We consider a multistage cancer model in which cells are arranged in a d-dimensional integer lattice. Starting with all wild-type cells, we prove results about the distribution of the first time when two neutral mutations have accumulated in some cell in dimensions d ≥ 2, extending work done by Komarova [12] for d = 1.
The Met Office operates several configurations of the Unified Model (UM) for forecasting the weather. Over the UK, there are three deterministic models: the global model, as its name suggests, covers the whole of our planet at a resolution of about 25km. It runs four times a day with two of those runs forecasting the weather out to five days and the other two to two-and-ahalf days. At a higher resolution, there is the North Atlantic and European (NAE) model, forecasting the weather over that region at a resolution of about 12km out to two days ahead. This also runs four times a day. Finally, centred over the UK is a high-resolution model forecasting the weather at a 4km scale for the next 36 hours. The 4km model also runs four times a day but, to spread the load on the Met Office's supercomputer, the run times are offset by three hours from the NAE and global model runs (Bell et al., 2002; Met Office, 2011). There is a second version of the 4km model which is nested in the longer range global model allowing high-resolution forecasts out to five days ahead based on that model's synoptic-scale forecast evolution. The next iteration of the Met Office model configuration is a 1.5km UK model. This is currently undergoing testing and is expected to be operational in 2012.If you managed to follow that paragraph from beginning to end without your eyes glazing over, you will see that there is a bit of a problem. How do you produce an automated forecast for the next five days? There are 24 different deterministic solutions for this afternoon's weather on different grids with different levels of skill and resolution. There are also sub-grid scale orographic features for which the models cannot provide a meaningful forecast. Indeed, even features that are larger than the grid-scale may not be resolved as the models require a slightly-smoothed orography field for numerical stability.There is another problem in that the global and regional models run on subtly different grids. They are both based on a standard latitude-longitude grid, but the regional models have a rotated pole. There is a good reason for this. The global model grid points are only 25km apart at around 50° from the equator (southern UK). As you move towards the poles, the lines of longitude move closer together: the longitudinal grid spacing over northern Scotland is around 20km. The regional models have the grid rotated so that the equator, where the grid is most square, passes through the centre of the region. Interpreting the forecast is much more difficult when the data are presented in several diverse ways.Solving these problems is where postprocessing comes in. That is, taking the resulting data from these different weather forecasting models and doing more work on it.This paper describes the post-processing methods currently in use in the Met Office to produce automated seamless forecast products for the next five days and to provide input to the manual forecaster decision-making process.
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