEP/NCAR index, the monthly reanalysis average rose from 0.299°C in July to 0.337°C in August, 2017. The results were late this month; for a few days NCEP/NCAR was not posting new results. It was a very up and down month; a dip at at the start, then quite a long warm period, and then a steep dip at the end. Now that a few days in September are also available, there is some recovery from that late dip. August 2017 was a bit cooler than Aug 2016, but warmer than 2015.

It was cool in Eastern US, but warm in the west and further north. Cool in Atlantic Europe, but warm further east. Mostly cool in Antarctica.







Wednesday, August 30, 2017

Gulf SST - warm before Harvey, cool after.

I maintain a page showing high resolution (1/4degree) AVHRR SST data from NOAA - in detail: "NOAA High Resolution SST data provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their Web site at http://www.esrl.noaa.gov/psd/". It renders it in WebGL and goes back with daily maps for a decade or so, then les frequently. It shows anomalies relative to 1971-2000. I have been tracking the effect of Hurricane Harvey. It was said to have grown rapidly because of warm Gulf waters; they were warm, but not exceptionally, as this extract from 15 August shows:



It remained much the same to 24th August, when Harvey grew rapidly, and gained Hurricane status late in the day (). But by 25th, there is some sign of cooling. 26th (not shown) was about the same. But by 27th, There was marked cooling, and by 28th more so. The cooling seems to show up rather belatedly alomg the path of the hurricane.



Here's is the latest day at higher resolution:



A few years ago, I developed a set of movies based on recent hurricanes of the time, showing their locations and SST at the time. Some showed a big effect, some not so much. Harvey was interesting in that it covered a fairly confined area of ocean, and moved slowly.



Thursday, August 24, 2017

Surface temperature sparsity error modes

This post follows last week's on temperature integration methods. I described a general method of regression fitting of classes of integrable functions, of which the most used to date is spherical harmonics (SH). I noted that the regression involved inverting a matrix HH consisting of all the scalar product integrals of the functions in the class. With perfect integration this matrix would be a unit matrix, but as the SH functions become more oscillatory, the integration method loses resolution, and the representation degrades with the condition number of the matrix HH. The condition number is the ratio of largest eigenvalue to smallest, so what is happening is that some eigenvectors become small, and the matrix is near singular. That means that the corresponding eigenvector might have a large multiplier in the representation.

I also use fitted SH for plotting each month's temperature. I described some of the practicalities here (using different functions). Increasing the number of functions improves resolution, but when HH becomes too ill-conditioned, artefacts intrude, which are multiples of these near null eigenvectors.

In the previous post, I discussed how the condition of HH depends on the scalar product integral. Since the SH are ideally orthogonal, better integration improves HH. I have been taking advantage of that in recent TempLS to increase the order of SH to 16, which implies 289 functions, using mesh integration. That might be overdoing it - I'm checking.

In this post, I will display those troublesome eigen modes. They are of interest because they are associated with regions of sparse coverage, and give a quantification of how much they matter. Another thing quantified is how much the integration method affects the condition number for a given order of SH. I'll develop that further in another post.

I took N=12 (169 functions), and looked at TempLS stations (GHCN+ERSST) which reported in May 2017. Considerations on choice of N are that if too low, the condition number is good, and the minimum modes don't show features emphasising sparsity. If the number is too high, each region like Antarctica can have several split modes, which confuses the issue.

The integration methods I chose were mostly described here
  • OLS - just the ordinary scalar product of the values
  • grid - integration by summing on a 5x5° latitude/longitude grid. This was the earliest TempLS method, and is used by HADCRUT.
  • infill - empty cells are infilled with an average of nearby values. Now the grid is a cubed sphere with 1536 cells
  • mesh - my generally preferred method using an irregular triangular grid (complex hull of stations) with linear interpolation.
OLS sounds bad, but works quite well at moderate resolution, and was used in TempLS until very recently.

I'll show the plots of the modes as an active lat/lon plot below, and then the OLS versions in WebGL, which gives a much better idea of the shapes. But first I'll show a table of the tapering eigenvalues, numbering from smallest up. They are scaled so that the maximum is 1, so reciprocal of the lowest is the condition number.
OLS grid infilled mesh
Eigen10.02110.01470.06950.135
Eigen20.03690.02750.1380.229
Eigen30.04230.04690.2120.283
Eigen40.05720.04990.2440.329
Eigen50.0840.0890.2480.461
Eigen60.1040.1070.3730.535
Eigen70.1080.1460.4060.571
Eigen80.1240.1640.4290.619
And here is a graph of the whole sequence, now largest first:


The hierarchy of condition numbers is interesting. I had expected that it would go in the order of the columns, and so it does until near the end. Then mesh drops below infilled grid, and OLS below grid, for the smallest eigenvalues. I think what determines this is the weighting of the nodes in the sparse areas. For grid, this is not high, because each just gets the area of its cell. For both infilled and mesh, the weight rises with the area, and apparently with infilled, more so.

Thursday, August 17, 2017

Temperature averaging and integration - the basics

I write a lot about spatial integration, which is at the heart of global temperature averaging. I'll write more here about the principles involved. But I'm also writing to place in context methods I use in TempLS, which I think are an advance on what is currently usual. I last wrote a comparison of methods in 2015 here, which I plan to update in a sequel. Some issues here arose in a discussion at Climate Audit.

It's a long post, so I'll include a table of contents. I want to start from first principles and make some math connections. I'll use paler colors for the bits that are more mathy or that are just to make the logic connect, but which could be skipped.