Space-time Interpolation Research Articles

4DVarNet-SSH: end-to-end learning of variational interpolation schemes for nadir and wide-swath satellite altimetry

Abstract. The reconstruction of sea surface currents from satellite altimeter data is a key challenge in spatial oceanography, especially with the upcoming wide-swath SWOT (Surface Water and Ocean and Topography) altimeter mission. Operational systems, however, generally fail to retrieve mesoscale dynamics for horizontal scales below 100 km and timescales below 10 d. Here, we address this challenge through the 4DVarnet framework, an end-to-end neural scheme backed on a variational data assimilation formulation. We introduce a parameterization of the 4DVarNet scheme dedicated to the space–time interpolation of satellite altimeter data. Within an observing system simulation experiment (NATL60), we demonstrate the relevance of the proposed approach, both for nadir and nadir plus SWOT altimeter configurations for two contrasting case study regions in terms of upper ocean dynamics. We report a relative improvement with respect to the operational optimal interpolation between 30 % and 60 % in terms of the reconstruction error. Interestingly, for the nadir plus SWOT altimeter configuration, we reach resolved space–timescales below 70 km and 7 d. The code is open source to enable reproducibility and future collaborative developments. Beyond its applicability to large-scale domains, we also address the uncertainty quantification issues and generalization properties of the proposed learning setting. We discuss further future research avenues and extensions to other ocean data assimilation and space oceanography challenges.

A non-intrusive space-time interpolation from compact Stiefel manifolds of parametrized rigid-viscoplastic FEM problems

This work aims to interpolate parametrized reduced order model (ROM) basis constructed via the proper orthogonal decomposition (POD) to derive a robust ROM of the system’s dynamics for an unseen target parameter value. A novel non-intrusive space-time (ST) POD basis interpolation scheme is proposed, for which we define ROM spatial and temporal basis curves on compact Stiefel manifolds. An interpolation is finally defined on a mixed part encoded in a square matrix directly deduced using the spacial part, the singular values and the temporal part, to obtain an interpolated snapshot matrix, keeping track of accurate space and temporal eigenvectors. Moreover, in order to establish a well-defined curve on the compact Stiefel manifold, we introduce a new procedure, the so-called oriented SVD. Such an oriented SVD produces unique right and left eigenvectors for generic matrices, for which all singular values are distinct. It is important to notice that the ST POD basis interpolation does not require the construction and the subsequent solution of a reduced-order FEM model as classically is done. Hence it is avoiding the bottleneck of standard POD interpolation which is associated with the evaluation of the nonlinear terms of the Galerkin projection on the governing equations. As a proof of concept, the proposed method is demonstrated with the adaptation of rigid-thermoviscoplastic finite element ROMs applied to a typical nonlinear open forging metal forming process. Strong correlations of the ST POD models with respect to their associated high-fidelity FEM counterpart simulations are reported, highlighting its potential use for near real-time parametric simulations using off-line computed ROM POD databases.

END-TO-END PHYSICS-INFORMED REPRESENTATION LEARNING FOR SATELLITE OCEAN REMOTE SENSING DATA: APPLICATIONS TO SATELLITE ALTIMETRY AND SEA SURFACE CURRENTS

Abstract. This paper addresses physics-informed deep learning schemes for satellite ocean remote sensing data. Such observation datasets are characterized by the irregular space-time sampling of the ocean surface due to sensors’ characteristics and satellite orbits. With a focus on satellite altimetry, we show that end-to-end learning schemes based on variational formulations provide new means to explore and exploit such observation datasets. Through Observing System Simulation Experiments (OSSE) using numerical ocean simulations and real nadir and wide-swath altimeter sampling patterns, we demonstrate their relevance w.r.t. state-of-the-art and operational methods for space-time interpolation and short-term forecasting issues. We also stress and discuss how they could contribute to the design and calibration of ocean observing systems.

