A multiresolution ensemble Kalman filter using the wavelet decomposition

We present a method of using classical wavelet-based multiresolution analysis to separate scales in model and observations during data assimilation with the ensemble Kalman filter. In many applications, the underlying physics of a phenomena involve the interaction of features at multiple scales. Blending of observational and model error across scales can result in large forecast inaccuracies since large errors at one scale are interpreted as inexact data at all scales due to the misrepresentation of observational error. Our method uses a partitioning of the range of the observation operator into separate observation scales. This naturally induces a transformation of the observation covariance and we put forward several algorithms to efficiently compute the transformed covariance. Another advantage of our multiresolution ensemble Kalman filter is that scales can be weighted independently to adjust each scale’s affect on the forecast. To demonstrate feasibility, we present applications to a one-dimensional Kuramoto-Sivashinsky (K–S) model with scale-dependent observation noise and an application involving the forecasting of solar photospheric flux. The solar flux application uses the Air Force Data Assimilative Photospheric Transport (ADAPT) model which has model and observation error exhibiting strong scale dependence. Results using our multiresolution ensemble Kalman filter show significant improvement in solar forecast error compared to traditional ensemble Kalman filtering.     

模式时间关联误差对集合平方根滤波估算土壤湿度的影响

为了定量评估模式时间关联误差对NOAH陆面模式同化表层土壤湿度观测估算土壤湿度廓线的影响,采用集合平方根滤波(En SRF)与状态增广相结合的技术,开展同时更新状态变量和订正模式偏差的观测系统模拟试验,结果表明:同化时若不对存在较大系统性偏差的模式时间关联误差进行处理,En SRF就不能有效估算土壤湿度廓线,而采用状态增广和En SRF相结合的技术,可以在更新土壤湿度时同步订正模式偏差,土壤湿度估算精度明显提高。敏感性试验进一步表明:模式偏差大小、同化时间间隔和观测误差会以不同方式对同化结果造成影响。     

A nonparametric view of generalized covariances for kriging

Fitting trend and error covariance structure iteratively leads to bias in the estimated error variogram. Use of generalized increments overcomes this bias. Certain generalized increments yield difference equations in the variogram which permit graphical checking of the model. These equations extend to the case where errors are intrinsic random functions of order k, k=1, 2, ..., and an unbiased nonparametric graphical approach for investigating the generalized covariance function is developed. Hence, parametric models for the generalized covariance produced by BLUEPACK-3D or other methods may be assessed. Methods are illustrated on a set of coal ash data and a set of soil pH data.     

Assessing the influence of assimilating radar-observed radial winds on the simulation of a tropical cyclone

Observations by Doppler weather radar are crucial for nowcasting and short-time forecasting of severe weather events as they bring in refined information of the atmosphere. However, due to the inevitable noises and non-meteorological signals, they cannot be assimilated straightforwardly into a numerical model. In the present study, assimilation of the radial component of wind velocity observed by two Doppler radars is performed in the numerical simulation of Supertyphoon Rammasun (2014) just before its landfall. After several quality-control steps, the radar-observed radial velocities are de-aliased, noise-reduced and assimilated into the model to improve initial conditions for the high-resolution simulation. Results show that only when using global background error covariance matrix can the observational increment be properly assimilated into the model, correcting large-scale background steering flow and yielding a simulated track close to the observed one. However, little improvement is found in simulating the TC core-scale structures by the assimilation of radar velocity as compared to the radar-observed flow, primarily due to the insufficient spatial resolution of the model that may lead to the incorrect representation of the TC core structure and the rejection of some core-region observations during the data assimilation procedure. Moreover, assimilation-induced asymmetries consume a certain portion of mean kinetic energy, preventing the simulated Rammasun from axisymmetrization and thus intensification as compared with the non-assimilated experiment.     

