The quality of precipitation forecasts is difficult to evaluate objectively because images with disjointed features surrounded by zero intensities cannot easily be compared pixel by pixel: any displacement between observed and predicted fields is punished twice, generally leading to better marks for coarser models. To answer the question of whether a highly resolved model truly delivers an improved representation of precipitation processes, alternative tools are thus needed. Wavelet transformations can be used to summarize high-dimensional data in a few numbers which characterize the field's texture. A comparison of the transformed fields judges models solely based on their ability to predict spatial structures. The fidelity of the forecast's overall pattern is thus investigated separately from potential errors in feature location. This study introduces several new waveletbased structure scores for the verification of deterministic as well as ensemble predictions. Their properties are rigorously tested in an idealized setting: a recently developed stochastic model for precipitation extremes generates realistic pairs of synthetic observations and forecasts with prespecified spatial correlations. The wavelet scores are found to react sensitively to differences in structural properties, meaning that the objectively best forecast can be determined even in cases where this task is difficult to accomplish by naked eye. Random rain fields prove to be a useful test bed for any verification tool that aims for an assessment of structure.Published by Copernicus Publications on behalf of the European Geosciences Union.

With the recent advent of a sound mathematical theory for extreme events in dynamical systems, new ways of analyzing a system's inherent properties have become available: Studying only the probabilities of extremely close Poincaré recurrences, we can infer the underlying attractor's local dimensionality-a quantity which is closely linked to the predictability of individual configurations, as well as the information gained from observing them. This study examines possible ways of estimating local and global attractor dimensions, identifies potential pitfalls, and discusses conceivable applications. The Portable University Model of the Atmosphere serves as a test subject of intermediate complexity between simple mathematical toys and truly realistic atmospheric data-sets. It is demonstrated that the introduction of a simple, analytical estimator can streamline the procedure and allows for additional tests of the agreement between theoretical expectation and observed data. We, furthermore, show how the newly gained knowledge about local dimensions can complement classical techniques like principal component analysis and may assist in separating meaningful patterns from mathematical artifacts.

One important attribute of meteorological forecasts is their representation of spatial structures. While several existing verification methods explicitly measure a structure error, they mostly produce a single value with no simple interpretation. Extending a recently developed wavelet‐based verification method, this study separately evaluates the predicted spatial scale, orientation and degree of anisotropy. The scale component has been rigorously tested in previous work and is known to assess the quality of a forecast similar to other, established methods. However, directional aspects of spatial structure are less frequently considered in the verification literature. Since important weather phenomena related to fronts, coastlines and orography have distinctly anisotropic signatures, their representation in meteorological models is clearly of interest. The ability of the new wavelet approach to accurately evaluate directional properties is demonstrated using idealized and realistic test cases from the MesoVICT project. A comparison of precipitation forecasts from several forecasting systems reveals that errors in scale and direction can occur independently and should be treated as separate aspects of forecast quality. In a final step, we use the inverse wavelet transform to define a simple post‐processing algorithm that corrects the structural errors. The procedure improves visual similarity with the observations, as well as the objective scores.

High‐resolution simulations (grid spacing 2.5 km) are performed with ICON‐LEM to characterize convective organization in the Tropics during August 2016 over a large domain ranging from northeastern South America, along the tropical Atlantic to Africa (8,000×3,000 km). The degree of organization is measured by a refined version of the wavelet‐based organization index (WOI), which is able to characterize the scale, the intensity and anisotropy of convection based on rain rates alone. Exploiting the localization of wavelets both in space and time, we define a localized version of the convective organization index (LWOI). We compare convection observed in satellite‐derived rain rates with the corresponding processes simulated by ICON‐LEM. Model and observations indicate three regions with different kinds of convective organization. Continental convection over West Africa has a predominantly meridional orientation and is more organized than over South America, because it acts on larger scales and is more intense. Convection over the tropical Atlantic is zonally oriented along the ITCZ and less intense. ICON and observations agree on the number and intensity of the African easterly waves during the simulation period. The waves are associated with strong vorticity anomalies and are clearly visible in a spatiotemporal wavelet analysis. The central speed and the wavelength of the waves is simulated well. Both the scale and intensity components of LWOI in ICON are significantly correlated with environmental variables. The scale of precipitation is related to wind shear, CAPE and its tendency, while the intensity strongly correlates with column‐integrated humidity, upper‐level divergence and maximum vertical wind speed. This demonstrates that the LWOI components capture important characteristics of convective precipitation.

Abstract. The quality of precipitation forecasts is difficult to evaluate objectively because images with disjoint features surrounded by zero intensities cannot easily be compared pixel by pixel: Any displacement between observed and predicted field is punished twice, generally leading to better marks for coarser models. To answer the question whether a highly resolved model truly delivers an improved representation of precipitation processes, alternative tools are thus needed. Wavelet transformations can be used to summarize high-dimensional data in a few numbers which characterize the field's texture. A comparison of the transformed fields judges models solely based on their ability to predict spatial correlations. The fidelity of the forecast's overall structure is thus investigated separately from potential errors in feature location. This study introduces several new wavelet based structure-scores for the verification of deterministic as well as ensemble predictions. Their properties are rigorously tested in an idealized setting: A recently developed stochastic model for precipitation extremes generates realistic pairs of synthetic observations and forecasts with prespecified spatial correlations. The wavelet-scores are found to react sensitively to differences in structural properties, meaning that the objectively best forecast can be determined even in cases where this task is difficult to accomplish by naked eye. Random rain fields prove to be a useful test-bed for any verification tool that aims for an assessment of structure.

Abstract. Recently developed verification tools based on local wavelet spectra can isolate errors in the spatial structure of quantitative precipitation forecasts, thereby answering the question of whether the predicted rainfall variability is distributed correctly across a range of spatial scales. This study applies the wavelet-based structure scores to real numerical weather predictions and radar-derived observations for the first time. After tackling important practical concerns such as uncertain boundary conditions and missing data, the behaviour of the scores under realistic conditions is tested in selected case studies and analysed systematically across a large data set. Among the two tested wavelet scores, the approach based on the so-called map of central scales emerges as a particularly convenient and useful tool: summarizing the local spectrum at each pixel by its centre of mass results in a compact and informative visualization of the entire wavelet analysis. The histogram of these scales leads to a structure score which is straightforward to interpret and insensitive to free parameters like wavelet choice and boundary conditions. Its judgement is largely the same as that of the alternative approach (based on the spatial mean wavelet spectrum) and broadly consistent with other, established structural scores.

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