# Introduction to particle filters and the significance of the Equivalent-Weight Particle Filter in geophysical science

**An online course for PhD students and early stage researchers developed by Bethan Perkins (Assimila), Peter Jan van Leeuwen (NCEO), Melanie Ades (NCEO), Azin Howells (University of Reading) and Guy Griffiths (University of Reading) for the European Space Agency Data Assimilation Projects.**

**Introduction**

A particle filter is a data assimilation technique. This means that it’s a method of combining observations of a system with a model of how that system works to give scientists and engineers the best possible estimate for the true state of the system.

Particle Filters are a relatively new method of data assimilation. There are numerous variants of particle filters – all with their merits and deficiencies – but the Equivalent-Weights Particle Filter is the only one that works well for systems with extremely high dimensions, so is most useful for geophysical scientists.

On these pages you will find many resources to help you learn about Particle Filters in general and the Equivalent-Weights Particle Filter specifically. There is an interactive demo of Particle Filters with accompanying exercises. And there is a set of light-hearted reading material which gives a non-technical introduction to Particle Filters and works through to provide more mathematical descriptions of the principles behind the Equivalent Weights Particle Filter.

### Why do we need Particle Filters?

There are a variety of different techniques for performing data assimilation, including Kalman Filters and Variational techniques; each of these techniques have their own merits. The strength of the particle filter is that it works well on assimilating data from non-linear systems.

Other data assimilation techniques are applied to the Earth’s non-linear environmental system, but generally assumptions and simplifications are made to translate non-linear equations into more simple linear ones and deal with those. Previously, the relatively low resolution of climate models has meant that these assumptions and simplifications have been perfectly reasonable and errors due to simplifications have been far outweighed by the increase in speed.

But as model resolutions increase it becomes increasingly important to account for non-linearity in the data assimilation process. Particle filters are designed for non-linear situations. The model equations are not simplified at all and no assumptions are made about the distribution of model results. There are other compromises which are made for the sake of this but – generally speaking – once a non-linear data assimilation method is essential then these compromises are worth the cost.

### What is a Particle Filter?

Particle Filters are a particular way of treating model ensembles. An ensemble is a set of model simulations of the same system e.g. many simulations of the weather all covering the same time and the geographical place; each model in the ensemble, however, is slightly different to the others, usually either in terms of boundary conditions or model calculations and parameterisations. These differences show us the sensitivity of the model to these small changes and give us an idea of the most likely outcomes.

Particle filters are essentially ensemble model simulations (where “particle” means “ensemble member”). The added value of a particle filter is in how the model results are collected and assimilated with observational data before the model is carried forward to simulate the system until the time of the next observation.

### How do Particle filters work?

The light-hearted story downloadable from the link on the RHS of this page will answer this question.

There are many different types of particle filters, each are useful in different circumstances. The type of Particle filter that we’re pioneering at NCEO is the Equivalent Weights Particle Filter. This advanced filter is a third evolutionary step from the basic particle filter. We talk through these evolutionary steps in the explanatory story.