土壤水分同化系统的敏感性试验研究

利用1998年7月6日至8月9日青藏高原GAME-Tibet试验区MS3608站点的4cm、20cm和100cm的土壤水分观测数据同化SiB2模型输出的表层、根区和深层土壤水分,探讨了一个基于集合卡尔曼滤波和简单生物圈模型的单点土壤水分同化方案。分析和评价了集合大小、同化周期、模型误差、背景场误差以及观测误差对同化系统性能的影响。结果表明:①增加集合数目可以减小土壤水分同化系统的误差,但同时又降低了运行效率;②对于集合卡尔曼滤波,初始场的估计是否准确对同化系统性能影响不大;③模型误差和观测误差的准确估计可以提高土壤水分的估计精度;④利用数据同化的方法对土壤水分的估计有显著提高。     

Nonstationarity of the Mean and Unbiased Variogram Estimation: Extension of the Weighted Least-Squares Method

When concerned with spatial data, it is not unusual to observe a nonstationarity of the mean. This nonstationarity may be modeled through linear models and the fitting of variograms or covariance functions performed on residuals. Although it usually is accepted by authors that a bias is present if residuals are used, its importance is rarely assessed. In this paper, an expression of the variogram and the covariance function is developed to determine the expected bias. It is shown that the magnitude of the bias depends on the sampling configuration, the importance of the dependence between observations, the number of parameters used to model the mean, and the number of data. The applications of the expression are twofold. The first one is to evaluate a priori the importance of the bias which is expected when a residuals-based variogram model is used for a given configuration and a hypothetical data dependence. The second one is to extend the weighted least-squares method to fit the variogram and to obtain an unbiased estimate of the variogram. Two case studies show that the bias can be negligible or larger than 20%. The residual-based sample variogram underestimates the total variance of the process but the nugget variance may be overestimated.     

Impact of assimilation of ATOVS temperature and humidity and SSM/I total precipitable water on the simulation of a monsoon depression

The present study explored the effect of assimilation of Advanced TIROS Vertical Sounder (ATOVS) temperature and humidity profiles and Spectral sensor microwave imager (SSM/I) total precipitable water (TPW) on the simulation of a monsoon depression which formed over the Arabian Sea during September 2005 using the Weather Research and Forecast model. The three-dimensional variational (3DVAR) data assimilation technique has been employed for the purpose of assimilation of satellite observations. Statistical scores like “equitable threat score,” “bias score,” “forecast impact,” and “improvement parameter” have been used to examine the impact of the above-mentioned satellite observations on the numerical simulation of a monsoon depression. The diagnostics of this study include verification of the vertical structure of depression, in terms of temperature anomaly profiles and relative vorticity profiles with observations/analysis. Additional diagnostics of the study include the analysis of the heat budget and moisture budget. Such budget studies have been performed to provide information on the role of cumulus convection associated with the depression. The results of this study show direct and good evidence of the impact of the assimilation of the satellite observations using 3DVAR on the dynamical and thermodynamical features of a monsoon depression along with the effect of inclusion of satellite observation on the spatial pattern of the simulated precipitation associated with the depression. The “forecast impact” parameter calculated for the wind speed provides good evidence of the positive impact of the assimilation of ATOVS temperature and humidity profiles and SSM/I TPW on the model simulation, with the assimilation of the ATOVS profiles showing better impact in terms of a more positive value of the “forecast impact” parameter. The results of the study also indicate the improvement of the forecast skill in terms of “equitable threat score” and “bias score” due to the assimilation of satellite observation.     

Sampling error distribution for the ensemble Kalman filter update step

In recent years, data assimilation techniques have been applied to an increasingly wider specter of problems. Monte Carlo variants of the Kalman filter, in particular, the ensemble Kalman filter (EnKF), have gained significant popularity. EnKF is used for a wide variety of applications, among them for updating reservoir simulation models. EnKF is a Monte Carlo method, and its reliability depends on the actual size of the sample. In applications, a moderately sized sample (40–100 members) is used for computational convenience. Problems due to the resulting Monte Carlo effects require a more thorough analysis of the EnKF. Earlier we presented a method for the assessment of the error emerging at the EnKF update step (Kovalenko et al., SIAM J Matrix Anal Appl, in press). A particular energy norm of the EnKF error after a single update step was studied. The energy norm used to assess the error is hard to interpret. In this paper, we derive the distribution of the Euclidean norm of the sampling error under the same assumptions as before, namely normality of the forecast distribution and negligibility of the observation error. The distribution depends on the ensemble size, the number and spatial arrangement of the observations, and the prior covariance. The distribution is used to study the error propagation in a single update step on several synthetic examples. The examples illustrate the changes in reliability of the EnKF, when the parameters governing the error distribution vary.     

