Data assimilation in 2D viscous Burgers equation using a stabilized explicit finite difference scheme run backward in time

An artificial example of a coupled system of three nonlinear partial differential equations generalizing 2D thermoelastic vibrations, is used to demonstrate the effectiveness, as well as the limitations, of a non iterative direct procedure in data assimilation. A stabilized explicit finite difference scheme, run backward in time, is used to find initial values, [ u ( . , 0 ) , v ( . , 0 ) , w ( . , 0 ) ] , that can evolve into a useful approximation to a hypothetical target result [ u ∗ ( . , T max ) , v ∗ ( . , T max ) , w ∗ ( T max ) ] , at some realistic 0 $ ]]> T max > 0 . Highly non smooth target data are considered, that may not correspond to actual solutions at time T max . Stabilization is achieved by applying a compensating smoothing operator at each time step. Such smoothing leads to a distortion away from the true solution, but that distortion is small enough to allow for useful results. Data assimilation is illustrated using 512 × 512 pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Computational experiments show that efficient FFT-synthesized smoothing operators, based on ( − Δ ) q with real q>3, can be successfully applied, even in nonlinear problems in non-rectangular domains. However, an example of failure illustrates the limitations of the method in problems where T max , and/or the nonlinearity, are not sufficiently small.

To improve the forecast quality of numerical weather prediction (NWP), the German Meteorological Service (Deutscher Wetterdienst, DWD) has initiated a project aimed at assessing data quality and assimilation of observations from ground-based remote sensing instruments that have not yet been exploited operationally.The objective of this initiative is to fill the observational gap in the atmospheric boundary layer, especially with respect to short time scales, by providing continuous, high-temporal-resolution profiles of thermodynamic variables, wind, and cloud properties. These observations are expected to be especially beneficial for weather forecasting applications. The DWD is evaluating various remote sensing systems with regard to the continuous data supply, their operational use, and their impact on NWP. In this contribution, we present first results of the assimilation of two ground-based remote sensing instruments into the kilometer-scale ensemble data assimilation system (KENDA): water vapour mixing ratio from a Differential Absorption Lidar (DIAL) and radar reflectivity from a cloud radar. For the integration of the DIAL observations into the data assimilation code environment, only small adjustments were necessary. In contrast, the cloud radar data required an adaptation of the complex forward operator EMVORADO (Efficient Modular Volume scan Radar Operator), which was originally developed and previously used only for precipitation radars.In an initial step, single observation data assimilation experiments and their observation minus first guess statistics have been shown to produce promising results. To assess the impact in an operational setting, dedicated data assimilation experiments were conducted and compared to reference experiments without these additional observations. Based on the successful data assimilation cycling experiments, first forecast experiments including the DIAL have been performed. Current results indicate a neutral to positive impact on humidity, temperature, and wind forecasts. The impact of cloud radar data in such experiments is currently under investigation by testing different settings.Our findings suggest that ground-based remote sensing data can provide valuable additional information for convective-scale data assimilation, and justify more extensive impact studies in the context of NWP.

To improve the forecast quality of numerical weather prediction (NWP), the German Meteorological Service (Deutscher Wetterdienst, DWD) has initiated a project aimed at assessing data quality and assimilation of profile observations from ground-based remote sensing instruments that have not yet been exploited operationally.The objective of this initiative is to fill the observational gap in the atmospheric boundary layer, especially with respect to short time scales, by providing continuous, high-temporal-resolution profiles of thermodynamic variables, wind, and cloud properties. These observations are expected to be especially beneficial for weather forecasting applications. The DWD is evaluating various remote sensing systems for their ability to provide continuous operational feasibility and impact on NWP. In this contribution, we present results of the assimilation of two ground-based remote sensing instruments into the kilometre-scale ensemble data assimilation system (KENDA): water vapour mixing ratio profiles from a Differential Absorption Lidar (DIAL) and radar reflectivity profiles from a cloud radar. For the integration of the DIAL observations into the data assimilation code environment, only small adjustments were necessary. In contrast, the cloud radar data required an adaptation of the complex forward operator EMVORADO (Efficient Modular Volume scan Radar Operator), which was originally developed and previously used only for precipitation radars.In an initial step, single observation data assimilation experiments and the corresponding observation minus first guess statistics showed promising results. To assess the impact in an operational setting, we performed dedicated data assimilation experiments with and without these additional observations. We considered both summer and winter periods, as well as different observation error specifications for the DIAL measurements. Based on the successful data assimilation cycling experiments, we conducted first forecast experiments, including DIAL water vapour mixing ratio observations. The results indicate a positive impact on humidity and temperature forecasts. We are currently investigating the impact of cloud radar reflectivity data in such experiments. Preliminary results show a neutral to slightly positive impact on the humidity first guess.Our findings suggest that ground-based remote sensing data can provide valuable additional information for convective-scale data assimilation and justify more extensive impact studies in the context of NWP.