Data gaps and degraded space-time resolution for modal decomposition: A compensator approach

High-resolution empirical mapping methods of dynamic oscillation responses for large power systems have recently been considered to treat the data gaps and its space-time compensation. Although virtually-designed data of signals are not a real condition for monitoring systems, these can be implemented to study objectively the moving features and dominant paths of inter-area oscillation modes. This paper demonstrates the application of the space-time interpolation of wide-area measurements to compensate data gaps and degraded space-time resolution for modal decomposition. Contributions of empirical mode coherence for splitting and clustering data in density factors are also presented to the classification area. The method involves the space-time high-resolution interpolation of empirical mode components for splitting closely-related oscillation modes at intrinsic timescales, preserving their multivariate waveform features. Therefore, high-resolution empirical viewing maps of remotely-sensed dynamic responses, with degraded space-time resolutions, are derived to study the observance and dominant paths of wide-area oscillation modes. The proposed approach is numerically applied to the IEEE 16-generator 68-bus test system for efficiently defining empirical viewing maps of inter-area oscillation modes.

Acoustic wave three-dimensional ground reflection and bouncing

This paper deals with the development of a noise propagation tool for the prediction of perceived noise due to aircraft/rotorcraft flying in a non-uniform atmosphere over a non-flat ground. The far-field noise is predicted through a direct ray tracing method, once the near-field aeroacoustic solution, used to define the aircraft/rotorcraft equivalent noise source, is known from a near-field aeroacoustic solver. The atmospheric effects are taken into account in the ray tracing algorithm through standard formulations. A new algorithm capable of taking into account the realistic orography and terrain properties handles the interaction between sound rays and ground. The ground orography is automatically generated by exploiting the open-source Application Programming Interface Open-Elevation, whereas the soil composition is defined using the CORINE Land Cover inventory. The new algorithm, applying the ground absorption on each ray independently and moving the waves interference evaluation to a post-processing space–time interpolation process, allows the inclusion of the contributions of multiple-reflecting rays in the evaluation of the receiver noise. The numerical investigation is aimed at validating the proposed tool against a Fast Field Program solver, as well as at assessing the effects of the realistic description of the terrain on the predictions of perceived noise.

Short-range clutter suppression method combining oblique projection and interpolation in airborne CFA radar

The airborne conformai array (CFA) radar's clutter ridges are range-modulated, which result in a bias in the estimation of the clutter covariance matrix (CCM) of the cell under test (CUT), further, reducing the clutter suppression performance of the airborne CFA radar. The clutter ridges can be effectively compensated by the space-time separation interpolation (STS-INT) method, which costs less computation than the space-time interpolation (STINT) method, but the performance of interpolation algorithms is seriously affected by the short-range clutter, especially near the platform height. Location distributions of CFA are free, which yields serious impact that range spaces of steering vector matrices are non-orthogonal complement and even no longer disjoint. Further, a new method is proposed that the short-range clutter is pre-processed by oblique projection with the intersected range spaces (OPIRS), and then clutter data after being pre-processed are compensated to the desired range bin through the STSINT method. The OPIRS also has good compatibility and can be used in combination with many existing methods. At the same time, oblique projectors of OPIRS can be obtained in advance, so the proposed method has almost the same computational load as the traditional compensation method. In addition, the proposed method can perform well when the channel error exists. Computer simulation results verify the effectiveness of the proposed method.

Minimum temperature mapping with spatial copula interpolation

Monitoring of variables like temperature, precipitation, and air quality is performed to determine their current situation, exhibit the presence of trends and occurrence of outliers. These variables are measured at specific locations and to obtain a full estimation map, we need to predict values at unknown locations. This study focuses on making a minimum air temperature map using copula interpolation with the spline family. Minimum temperature observations for January 2017, all months of 2017, and seasonal averages are analysed over the Euphrates Basin in Turkey. The minimum temperature observations have a high variability due to the varying topography of the area, ranging between -2 C∘ and +14 C∘ in whole of 2017. The interpolation methods incorporated the above mean sea level elevation map and remotely-sensed land surface temperature. We evaluated the accuracy of the predictions using ten-fold cross-validation and compare copula interpolation with External Drift Kriging (KED). The study shows that copulas provided more accurate predictions than KED for most of the months, and for the summer and autumn seasons, whereas KED produced high accurate predictions for January 2017. Results of this study indicate that copulas are able to detect variation in minimum temperatures accurately in areas where topography and observation values are highly variable. Further work will focus on copula-based space–time interpolation techniques for the minimum temperature mapping.