Influence of numerical precision on the calibration of AEM-based groundwater flow models

Groundwater modelers have embraced the use of automated calibration tools based on classical nonlinear regression techniques. While clearly an improvement over trial-and-error calibration, it is not clear to what extent these popular inverse modeling tools yield accurate parameter sets for groundwater flow models. The impact of model configuration and precision upon automated parameter estimation is also unclear. An extensive set of numerical experiments was performed to explore the influence of model configuration on the calibration of a regional groundwater flow model developed using the analytic element method. The results provided insight into the manner in which the specified level of model precision and the location of observation points influence the results of inverse modeling based on nonlinear regression. While the importance of these issues is application-specific, obtaining an accurate model calibration for the case study required both a careful placement of test observations and a greater-than-anticipated level of model precision. The required level of model precision for calibration was more than necessary to produce an acceptable flow solution.     

基于实测资料对日蒸散发估算模型的比较

利用设置于江西省南昌县的新型高精度自动蒸渗仪,于2007年9月1日至2008年8月31日的实测陆面实际蒸散发过程,检验了面蒸散发互补关系模型CRAE (Complementary Relationship Areal Evapotranspiration)、GG模型 (Granger-Gray)、平流-干旱模型AA (Advection-Aridity)3个逐日路面实际蒸散发模型在不同时间尺度上的计算精度,并对计算误差的影响因素进行了讨论.结果表明:该地区实测年蒸散发量为746.1 mm,采用各模型的推荐经验参数对该地区蒸散发的估算结果误差较大,普遍干旱条件下蒸散发的计算值比观测值偏小,而湿润条件下的计算值偏大.通过对各模型的经验参数进行调整,各模型对年蒸散发量的计算精度大为提高,但逐日蒸散发过程的计算精度改进效果有限,在7日的时间尺度上,计算结果显著优于逐日的计算结果,在此时间尺度下,AA模型仍存在一定的系统误差,CRAE模型的估算精度相对较差,GG模型的总体计算效果相对最好.根据与蒸渗仪观测结果的对比分析,根据区域特征进行参数调整后的模型,需要在7日及更长时间尺度上,蒸散发模型的估算结果较为可靠.上述研究对全面认识陆面实际蒸散发特征、理解各蒸散发模型在不同时间尺度上的模拟能力、正确认识气候变化条件下的水循环特征具有重要意义.     

The Second-Order Stationary Universal Kriging Model Revisited

Universal kriging originally was developed for problems of spatial interpolation if a drift seemed to be justified to model the experimental data. But its use has been questioned in relation to the bias of the estimated underlying variogram (variogram of the residuals), and furthermore universal kriging came to be considered an old-fashioned method after the theory of intrinsic random functions was developed. In this paper the model is reexamined together with methods for handling problems in the inference of parameters. The efficiency of the inference of covariance parameters is shown in terms of bias, variance, and mean square error of the sampling distribution obtained by Monte Carlo simulation for three different estimators (maximum likelihood, bias corrected maximum likelihood, and restricted maximum likelihood). It is shown that unbiased estimates for the covariance parameters may be obtained but if the number of samples is small there can be no guarantee of good estimates (estimates close to the true value) because the sampling variance usually is large. This problem is not specific to the universal kriging model but rather arises in any model where parameters are inferred from experimental data. The validity of the estimates may be evaluated statistically as a risk function as is shown in this paper.     

On maximum likelihood estimation of parameters in incorrectly specified models of covariance for spatial data

The analysis of, and from, models of spatial data usually proceeds under the assumption, often implicit, that the correct model has been specified. However, any model identification procedures based on sample data are subject to error, and consequences of such errors then permeate subsequent analysis. Thus, an attempt to quantify some of these consequences is of interest. A standard framework for analysis is extended here, by introduction of information theory, to permit the study of effects of model misspecification on maximum likelihood estimators of parameters of model covariance. Asymptotically valid theoretical results are presented, and the relevance of these results to samples of finite sizes met in practice is assessed in a series of simulation experiments. The effect of model misspecification, and use of estimators of parameters of misspecified covariance models, on the practical problem of prediction at a previously unsampled location is considered briefly, and further areas for possible investigation are outlined.     

