Singular vectors

The initial conditions for the EPS are designed to represent the uncertainties inherent in the operational analysis. They are created by adding to the operational ECMWF analysis perturbations which produce the fastest energy growth during the first two days of the forecast period, defined using the singular vector technique.

• singular vectors
• linear perturbation growth
• time interval
• initial and final norms
• target areas
• summary
• Initial perturbations - combining the singular vectors

Singular vectors

The aim is to find those perturbations to a given initial state which grow most rapidly. The problem can be formalised mathematically and solutions found if certain assumptions and choices are made. The main assumption is that the perturbations grow linearly in time. The choices include deciding how to measure the size of a perturbation (choice of norms), what regions to focus on (target areas) and how long to allow the perturbations to grow for. The singular vectors are the perturbations with the greatest linear growth over the specified time interval for given norms and target areas. Different choices may lead to different sets of singular vectors.

Linear perturbation growth

The calculation of the singular vectors only becomes feasible if the evolution of the perturbations is linear (the rate of growth is constant over time). In the atmosphere this assumption is generally valid for small perturbations for one or two days ahead. To determine the growth of a perturbation over time a forecast model is needed, but this model must be specifically for the linear evolution of small perturbations. Technically this model is the tangent-linear version of the full (non-linear) ECMWF operational forecast model. A second model of equivalent complexity (the adjoint of the tangent linear model) is also needed in the singular vector calculation.

Time interval

The method finds perturbations which grow over a specified time interval. The assumption that the perturbations grow linearly limits the time interval to around two days. The EPS singular vectors are calculated over a 48-hour period. This is a compromise between the constraints of the linearity approximation and the requirement to provide an operational EPS for medium-range forecasting (three to ten days ahead).

Perturbation growth - initial and final norms

Singular vectors are the perturbations with the largest (linear) growth over a given period, from initial (analysis) time to final time. Before we can find the fastest growing perturbations we need to define how we will measure the size of the perturbations at both initial and final time: appropriate choices (norms) must be made for each. Ideally the initial norm would be related to the spatial distribution of expected errors in the analysis while the final-time norm would reflect the forecast errors of interest. In practice, the use of total energy for both initial and final time norms was found to to be a suitable pragmatic choice for the general requirements of the ECMWF EPS.

Projection operator - spatial target area

In the singular vector calcualtions it is possible to select particular regions of interest at either initial or final time. So, singular vectors can be calculated which originate in one specified region and have the greatest impact in another specified target region. Perturbations originating anywhere in the northern hemisphere may have an impact on forecasts over Europe at some time during the ten day range of the EPS. Also the EPS is used for forecasts elsewhere around the globe, including in the southern hemisphere. Therefore the EPS singular vectors are calculated with initial and target regions covering the whole of the extra-tropical hemisphere (separate calculations are made for the orthern and southern hemisphere). The mechanisms for perturbation growth are different in the tropical and extra-tropical regions (physical processes play a more significant role in the tropics) so singular vectors are calculated separately for the tropics.

• more details of tropical singular vector perturbations

Summary of EPS singular vectors

The singular vectors calculated for the EPS are the perturbations to the best estimate analysis which show the greatest linear growth in total energy over the extra-tropical northern and southern hemisphere over a 48 hour period following the analysis time.

Initial perturbations - combining the singular vectors

25 singular vectors are used to generate the 50 perturbed initial states for the EPS. The singular vectors tend to be rather localised in space. To make perturbations which have a more uniform spatial distribution and to ensure that the perturbed states are all equally likely, the initial perturbations are formed from linear combinations of the 25 singular vectors.