Until recent times, weather forecasts were deterministic in nature. For example, a forecast might state ``The temperature tomorrow will be $20^\circ$C.'' More recently, however, increasing interest has been paid to the uncertainty associated with such predictions. By quantifying the uncertainty of a forecast, for example with a probability distribution, users can make risk-based decisions. The uncertainty in weather forecasts is typically based upon `ensemble forecasts'. Rather than issuing a single forecast from a numerical weather prediction (NWP) model, ensemble forecasts comprise multiple model runs that differ in either the model physics or initial conditions. Ideally, ensemble forecasts would provide a representative sample of the possible outcomes of the verifying observations. However, due to model biases and inadequate specification of initial conditions, ensemble forecasts are often biased and underdispersed. As a result, estimates of the most likely values of the verifying observations, and the associated forecast uncertainty, are often inaccurate. It is therefore necessary to correct, or post-process ensemble forecasts, using statistical models known as `ensemble post-processing methods'. To this end, this thesis is concerned with the application of statistical methodology in the field of probabilistic weather forecasting, and in particular ensemble post-processing. Using various datasets, we extend existing work and propose the novel use of statistical methodology to tackle several aspects of ensemble post-processing. Our novel contributions to the field are the following. In chapter~3 we present a comparison study for several post-processing methods, with a focus on probabilistic forecasts for extreme events. We find that the benefits of ensemble post-processing are larger for forecasts of extreme events, compared with forecasts of common events. We show that allowing flexible corrections to the biases in ensemble location is important for the forecasting of extreme events. In chapter~4 we tackle the complicated problem of post-processing ensemble forecasts without making distributional assumptions, to produce recalibrated ensemble forecasts without the intermediate step of specifying a probability forecast distribution. We propose a latent variable model, and make a novel application of measurement error models. We show in three case studies that our distribution-free method is competitive with a popular alternative that makes distributional assumptions. We suggest that our distribution-free method could serve as a useful baseline on which forecasters should seek to improve. In chapter~5 we address the subject of parameter uncertainty in ensemble post-processing. As in all parametric statistical models, the parameter estimates are subject to uncertainty. We approximate the distribution of model parameters by bootstrap resampling, and demonstrate improvements in forecast skill by incorporating this additional source of uncertainty in to out-of-sample probability forecasts. In chapter~6 we use model diagnostic tools to determine how specific post-processing models may be improved. We subsequently introduce bias correction schemes that move beyond the standard linear schemes employed in the literature and in practice, particularly in the case of correcting ensemble underdispersion. Finally, we illustrate the complicated problem of assessing the skill of ensemble forecasts whose members are dependent, or correlated. We show that dependent ensemble members can result in surprising conclusions when employing standard measures of forecast skill.

SMHI:s hydrologiska prognos- och varningstjänst använder sig av meteorologiska ensembleprognoser som indata i hydrologiska modeller. De hydrologiskaensembleprognoserna tar därmed hänsyn till framtida osäkerhet i temperatur och nederbördoch används som underlag vid utfärdandet av risker och varningar för höga flöden. För närvarande beaktas dock inte osäkerheten i modellens starttillstånd, vilket består av de tillståndsvariabler i modellen som beskriver bland annat markvattenhalt och snötäcke. I dennastudie undersöktes hur starttillståndet i den hydrologiska modellen HYPE inverkar på prognoser i syfte att kvantifiera osäkerheten och på sikt möjliggöra säkrare prognoser.Studien hade tre mål: 1) Ta fram ett förslag på hur starttillståndet kan varieras för att ge en god uppskattning av prognososäkerheten relaterat till det hydrologiska starttillståndet. 2) Undersöka sambandet mellan starttillståndens spridning och det hydrologiska prognosfelet. 3) Analysera hur årstider, avrinningsområdens area, sjöprocent, skogsprocent och höjd över havet inverkar på prognososäkerheten. En central hypotes var att mindre skillnad mellan starttillståndets vattenföring och den observerade vattenföringen vid prognosstart resulterar i mer träffsäkra prognoser. Studien begränsades av att starttillstånden endast genererades med hjälp av störningar i drivdata.Indata till HYPE-modellen var femton temperatur- och nederbördsserier som manipulerats i syfte att skapa en ensemble av olika starttillstånd. Denna ensemble användes sedan för att göra vattenföringsprognoser med observerad temperatur och nederbörd som drivdata. Studien omfattade 76 avrinningsområden från hela Sverige med data för perioden 1999-2008. Prognoser utfördes varje dygn och ensemblespridningen utvärderades 2, 4 och 10 dygn in i prognosen. Samma utvärderingar utfördes även på autoregressiva prognoser, vilket innebär att modellerad rättas utefter observerad vattenföring.Resultaten indikerade ett samband mellan ensemblespridning och prognosfel, vilket innebär att spridning kan användas som ett mått på starttillståndets osäkerhet. Prognosfelet korrelerade positivt med skogsprocent och negativt med avrinningsområdenas area, sjöprocent och höjd över havet. Samma samband uppvisades mellan dessa områdesvariableroch spridning. Spridningen var störst på vintern och våren då normalisering skett med medelvattenföring över tio år, och under vår och sommar då normalisering skett med medelvattenföring per månad. Hypotesen att mindre skillnad mellan starttillståndets vattenföring och den observerade vattenföringen vid prognosstart resulterar i mer träffsäkraprognoser bekräftades av resultaten. Implementering av en ensemble av olika starttillstånd i operationella prognoser vid SMHIs hydrologiska prognos- och varningstjänst