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.
The Met Office in the UK has developed a completely new probabilistic post-processing system called IMPROVER to operate on outputs from its operational Numerical Weather Prediction (NWP) forecasts and precipitation nowcasts. The aim is to improve weather forecast information to the public and other stakeholders whilst better exploiting the current and future generations of underpinning kilometer-scale NWP ensembles. We wish to provide seamless forecasts from nowcasting to medium range, provide consistency between gridded and site-specific forecasts and be able to verify every stage of the processing. The software is written in a modern modular framework that is easy to maintain, develop and share. IMPROVER allows forecast information to be provided with greater spatial and temporal detail and a faster update frequency than previous post-processing. Independent probabilistic processing chains are constructed for each meteorological variable consisting of a series of processing stages that operate on pre-defined grids and blend outputs from several NWP inputs to give a frequently updated, probabilistic forecast solution. Probabilistic information is produced as standard, with the option of extracting a most likely or yes/no outcome if required. Verification can be performed at all stages, although it is only currently switched on for the most significant stages when run in real time. IMPROVER has been producing real-time output since March 2021 and became operational in Spring 2022.
No abstract
<p>The UK Met Office is developing an open-source probability-based post-processing system called IMPROVER to exploit convection permitting, hourly cycling ensemble forecasts. The system is tasked with blending these forecasts with both deterministic nowcast data, and coarser resolution global ensemble model data, to produce seamless probabilistic forecasts from the very short to medium range.</p><p>A majority of the post-processing within IMPROVER is performed on gridded forecasts, with site-specific forecasts extracted as a final step, helping to ensure consistency. IMPROVER delivers a wide range of probabilistic products to both operational meteorologists and as input to automated forecast production. and this presentation will detail some of the work that has been undertaken in the past year to prepare, with a focus on the use of statistical post-processing.</p><p>Statistical post-processing plays two complimentary roles within IMPROVER; ensuring forecasts better reflect reality, and in so doing, bringing different models into better alignment, which improves the seamlessness of model transitions. For a selection of diagnostics, the gridded forecasts from different source models are calibrated independently using ensemble model output statistics (EMOS). Results of experiments looking at the calibration of gridded forecasts will be discussed briefly.</p><p>More recently calibration of site forecasts has been introduced as a final step for temperature and wind speed forecasts. Results of experiments using EMOS to perform calibration in a variety of different ways will be presented, including justifications and trade-offs made in choosing a final approach.</p><ul><li>This will include some discussion of the remaking of weather symbol products as period, rather than instantaneous, forecasts and the implications for their verification.</li> </ul>
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