Optimal Simulation of Wind Field under Disaster Conditions based on PSO and Entropic Lattice Boltzmann Method

The entropic lattice Boltzmann method (ELBM) is an excellent method of numerical stability among of different versions of the lattice Boltzmann method for the simulation of hydrodynamics, especially for wind field simulation application of power grid disaster conditions. In this paper, an efficient improved particle swarm optimization (PSO) algorithm is studied for optimizing calculation parameters to achieve load balancing of ELBM on nonuniform grids in a heterogeneous computing platform. We also introduce a new concept of multi-block ELBM on composite grids for realization of the ELBM simulations of incompressible driven cavity flow. These new approaches rely on a two-dimensional space-time interpolation and solving the relaxation time parameter by direct approximation optimization strategy to guarantee conservation. Our CPU–GPU implementation of multi-block ELBM based on the improved PSO algorithm not only exploits adequately multi-core CPU computing resources for load balancing, but also follows carefully optimized storage to increase coalesced access on a GPU platform. The three-dimensional-driven cavity flow simulations validate the proposed multi-block ELBM even with severely under-resolved grids. In addition, some performance metrics are investigated based on the implementations of different refined grids and threading blocks. These results exhibit the improved PSO algorithm of the ELBM method which can optimize computing resource parameters in heterogeneous platforms, and the present multi-block ELBM can substantially improve the accuracy and computational efficiency for viscous flow computations.

A method of establishing an instantaneous water level model for tide correction

Tide correction is important in ship-board bathymetric data. Currently, most tide correction algorithms use a space-time interpolation method to regain the composite sea surface morphology. These interpolation algorithms are mostly based on geometric interpolation. However, when the tide stations are insufficient in the survey region, the spatial pattern of tide change may not comply with the geometric trend. Some of the algorithms use tide simulation to obtain a space-time astronomical tide model for tide correction in tide-station-insufficient regions, because the instantaneous water level changes are mainly caused by astronomical tides in normal conditions. However, in some cases, the instantaneous water level effect of the short-period stochastic meteorological factors results in short-term water level anomalies, which can be difficult to simulate using a tide simulation method. Thus, in this paper, an instantaneous water level model for tide correction in tide-station-insufficient regions is proposed. The model includes the simulation of astronomical tide and deviation-tidal components. We first simulate the astronomical tidal model using a two-dimensional MIKE21 Flow Model. Then, we propose a deviation correction method to mitigate the deviation-tidal components. Using the revised instantaneous water level model, we present the instantaneous tide correction (ITC) algorithm. Then, we compare the ITC algorithm with the two commonly used algorithms of Discrete Tidal Zoning and TCARI. The results show that the ITC algorithm is superior to the common algorithms in terms of accuracy and applicability with respect to a tide-station-insufficient survey region. Our conclusion is that ITC algorithm is a feasible tide correction algorithm when tide stations are not sufficiently measured.

A space–time parallel framework for fine-scale visualization of pollen levels across the Eastern United States

ABSTRACTAllergic rhinitis (hay fever) resulting from seasonal pollen affects 15–30% of the population in the United States, and can exacerbate several related conditions, including asthma, atopic eczema, and allergic conjunctivitis. Timely monitoring, accurate prediction, and visualization of pollen levels are critical for public health prevention purposes, such as limiting outdoor exposure or physical activity. The low density of pollen detecting stations and complex movement of pollen represent a challenge for accurate prediction and modeling. In this paper, we reconstruct the dynamics of pollen variation across the Eastern United States for 2016 using space–time interpolation. Pollen levels were extracted according to a stratified spatial sampling design, augmented by additional samples in densely populated areas. These measurements were then used to estimate the space–time cross-correlation, inferring optimal spatial and temporal ranges to calibrate the space–time interpolation. Given the computational requirements of the interpolation algorithm, we implement a spatiotemporal domain decomposition algorithm, and use parallel computing to reduce the computational burden. We visualize our results in a 3D environment to identify the seasonal dynamics of pollen levels. Our approach is also portable to analyze other large space–time explicit datasets, such as air pollution, ash clouds, and precipitation.

