The Ensemble Kalman Filter: theoretical formulation and practical implementation

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.Responsible Editor: Jörg-Olaf Wolff     

La prévision du climat : de l'échelle saisonnière à l'échelle décennale

The atmosphere and the ocean are subject to many dynamical instabilities, which limit the time during which their behaviour can be deterministically forecasted. At longer timescales, the atmosphere can be predicted at best using statistical methods, as a response to external forcing linked to sea- and land-surface anomalies. Climate being defined as the mean of atmospheric states, it appears that it can be predicted up to a few months in advance, which is the characteristic time of the so-called slow components of the climate system. Forecasting can sometimes be extended to longer time ranges, especially when the coupled ocean–atmosphere system exhibits internal variability modes, with characteristic times of a few years. Seasonal climate forecasting is most often based upon Monte-Carlo simulations, where the various realisations correspond to slightly different initial conditions. The present sate-of-the-art in Europe (ECMWF) and/or in the USA (IRI) allows to forecast such major phenomena, as El Niño, up to six months in advance. Finally, some parameters may exhibit predictability at still longer time-ranges (inter-annual to decadal), but only for certain regions. The example of electricity production is used to underline the potentially large economical benefit of seasonal climate forecasting. To cite this article: J.-C. André et al., C. R. Geoscience 334 (2002) 1115–1127.

Résumé

L'atmosphère et l'océan sont le siège d'instabilités dynamiques, qui limitent la durée pendant laquelle il est possible d'en prévoir l'évolution de façon déterministe. Au-delà, l'atmosphère n'est plus prévisible, au mieux, que de façon statistique, en fonction du forçage externe qu'exerce(nt) sur elle l'océan et/ou la surface des continents. Le climat (au sens d'une moyenne des états atmosphériques) se révèle ainsi prévisible jusqu'à des échéances temporelles de quelques mois, échelle de temps caractéristique des composantes dites « lentes » du système climatique. La prévision peut s'étendre à des échéances parfois plus longues, dans le cas où le système couplé océan–atmosphère posséderait des modes de variabilité temporelle de périodes caractéristiques de quelques années. La prévision climatique saisonnière est très souvent construite à partir de simulations de type Monte-Carlo, avec des ensembles de réalisations utilisant des conditions initiales légèrement différentes. Dans l'état actuel de ces prévisions, qu'elles soient réalisées en Europe (CEPMMT) ou aux États-Unis (IRI), il est possible de prévoir environ six mois à l'avance un certain nombre de phénomènes climatiques, en particulier ceux liés aux épisodes dits « El Niño », pour lesquels l'amplitude des variations est suffisamment importante. Il existe, par ailleurs, une prévisibilité à encore plus longue échéance (inter-annuelle à décennale), mais seulement pour certains paramètres et certaines régions. L'exemple de la production d'électricité montre l'importance économique potentielle très grande de la prévision climatique saisonnière. Pour citer cet article : J.-C. André et al., C. R. Geoscience 334 (2002) 1115–1127.     

Steady- and transient-state inversion in hydrogeology by successive flux estimation

A calibration method to solve the groundwater inverse problem under steady- and transient-state conditions is presented. The method compares kriged and numerical head field gradients to modify hydraulic conductivity without the use of non-linear optimization techniques. The process is repeated iteratively until a close match with piezometric data is reached. The approach includes a damping factor to avoid divergence and oscillation of the solution in areas of low hydraulic gradient and a weighting factor to account for temporal head variation in transient simulations. The efficiency of the method in terms of computing time and calibration results is demonstrated with a synthetic field. It is shown that the proposed method provides parameter fields that reproduce both hydraulic conductivity and piezometric data in few forward model solutions. Stochastic numerical experiments are conducted to evaluate the sensitivity of the method to the damping function and to the head field estimation errors.     

Study on the Variational Assimilation Technique for the Retrieval of Wind Fields from Doppler Radar Data

This paper introduces a variational assimilation technique for the retrieval of wind fields from Doppler radar data. The assimilated information included both the radial velocity (RV) and the movement of radar echo. In this assimilation technique, the key is transforming the movement of radar echo to a new radar measuring variable- "apparent velocity" (AV). Thus, the information of wind is added, and the indeterminacy of recovering two-dimensional wind only by AV was overcome effectively by combining RV with AV. By means of CMA GRAPES-3Dvar and CINRAD data, some experiments were performed. The results show that the method of retrieval of wind fields is useful in obtaining the construction of the weather system.     