Application of ensemble Kalman filter in WRF model to forecast rainfall on monsoon onset period in South Vietnam

To improve the forecast quality of numerical weather prediction (NWP), the German Meteorological Service (Deutscher Wetterdienst, DWD) has initiated a project aimed at assessing data quality and assimilation of observations from ground-based remote sensing instruments that have not yet been exploited operationally.The objective of this initiative is to fill the observational gap in the atmospheric boundary layer, especially with respect to short time scales, by providing continuous, high-temporal-resolution profiles of thermodynamic variables, wind, and cloud properties. These observations are expected to be especially beneficial for nowcasting and (short-term) forecasting applications. The DWD is evaluating various remote sensing systems with regard to the continuous data supply, their operational use and their impact on NWP. In this contribution, we present the integration and assimilation of two such data sources into the kilometer-scale ensemble data assimilation system (KENDA): radar reflectivity from a cloud radar and water vapour mixing ratio from a Differential Absorption Lidar (DIAL). The complex forward operator EMVORADO (Efficient Modular Volume scan Radar Operator), originally developed and previously used only for precipitation radars, has been adapted for cloud radar data. In contrast, the DIAL observations do not require a complex forward operator, and only minor adjustments have been made to the data assimilation code environment of the DWD.Observation minus first guess statistics, as well as first single observation data assimilation experiments have been shown to produce promising results. To assess the overall impact, dedicated data assimilation experiments were conducted and compared to reference experiments without these additional observations. First results indicate a positive impact of the DIAL data on first guess humidity and temperature fields in the analysis cycle, while the impact of cloud radar data appears neutral at this stage. These findings suggest that these ground-based remote sensing data can provide valuable additional information for convective-scale data assimilation and form a sound basis for further impact studies in the context of NWP.

With an artificial example of a 2D nonlinear advection diffusion equation on the unit square this paper considers the data assimilation problem of finding initial values that can evolve into a close approximation to a desired target result at some realistic T > 0. Highly non smooth target data are considered, that may not correspond to actual solutions at time T, and it may not be possible to find such initial values. The aim is to illustrate the inherent difficulties of the ill-posed data assimilation problem.

For the 2D incompressible Navier-Stokes equations, with given hypothetical non smooth data at time 𝑇 > 0 that may not correspond to an actual solution at time 𝑇, a previously developed stabilized backward marching explicit leapfrog finite difference scheme is applied to these data, to find initial values at time 𝑡 = 0 that can evolve into useful approximations to the given data at time 𝑇. That may not always be possible. Similar data assimilation problems, involving other dissipative systems, are of considerable interest in the geophysical sciences, and are commonly solved using computationally intensive methods based on neural networks informed by machine learning. Successful solution of ill-posed timereversed Navier-Stokes equations is limited by uncertainty estimates, based on logarithmic convexity, that place limits on the value of 𝑇 > 0. In computational experiments involving satellite images of hurricanes and other meteorological phenomena, the present method is shown to produce successful solutions at values of 𝑇 > 0, that are several orders of magnitude larger than would be expected, based on the best-known uncertainty estimates. However, unsuccessful examples are also given. The present self-contained paper outlines the stabilizing technique, based on applying a compensating smoothing operator at each time step, and stressesthe important differences between data assimilation, and backward recovery, in ill-posed time reversed problems for dissipative equations. While theorems are stated without proof, the reader is referred to a previous paper, on Navier-Stokes backward recovery, where these proofs can be found.

The unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann–Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrödinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial boundary value problem on a finite interval with x-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle) belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg–de Vries (KdV) and modified KdV equations.

Modern Approaches to Data Assimilation in Ocean Modeling

Four-dimensional variational (4D-Var) data assimilation (DA) experiments using Global Positioning System (GPS)-derived precipitable water vapor (PWV) were conducted for the tropical cyclone (TC) Nargis in 2008. In order to analyze the initial field at 1200 UTC 30 April 2008, 12, 24, 36, and 48 h sequential DA experiments with 3 h assimilation windows were performed. The initial fields made by these DA experiments were applied to subsequent forecast experiments using a nonhydrostatic model (NHM) with a horizontal resolution of 10 km. NHM predictions using initial fields produced by DA experiments that used only ordinary observational data (without GPS PWV) exhibited a large variation of predicted maximum TC intensity (958 to 983 hPa) for each experiment. In these experiments, a longer assimilation period did not necessarily result in better prediction. The DA of GPS PWV yielded a smaller variation of predicted maximum TC intensity (964 to 974 hPa), and a longer assimilation period tended to bring deeper depression of TC central pressure. Overall, TC intensities determined by DA experiments with GPS data were closer to the best track produced by the Regional Specialized Meteorological Centre (RSMC) New Delhi than the DA experiments without GPS data. The 48 h DA without GPS PWV resulted in the weakest prediction of TC development with the deepest TC central pressure of 983 hPa, while 48 h DA with GPS PWV successfully predicted rapid TC development with the deepest pressure of 967 hPa. One cause of the incomplete development of Nargis in the 48 h DA experiment without GPS PWV was insufficient observations in the Bay of Bengal, especially in the first 12 h. Underestimation of precipitation was conspicuous in the first 12 h of the DA. Implementation of GPS PWV into the DA contributed to increasing the precipitation and changed the fields of pressure and wind in the bay. In the first several hours, modifications of the fields of pressure and wind around the Andaman Islands were conspicuous. These affected areas extended with time and created a more favorable environment for TC development.