Time-dependent Inversion of Surface Subsidence due to Dynamic Reservoir Compaction

We introduce a novel, time-dependent inversion scheme for resolving temporal reservoir pressure drop from surface subsidence observations (from leveling or GPS data, InSAR, tiltmeter monitoring) in a single procedure. The theory is able to accommodate both the absence of surface subsidence estimates at sites at one or more epochs as well as the introduction of new sites at any arbitrary epoch. Thus, all observation sites with measurements from at least two epochs are utilized. The method uses both the prior model covariance matrix and the data covariance matrix, which incorporates the spatial and temporal correlations between model parameters and data, respectively. The incorporation of the model covariance implicitly guarantees smoothness of the model estimate, while maintaining specific geological features like sharp boundaries. Taking these relations into account through the model covariance matrix enhances the influence of the data on the inverted model estimate. This leads to a better defined and interpretable model estimate. The time-dependent aspect of the method yields a better constrained model estimate and makes it possible to identify non-linear acceleration or delay in reservoir compaction. The method is validated by a synthetic case study based on an existing gas reservoir with a highly variable transmissibility at the free water level. The prior model covariance matrix is based on a Monte Carlo simulation of the geological uncertainty in the transmissibility.     

Automatic calibration tool for river models based on the MHYSER software

Due to their complex nature, river models require extensive calibration in order to achieve reliable model predictions. Manually fitting the numerous parameters included in this procedure can be a laborious and repetitive process. This paper presents a new instrument, developed specifically for the automatic calibration of river models based on the software MHYSER. The instrument is completely autonomous and returns the model with the parameter values giving rise to the smallest difference between the model-generated observations and the measured observations. It utilises the software PEST to fit continuous calibration parameters and exceeds the program’s capabilities in order to also fit discontinuous calibration parameters. Testing of the instrument is accomplished using three models, one of which was developed during a study on the dynamics of sediments on the Romaine River, situated in the Eastern region of the Province of Quebec.     

通用模型参数率定技术研究

在介绍当前模型参数优选技术的基础上，设计了模型参数同优选方法相耦合的一系列数据接口定义，实现了人工试错和自动优选相耦合、多模型多参数同时自动优选的模型参数率定功能。应用实例表明，通用模型参数率定功能具有简便、快捷、准确的优点，大大提高了参数率定的效率。     

Gravity field recovery from airborne gradiometer data using collaction and taking into account correlated errors

Least squares collocation is a very comprehensive method for gravity field modelling, since it may use known noise characteristics of the data. In many earlier applications the errors affecting the data were considered uncorrelated, mainly due to the difficulty in estimating the systematic character of such kind of errors. In this study, error covariance functions of airborne gravity gradiometer data are estimated by comparing model covariance functions with empirical covariance functions of the gravity gradiometer data. The model covariance functions were estimated from accurate surface gravity data and continuated upward to the height of the airborne measurements using the covariance propagation law. The estimated error covariance functions were modeled as finite ones and used as an additional information for the prediction of gravity anomalies from gravity gradiometer data. The assessment of the prediction results was made by comparing the gravity values predicted from the airborne gradient data and showed up to 25% improvement compared to not using correlated errors.     

Estimating observation error covariance matrix of seismic data from a perspective of image denoising

Estimating observation error covariance matrix properly is a key step towards successful seismic history matching. Typically, observation errors of seismic data are spatially correlated; therefore, the observation error covariance matrix is non-diagonal. Estimating such a non-diagonal covariance matrix is the focus of the current study. We decompose the estimation into two steps: (1) estimate observation errors and (2) construct covariance matrix based on the estimated observation errors. Our focus is on step (1), whereas at step (2) we use a procedure similar to that in Aanonsen et al. 2003. In Aanonsen et al. 2003, step (1) is carried out using a local moving average algorithm. By treating seismic data as an image, this algorithm can be interpreted as a discrete convolution between an image and a rectangular window function. Following the perspective of image processing, we consider three types of image denoising methods, namely, local moving average with different window functions (as an extension of the method in Aanonsen et al. 2003), non-local means denoising and wavelet denoising. The performance of these three algorithms is compared using both synthetic and field seismic data. It is found that, in our investigated cases, the wavelet denoising method leads to the best performance in most of the time.     