föreslås i syfte att kvantifiera osäkerheten och därigenom utöka bedömningsunderlaget vid utfärdande av risker och varningar. / The Hydrological Forecast and Warning Service of The Swedish Meteorological and Hydrological Institute (SMHI) use meteorological ensemble forecasts as input in hydrological models. The hydrological ensemble forecasts take the uncertainty of future temperature and precipitation into account and serve as the basis of issued risks and warnings of high flows. Currently not considered is the uncertainty of the initial state, which consists of state variables in the model describing for instance soil water content and snow pack. This study assessed the impact of the initial state on forecasts in the hydrological model HYPE aiming to quantify the uncertainty and eventually enable more accurate forecasts.There were three aims of this study : 1) Evaluate a suggestion about how the initial state can be varied to give a good estimation of forecast uncertainty related to the hydrological initial state. 2) Examine the relationship between the spread of initial states and the hydrological forecast error. 3) Analyze the impact of seasons, catchment area, lake percentage, forest percentage and elevation on forecast uncertainty. A central hypothesis was that a smaller difference between the discharge of the initial state and the observed discharge results in more accurate forecasts. A restriction of the study was that the initial states only could be generated by disturbances of forcing data in before the forecast.Input data to the HYPE model were fifteen temperature and precipitation series, manipulated to generate an ensemble of different initial states. This ensemble was then used to make discharge forecasts with observed temperature and precipitation as forcing data. The study was performed on 76 catchments all over Sweden with data from the time period 1999-2008. Forecasts were made every day and the ensemble spread was evaluated 2, 4 and 10 days into the forecast. Autoregressive forecasts where the modelled discharge is corrected after the observed discharge were executed and evaluated as well. The results indicated a relationship between ensemble spread and forecast error, which implies that the spread can be used as a measure of the uncertainty of the initial state. The forecast error and ensemble spread correlated positively to forest percentage and negatively to catchment area, lake percentage and elevation. The same trend was detected between spread and catchment characteristics. The spread was biggest in winter and spring when normalization was made with mean discharge for the ten-year period and in spring and summer when normalization was done with mean discharge per month. The hypothesis that a smaller difference between the discharge of the initial state and the observed discharge results in more accurate forecasts was confirmed by the results. An implementation of an ensemble of different initial states in operational forecasts at SMHI’s Hydrological Forecast and Warning Service is suggested in order to further quantify the uncertainty of hydrological forecasts, and thereby improve the basis of judgment when issuing risks and warnings.

Dans cette thèse, nous nous intéressons à la représentation et à la prise en compte des incertitudes dans les systèmes de prévision hydrologique probabilistes à moyen-terme. Ces incertitudes proviennent principalement de deux sources : (1) de l’imperfection des prévisions météorologiques (utilisées en intrant de ces systèmes) et (2) de l’imperfection de la représentation du processus hydrologique par le simulateur pluie-débit (SPQ) (au coeur de ces systèmes).La performance d’un système de prévision probabiliste s’évalue par la précision de ses prévisions conditionnellement à sa fiabilité. L’approche statistique que nous suivons procure une garantie de fiabilité à condition que les hypothèses qu’elle implique soient réalistes. Nous cherchons de plus à gagner en précision en incorporant des informations auxiliaires.Nous proposons, pour chacune des sources d’incertitudes, une méthode permettant cette incorporation : (1) un post-traitement des prévisions météorologiques s’appuyant sur la propriété statistique d’échangeabilité et permettant la prise en compte de plusieurs sources de prévisions, ensemblistes ou déterministes ; (2) un post-traitement hydrologique utilisant les variables d’état des SPQ par le biais d’un modèle Probit arbitrant entre deux régimes hydrologiques interprétables et permettant ainsi de représenter une incertitude à variance hétérogène.Ces deux méthodes montrent de bonnes capacités d’adaptation aux cas d’application variés fournis par EDF et Hydro-Québec, partenaires et financeurs du projet. Elles présentent de plus un gain en simplicité et en formalisme par rapport aux méthodes opérationnelles tout en montrant des performances similaires. / In this thesis, we are interested in representing and taking into account uncertainties in medium term probabilistic hydrological prediction systems.These uncertainties mainly come from two sources: (1) from the imperfection of meteorological forecasts (used as inputs to these systems) and (2) from the imperfection of the representation of the hydrological process by the rainfall-runoff simulator (RRS) (at the heart of these systems).The performance of a probabilistic forecasting system is assessed by the sharpness of its predictions conditional on its reliability. The statistical approach we follow provides a guarantee of reliability if the assumptions it implies are complied with. We are also seeking to incorporate auxilary information to get sharper.We propose, for each source of uncertainty, a method enabling this incorporation: (1) a meteorological post-processor based on the statistical property of exchangeability and enabling to take into account several (ensemble or determistic) forecasts; (2) a hydrological post-processor using the RRS state variables through a Probit model arbitrating between two interpretable hydrological regimes and thus representing an uncertainty with heterogeneous variance.These two methods demonstrate adaptability on the various application cases provided by EDF and Hydro-Québec, which are partners and funders of the project. Those methods are moreover simpler and more formal than the operational methods while demonstrating similar performances.