The Cross‐track Infrared Sounder (CrIS) radiance data plays a crucial role in numerical weather prediction (NWP) models by providing essential atmospheric sounding information through data assimilation. However, challenges arise in handling subpixel cloud contamination within CrIS fields of view (FOVs), which can impact the accuracy of radiance simulations. To address this, the Visible Infrared Imaging Radiometer Suite (VIIRS) Radiances Cluster analysis within the CrIS FOVs is developed to characterize subpixel scene homogeneity. This paper describes the algorithms and data processing procedures for this cluster analysis. A fast and accurate collocation method was developed to directly align VIIRS measurements within CrIS FOVs using line‐of‐sight (LOS) pointing vectors. This method supports both terrain‐corrected and non‐terrain‐corrected VIIRS geolocation data sets as inputs. The K‐means clustering method is used to group collocated VIIRS radiance within CrIS FOVs into seven (7) clusters based on their radiance values. The mean, standard deviation, and coverage of each cluster are output for each CrIS FOV. Comparisons with the Infrared Atmospheric Sounding Interferometer cluster analysis demonstrate similar performance, confirming the validity of the CrIS‐VIIRS approach. Data assimilation experiments at the European Centre for Medium‐Range Weather Forecasts indicate that the VIIRS radiance cluster data can be effectively integrated into NWP models, aiding in cloud detection and improving data quality. These findings highlight the potential of CrIS‐VIIRS clustering for enhancing data thinning, quality control, and assimilation of cloudy radiance observations in operational NWP systems.

ABSTRACTThis paper constructs an unconditionally stable explicit difference scheme, marching backward in time, that can solve a limited, but important class of time-reversed 2D Burgers' initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. This leads to a distortion away from the true solution. However, in many interesting cases, the cumulative error is sufficiently small to allow for useful results. Effective smoothing operators based on , with real p>2, can be efficiently synthesized using FFT algorithms, and this may be feasible even in non-rectangular regions. Similar stabilizing techniques were successfully applied in other ill-posed evolution equations. The analysis of numerical stability is restricted to a related linear problem. However, extensive numerical experiments indicate that such linear stability results remain valid when the explicit scheme is applied to a significant class of time-reversed nonlinear 2D Burgers' initial value problems. As illustrative examples, the paper uses fictitiously blurred pixel images, obtained by using sharp images as initial values in well-posed, forward 2D Burgers' equations. Such images are associated with highly irregular underlying intensity data that can seriously challenge ill-posed reconstruction procedures. The stabilized explicit scheme, applied to the time-reversed 2D Burgers' equation, is then used to deblur these images. Examples involving simpler data are also studied. Successful recovery from severely distorted data is shown to be possible, even at high Reynolds numbers.

Solution of nonlinear partial differential equations from base equations

A weak constraint inverse for a zero-dimensional marine ecosystem model

Based on the GRAPES (Global/Regional Assimilation and Prediction System) regional ensemble prediction system and 3DVAR (three-dimensional variational) data assimilation system, which are implemented operationally at the Numerical Weather Prediction Center of the China Meteorological Administration, an ensemble-based 3DVAR (En-3DVAR) hybrid data assimilation system for GRAPES_Meso (the regional mesoscale numerical prediction system of GRAPES) was developed by using the extended control variable technique to implement a hybrid background error covariance that combines the climatological covariance and ensemble-estimated covariance. Considering the problems of the ensemble-based data assimilation part of the system, including the reduction in the degree of geostrophic balance between variables, and the non-smooth analysis increment and its obviously smaller size compared with the 3DVAR data assimilation, corresponding measures were taken to optimize and ameliorate the system. Accordingly, a single pressure observation ensemble-based data assimilation experiment was conducted to ensure that the ensemble-based data assimilation part of the system is correct and reasonable. A number of localization-scale sensitivity tests of the ensemble-based data assimilation were also conducted to determine the most appropriate localization scale. Then, a number of hybrid data assimilation experiments were carried out. The results showed that it was most appropriate to set the weight factor of the ensemble-estimated covariance in the experiments to be 0.8. Compared with the 3DVAR data assimilation, the geopotential height forecast of the hybrid data assimilation experiments improved very little, but the wind forecast improved slightly at each forecast time, especially over 300 hPa. Overall, the hybrid data assimilation demonstrates some advantages over the 3DVAR data assimilation.