基于变动饱和带的产汇流模型及其参数确定方法

当前分布式水文模型的参数确定仍主要依赖率定方式,在缺资料地区应用受到限制。建立一种基于变动饱和带产流模式和网格水滴汇流方法的分布式产汇流模型,提出利用下垫面特征来确定模型参数的方法。结合野外入渗试验和参数敏感性分析,建立地表饱和水力传导度(K_0z)和饱和水力传导度随深度衰减系数(f)2个敏感性产流参数与地形参数、土壤类型数据的定量统计关系,利用野外坡面流观测试验确定坡面汇流参数,并在多个实际流域进行应用验证。结果表明：(1)利用地形参数确定K_0z与使用遥感资料确定K_0z的模型精度进行对比,在姜湾实验流域场次洪水模拟的平均确定性系数从0.82提高至0.86,洪峰与洪量误差的平均绝对值分别降低了2.2%和0.95%,但峰现时间误差平均绝对值增大了4%(仍控制在2 h内)。(2)建立姜湾等14个流域参数f率定值与不同深度土壤类型数据的定量关系,移用至七邻等6个流域进行验证,表明参数f关系式与模型率定的精度非常接近,相对误差的平均绝对值为2.8%,场次洪水模拟的平均确定性系数为0.83,洪峰与洪量误差的平均绝对值为10.07%和...     

Comparing multi-objective optimization techniques to calibrate a conceptual hydrological model using in situ runoff and daily GRACE data

Hydrological models are necessary tools for simulating the water cycle and for understanding changes in water resources. To achieve realistic model simulation results, real-world observations are used to determine model parameters within a “calibration” procedure. Optimization techniques are usually applied in the model calibration step, which assures a maximum similarity between model outputs and observations. Practical experiences of hydrological model calibration have shown that single-objective approaches might not be adequate to tune different aspects of model simulations. These limitations can be as a result of (i) using observations that do not sufficiently represent the dynamics of the water cycle, and/or (ii) due to restricted efficiency of the applied calibration techniques. To address (i), we assess how adding daily Total Water Storage (dTWS) changes derived from the Gravity Recovery And Climate Experiment (GRACE) as an extra observations, besides the traditionally used runoff data, improves calibration of a simple 4-parameter conceptual hydrological model (GR4J, in French: modèle du Génie Rural à 4 paramètres Journalier) within the Danube River Basin. As selecting a proper calibration approach (in ii) is a challenging task and might have significant influence on the quality of model simulations, for the first time, four evolutionary optimization techniques, including the Non-dominated Sorting Genetic Algorithm II (NSGA-II), the Multi-objective Particle Swarm Optimization (MPSO), the Pareto Envelope-Based Selection Algorithm II (PESA-II), and the Strength Pareto Evolutionary Algorithm II (SPEA-II) along with the Combined objective function and Genetic Algorithm (CGA) are tested to calibrate the model in (i). A number of quality measures are applied to assess cardinality, accuracy, and diversity of solutions, which include the Number of Pareto Solutions (NPS), Generation Distance (GD), Spacing (SP), and Maximum Spread (MS). Our results indicate that according to MS and SP, NSGA-II performs better than other techniques for calibrating GR4J using GRACE dTWS and in situ runoff data. Considering GD as a measure of efficiency, MPSO is found to be the best technique. CGA is found to be an efficient method, while considering the statistics of the GR4J’s 4 calibrated parameters to rank the optimization techniques. The Nash-Sutcliffe model efficiency coefficient is also used to assess the predictive power of the calibrated hydrological models, for which our results indicate satisfactory performance of the assessed calibration experiments.     

How can remote sensing contribute in groundwater modeling?

Groundwater resources assessment, modeling and management are hampered considerably by a lack of data, especially in semi-arid and arid environments with a weak observation infrastructure. Usually, only a limited number of point measurements are available, while groundwater models need spatial and temporal distributions of input and calibration data. If such data are not available, models cannot play their proper role in decision support as they are notoriously underdetermined and uncertain. Recent developments in remote sensing have opened new sources for distributed spatial data. As the relevant entities such as water fluxes, heads or transmissivities cannot be observed directly by remote sensing, ways have to be found to link the observable quantities to input data required by the model. An overview of the possibilities for employing remote-sensing observations in groundwater modeling is given, supported by examples in Botswana and China. The main possibilities are: (1) use of remote-sensing data to create some of the spatially distributed input parameter sets for a model, and (2) constraining of models during calibration by spatially distributed data derived from remote sensing. In both, models can be improved conceptually and quantitatively.