APPLICATION OF BDA SCHEME IN TYPHOON TRACK PREDICTION

The MM5, which is the PSU/NCAR mesoscale nonhydrostatic limited-area model, and its adjoining modeling system are used in this paper. Taking T106 analysis data as background field the authors generate an optimal initial condition of a typhoon by using two bogus data assimilation schemes, and conduct some numerical simulating experiments. The results of No.9608 typhoon (Gloria) show that the optimal initial field have some dramatic improvements, such as inaccurate position of typhoon center, weaker typhoon circulation and incomplete inner structure of the typhoon, which are caused by shortage of data over the sea. Some improvements have been made in the track forecast. Through several comparing experiments, the initialization optimized by BDA scheme is found to be more reasonable than GFDL scheme and its typhoon track forecast is better.     

多普勒天气雷达资料对中尺度模式短时预报的影响

利用中尺度模式ARPS(The Advanced Regional Prediction System)及其资料分析系统ADAS(ARPS Data Analysis System),将国内新一代多普勒雷达(CINRAD)反射率及径向风资料直接用于中尺度数值模拟,通过一次华北地区暴雨过程的模拟对比试验,分析了雷达资料对初始场的改进效果及其对模拟结果的影响,结果表明:(1)利用雷达径向风资料对初始风场进行调整后,自近地面到对流层顶的u,v,w都发生了变化,调整后的初始风场在对流层中层变化最大.(2)利用雷达反射率进行微物理调整和云分析能调整初始场中的云水信息,使得雷达回波附近3 km以下的水汽混合比(qv)增加,4 km以下的雨水混合比(qr)增加,对流层(约10 km以下)的云水混合比(qc)增加,4～9 km的对流层上部云冰混合比(qi)和雪混合比(qs)增加.ADAS通过非绝热初始化调整温度场,从而得到了一个动力和热力上平衡的初始场.(3)模拟的1 h雨量与实况的对比表明,同时利用雷达反射率和径向风改进过的初始场能明显增强3 h内的降水强度和落区预报,改善中尺度数值模式短时定量降水预报.模拟的1 h流场对比分析表明,经雷达径向风调整后,能够在初始场中增加气旋性涡旋等中小尺度风信息,明显减少模式的spin-up时间.(4)通过对雷达径向风和反射率对模式初始场和模拟结果影响的对比分析发现,雷达径向风主要是改进初始风场,而雷达反射率主要是改进初始场中的湿度参数,增加初始场中云水等的含量,调整温度场.通过模拟的6 h降水对比发现,利用雷达径向风调整初始场后,对降水模拟有一定的改进,但效果不甚明显,而雷达反射率资料对定量降水预报改进效果明显,同时使用雷达径向风和反射率资料改进初始场后对降水的模拟效果最明显.     

A review on reflective remote sensing and data assimilation techniques for enhanced agroecosystem modeling

During the last 50 years, the management of agroecosystems has been undergoing major changes to meet the growing demand for food, timber, fibre and fuel. As a result of this intensified use, the ecological status of many agroecosystems has been severely deteriorated. Modeling the behavior of agroecosystems is, therefore, of great help since it allows the definition of management strategies that maximize (crop) production while minimizing the environmental impacts. Remote sensing can support such modeling by offering information on the spatial and temporal variation of important canopy state variables which would be very difficult to obtain otherwise.In this paper, we present an overview of different methods that can be used to derive biophysical and biochemical canopy state variables from optical remote sensing data in the VNIR-SWIR regions. The overview is based on an extensive literature review where both statistical–empirical and physically based methods are discussed. Subsequently, the prevailing techniques of assimilating remote sensing data into agroecosystem models are outlined. The increasing complexity of data assimilation methods and of models describing agroecosystem functioning has significantly increased computational demands. For this reason, we include a short section on the potential of parallel processing to deal with the complex and computationally intensive algorithms described in the preceding sections.The studied literature reveals that many valuable techniques have been developed both for the retrieval of canopy state variables from reflective remote sensing data as for assimilating the retrieved variables in agroecosystem models. However, for agroecosystem modeling and remote sensing data assimilation to be commonly employed on a global operational basis, emphasis will have to be put on bridging the mismatch between data availability and accuracy on one hand, and model and user requirements on the other. This could be achieved by integrating imagery with different spatial, temporal, spectral, and angular resolutions, and the fusion of optical data with data of different origin, such as LIDAR and radar/microwave.     

