## 1. Introduction

In recent years, numerical weather prediction (NWP) models have gone to smaller and smaller grid sizes. Within the last 5 years, the grid size of many limited-area models has become fine enough to allow an explicit representation of convective processes (Baldauf *et al.*, 2011, and references within). This aims at an improved simulation of convection-related weather such as strong wind gusts and heavy precipitation. These improvements are highly relevant for the quality of severe weather warnings.

However, apart from the obvious benefits, the smaller grid size creates a challenge in terms of predictability. The ability of a system to resolve small scales results in forecast errors that grow rapidly (Lorenz, 1969). Convective processes are non-linear and strongly affected by uncertainties. Therefore, precipitation-related forecasts of convection-permitting models should be produced and interpreted within a probabilistic framework.

On the numerical weather prediction side, ensemble forecasting is today a standard strategy adopted to deal with forecast uncertainties (Lewis, 2005). For limited-area models, variations in boundary condition, initial condition, physics parameterization and/or dynamics formulation aim to reflect the uncertainties related to the forecasting process. Ensemble forecasting derives a probabilistic view from a sample of deterministic forecasts, thereby providing information about the degree of predictability.

Many weather prediction centres are therefore developing ensemble prediction systems (EPS) at the convective scale (Clark *et al.*, 2009; Vié *et al.*, 2011). At the German weather service (DWD), an EPS based on the convection-permitting model COSMO-DE has been operational since May 2012 (Gebhardt *et al.*, 2011). This is one of the first operational convection-permitting EPSs worldwide. Because it is a new development and since the ensemble forecasts certainly do not have perfect quality, it is now necessary to learn how to use the forecasts optimally for weather warnings.

On the verification side, spatial verification methods have been developed, to assess precipitation forecasts from high-resolution models, accounting for limited predictability (Ebert, 2008). They put a grid point forecast into its spatial context. The idea is to relax the necessity of exact matching between forecast and observation. The uncertainty inherent to the forecast is integrated *a posteriori* following approaches inspired by ‘fuzzy logic’ (Zadeh, 1965). An event is seen as occurring somewhere within an area rather than at an exact location or with a certain probability of occurrence rather than in binary terms (yes or no). Neighbourhood approaches, which compare statistical properties of forecast and observation fields within a spatial neighbourhood (Gilleland *et al.*, 2009), are explored in this paper.

The same spatial technique can be used in the context of verifying a forecast and in the context of providing forecast guidance to the forecaster. For example, comparing the fractional occurrence of events within a spatial window, Roberts and Lean (2008) define a new metric to assess the performance of a deterministic forecast: the Fractions Skill Score. Their approach derives a ‘scale of usefulness’ in order to avoid a naive point-based interpretation of a deterministic precipitation forecast (Roberts, 2008; Roberts and Lean, 2008). This is a syncretic example of the duality of the spatial techniques applications: forecast verification and forecast guidance.

Moreover, the generically similar technique (statistics within a spatial window) can be applied in a different manner leading to complementary information. Considering the near neighbourhood forecasts as possible realizations of a local grid point forecast, Theis *et al.* (2005) derive a probabilistic forecast guidance from a single forecast. The spatial technique aims here at representing the spatial uncertainty at the grid-scale while the Roberts and Lean (2008) approach estimates the smallest scale at which the spatial variability can be considered as useful. Scale of usefulness and grid point probabilistic forecasts derived from spatial neighbourhoods are complementary guidances that contribute to the forecast interpretation. However, Theis *et al.* (2005) go one step further in terms of forecast guidance by generating refined forecast products presented to the forecaster. The present paper follows along this line and explores spatial techniques in terms of verification and refined forecast products, now with the focus on a convective-scale ensemble system.

Schwartz *et al.* (2010) have already taken the step to apply the Theis *et al.* (2005) approach to ensemble forecasts. The resulting probabilities correspond to a spatially smoothed version of the raw probabilities which are directly derived from the ensemble members at each grid point. This smoothing procedure can be seen as a computationally inexpensive method to enlarge the ensemble sample size by including the spatial neighbourhood forecasts of all members in the probability computation (Ben Bouallègue *et al.*, 2013). It has been shown (Schwartz *et al.*, 2010; Ben Bouallègue *et al.*, 2013) that the smoothing has a positive impact on the probabilistic forecast skill, in particular in terms of reliability but also to some extent in terms of resolution. However, the relationship between benefits and size of the spatial neighbourhood as well as the limit of the method still have to be explored.

Smoothing inevitably reduces the sharpness in the forecasts, i.e. probabilities close to 0 or 100% will occur in fewer cases. This may be the correct thing to do, because it corresponds to the inherently low predictability. However, in many situations the forecast needs to reach some level of certainty before it may be used for a weather warning in practice. Therefore, this paper investigates yet another spatial technique which we call ‘upscaling’.

‘Upscaling’ aims to alleviate the problem of low predictability by changing the spatial scale of the forecast output. In weather forecasting, a spatial scale and a time window are often associated with the prediction, e.g. the probability that it will rain anywhere within a specific region and anytime within a specific time interval. This reference area and time must be known in order to interpret the forecast correctly (cf Gigerenzer *et al.*, 2005). For example, Epstein (1966) described the relationship between point and area probabilities for idealized cases and warned against confusion between these two kinds of forecast. The reference area of an ensemble probabilistic forecast can be modified through an upscaling procedure as described for verification purposes by Marsigli *et al.* (2008). Choosing the maximum value of each member within pre-defined spatial windows, new probabilistic products can be derived and interpreted as the probabilities that an event occurs anywhere within the selected windows. The forecast is still produced by the fine-scale model and still retains its benefits such as the occurrence of heavy precipitation which may only be captured by a convection-permitting model. However, the resulting forecast is formulated for a larger area and time window than the original grid size and the original time interval of the model output. For example, one could look at the probability of heavy precipitation anywhere within the region of Berlin and anytime within the afternoon.

This paper explores how two spatial techniques can better characterize the performance of an ensemble forecasting system and how they can be used to provide guidelines for the generation of more skillful probabilistic products. The first technique is the spatial neighbourhood and aims at improving the probabilistic forecast at grid scale. The second technique is the spatial upscaling procedure and aims at omitting the fine-scale information when issuing a probabilistic forecast product. The techniques are applied independently and are meant to provide two separate types of products. These products may then combine well to form a consolidated forecast guidance.

Smoothing and upscaling are applied here to precipitation forecasts derived from the COSMO-DE-EPS, an ensemble prediction system at the convective scale. Verification is performed for a range of spatial parameters, i.e. neighbourhood environment and window sizes, over a 3 month period covering summer 2011.

The rest of this manuscript is organized as follows: Section 2 describes the convection permitting ensemble COSMO-DE-EPS and the application of the two spatial techniques. Section 3 presents the dataset and verification methodology. Section 4 shows and discusses the results. Section 5 concludes and gives an outlook.