A spatiotemporal regression-kriging model for space-time interpolation: a case study of chlorophyll-a prediction in the coastal areas of Zhejiang, China

ABSTRACTSpatiotemporal kriging (STK) is recognized as a fundamental space-time prediction method in geo-statistics. Spatiotemporal regression kriging (STRK), which combines space-time regression with STK of the regression residuals, is widely used in various fields, due to its ability to take into account both the external covariate information and spatiotemporal autocorrelation in the sample data. To handle the spatiotemporal non-stationary relationship in the trend component of STRK, this paper extends conventional STRK to incorporate it with an improved geographically and temporally weighted regression (I-GTWR) model. A new geo-statistical model, named geographically and temporally weighted regression spatiotemporal kriging (GTWR-STK), is proposed based on the decomposition of deterministic trend and stochastic residual components. To assess the efficacy of our method, a case study of chlorophyll-a (Chl-a) prediction in the coastal areas of Zhejiang, China, for the years 2002 to 2015 was carried out. The results show that the presented method generated reliable results that outperform the GTWR, geographically and temporally weighted regression kriging (GTWR-K) and spatiotemporal ordinary kriging (STOK) models. In addition, employing the optimal spatiotemporal distance obtained by I-GTWR calibration to fit the spatiotemporal variograms of residual mapping is confirmed to be feasible, and it considerably simplifies the residual estimation of STK interpolation.

Implementation of a Parallel GPU-Based Space-Time Kriging Framework

In the study of spatiotemporal geographical phenomena, the space–time interpolation method is widely applied, and the demands for computing speed and accuracy are increasing. For nonprofessional modelers, utilizing the space–time interpolation method quickly is a challenge. To solve this problem, the classical ordinary kriging algorithm was selected and expanded to a spatiotemporal kriging algorithm. Using the OpenCL framework to integrate central processing unit (CPU) and graphic processing unit (GPU) computing resources, a parallel spatiotemporal kriging algorithm was implemented, and three experiments were conducted in this work to verify the results. The results indicated the following: (1) when the size of the prediction point dataset is consistent, the performance of the method is robust with the increasing size of the observation point dataset; (2) the acceleration effect of the parallel method increases with an increased number of predicted points. Compared with the original sequential program, the implementation of the improved parallel framework showed a 3.23 speedup, which obviously shortens the interpolation time; (3) when cross-validating the temperature data in the Beijing Tianjin Hebei region, the space–time acceleration model provides a better fit than traditional pure space interpolation.

Finely Resolved On-Road PM2.5 and Estimated Premature Mortality in Central North Carolina.

To quantify the on-road PM2.5 -related premature mortality at a national scale, previous approaches to estimate concentrations at a 12-km × 12-km or larger grid cell resolution may not fully characterize concentration hotspots that occur near roadways and thus the areas of highest risk. Spatially resolved concentration estimates from on-road emissions to capture these hotspots may improve characterization of the associated risk, but are rarely used for estimating premature mortality. In this study, we compared the on-road PM2.5 -related premature mortality in central North Carolina with two different concentration estimation approaches-(i) using the Community Multiscale Air Quality (CMAQ) model to model concentration at a coarser resolution of a 36-km × 36-km grid resolution, and (ii) using a hybrid of a Gaussian dispersion model, CMAQ, and a space-time interpolation technique to provide annual average PM2.5 concentrations at a Census-block level (∼105,000 Census blocks). The hybrid modeling approach estimated 24% more on-road PM2.5 -related premature mortality than CMAQ. The major difference is from the primary on-road PM2.5 where the hybrid approach estimated 2.5 times more primary on-road PM2.5 -related premature mortality than CMAQ due to predicted exposure hotspots near roadways that coincide with high population areas. The results show that 72% of primary on-road PM2.5 premature mortality occurs within 1,000 m from roadways where 50% of the total population resides, highlighting the importance to characterize near-road primary PM2.5 and suggesting that previous studies may have underestimated premature mortality due to PM2.5 from traffic-related emissions.