Three-dimensional mapping of salt stores in the Murray-Darling Basin,Australia: 2. Calculating landscape salt loads from airborne electromagnetic and laboratory data

Salt is widespread in the Australian landscape – in soil, regolith and groundwater – leading to concerns that land management practices may be putting much agricultural land and important water resources at risk of salinisation. Defining the location and nature of salt stores is an important first step in understanding the processes leading to salinity of soils, streams and groundwater resources, and predicting areas that may be at risk. Airborne geophysics can define subsurface salt stores and mobilisation pathways. Airborne electromagnetics (AEM) can map the three-dimensional conductivity structure of the landscape but does not, intrinsically, quantify the amount of salt. Salinity, moisture content, porosity and mineralogy all contribute to the electromagnetic signal, and each can vary significantly throughout the landscape.In the Lower Balonne catchment, Queensland, the relationship between AEM and the amount of salt in the landscape was quantified using laboratory analyses of pore fluids from core samples. A general statistical relationship was established between AEM conductivity and salt load (defined as the product of pore fluid salinity, porosity and moisture content)—with a significant positive correlation although data were generally widely dispersed. Comparison of calculated salt load with borehole electromagnetic logs gives insight into the factors contributing to dispersion in the AEM data.The relationship transforms bulk conductivity to salt load in 5 m layers, allowing the generation of a three-dimensional map of the salt load. This is a powerful tool for identifying areas that may require monitoring and management interventions to reduce salinity risk. An example is given of salt loads beneath an established irrigation area in the Lower Balonne catchment, Queensland.     

Thermoelastic stress due to a rectangular heat source in a semi-infinite medium. Presentation of an analytical solution

The thermoelastic response due to a time-dependent rectangular heat source in a semi-infinite medium is analyzed. The problem originates from studies of nuclear waste repositories in rock. Canisters containing heat-emitting nuclear waste are deposited over a large rectangular area deep below the ground surface. The solution for a time-dependent heat source is obtained from the corresponding instantaneous heat source by superposition. The thermoelastic problem for the instantaneous rectangular heat source in a infinite surrounding is solved exactly. An important step is the introduction of so-called quadrantal heat sources. The solution for the rectangle is obtained from four quadrantal solutions. The solution for the quadrantal heat source depends on the three dimelasionless coordinates only. Time occurs in the scale factors only. The condition of zero normal and shear stresses at the ground surface is fulfilled by using a mirror heat source and a boundary solution. The boundary solution accounts for the residual normal stress at the ground surface. Using a Hertzian potential, a surprisingly simple solution is obtained. The final analytical solution is quite tractable considering the complexity of the initial problem. The solution may be used to test numerical models for coupled thermoelastic processes. It may also be used in more detailed numerical simulations of the process near the heat sources as boundary conditions to account for the three-dimensional global process.     

Crustal contamination and fluid/rock interaction in the carbonatites of Fuerteventura (Canary Islands, Spain): a C, O, H isotope study

Fuerteventura—the second largest of the Canary Islands consists of Mesozoic sediments, submarine volcanic rocks, dike swarms and plutons of the Basal Complex, and younger subaerial basaltic and trachytic series. Carbonatites are found in two Basal Complex exposures: the Betancuria Massif in the central part of the island and the Esquinzo area in the north. values of the carbonatites increase progressively from south to north of the island. This phenomenon is attributed to different degrees of assimilation of sedimentary carbonate. Homogeneous, typically magmatic values for carbonatites which have preserved primary igneous textures and minerals suggest a well-mixed reservoir where changes in values result from the storage of carbonate magmas at different structural levels. The magma storage allowed assimilation of sediment to varying degrees before final emplacement of carbonatites. Shifts in towards more positive and negative values from presumed primary compositions are observed in the carbonatites. On the basis of the oxygen isotope compositions of calcite, mica and K-feldspar, and the hydrogen isotope compositions of micas, the changes in the values of the carbonatites can be related to fluid/rock interactions.