A Hybrid Method for Interpolating Missing Data in Heterogeneous Spatio-Temporal Datasets

Space-time interpolation is widely used to estimate missing or unobserved values in a dataset integrating both spatial and temporal records. Although space-time interpolation plays a key role in space-time modeling, existing methods were mainly developed for space-time processes that exhibit stationarity in space and time. It is still challenging to model heterogeneity of space-time data in the interpolation model. To overcome this limitation, in this study, a novel space-time interpolation method considering both spatial and temporal heterogeneity is developed for estimating missing data in space-time datasets. The interpolation operation is first implemented in spatial and temporal dimensions. Heterogeneous covariance functions are constructed to obtain the best linear unbiased estimates in spatial and temporal dimensions. Spatial and temporal correlations are then considered to combine the interpolation results in spatial and temporal dimensions to estimate the missing data. The proposed method is tested on annual average temperature and precipitation data in China (1984–2009). Experimental results show that, for these datasets, the proposed method outperforms three state-of-the-art methods—e.g., spatio-temporal kriging, spatio-temporal inverse distance weighting, and point estimation model of biased hospitals-based area disease estimation methods.

Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities

In this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\rho_0 \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.

IRSeL—An approach to enhance continuity and accuracy of remotely sensed land cover data

Land cover data gives the opportunity to study interactions between land cover status and environmental issues such as hydrologic processes, soil properties, or biodiversity. Land cover data often are based on classification of remote sensing data that seldom provides the requisite accuracy, spatial availability and temporal observational frequency for environmental studies. Thus, there is a high demand for accurate and spatio-temporal complete time series of land cover. In the past considerable research was undertaken to increase land cover classification accuracy, while less effort was spent on interpolation techniques. The purpose of this article is to present a space–time interpolation and revision approach for remotely sensed land cover data. The approach leverages special properties known for agricultural areas such as crop rotations or temporally static land cover classes. The newly developed IRSeL-tool (Interpolation and improvement of Remotely Sensed Land cover) corrects classification errors and interpolates missing land cover pixels. The easy-to-use tool solely requires an initial land cover data set. The IRSeL specific interpolation and revision technique, the data input requirements and data output structure are described in detail. A case study in an area around the city of Neumünster in Northern Germany from 2006 to 2012 was performed for IRSeL validation with initial land cover data sets (Landsat TM image classifications) for the years 2006, 2007, 2009, 2010 and 2011. The results of the case study showed that IRSeL performs well; including years with no classification data overall accuracy values for IRSeL interpolated pixels range from 0.63 to 0.81. IRSeL application significantly increases the accuracy of the land cover data; overall accuracy values rise 0.08 in average resulting in overall accuracy values of at least 0.86. Considering estimated reliabilities, the IRSeL tool provides a temporally and spatially completed and revised land cover data set that allows drawing conclusions for land cover related studies.

Interpolation methods and the accuracy of lattice-Boltzmann mesh refinement

A lattice-Boltzmann model to solve the equivalent of the Navier–Stokes equations on adaptively refined grids is presented. A method for transferring information across interfaces between different grid resolutions was developed following established techniques for finite-volume representations. This new approach relies on a space–time interpolation and solving constrained least-squares problems to ensure conservation. The effectiveness of this method at maintaining the second order accuracy of lattice-Boltzmann is demonstrated through a series of benchmark simulations and detailed mesh refinement studies. These results exhibit smaller solution errors and improved convergence when compared with similar approaches relying only on spatial interpolation. Examples highlighting the mesh adaptivity of this method are also provided.

Searching for an energy source of the intensification of tropical cyclone Katrina using microwave satellite sensing data

The possibility of utilizing a specialized approach to the space-time interpolation of satellite data that takes the kinematics of multiscale processes into account in solving the problems of remote sensing of fast processes in the Earth’s atmosphere is considered. The approach is borrowed from the rapidly progressing disciplines of machine vision and video data compression. The main advantages of this method are its completeness with respect to the input (interpolated) satellite data, its capacity to estimate the kinematics of multiscale processes and take it into account, a relatively low computational cost, and the ease of its software-algorithmic implementation. The problem of an “instantaneous” source of hidden energy for the intensification of Tropical Cyclone (TC) Katrina is solved based on the developed software-algorithmic approach using data from the SSM/I satellite radiometers of the DMSP mission.

An approach to suppress short-range clutter for non-side looking airborne radar

When the Airborne Early Warning (AEW) radar transmits medial or high Pulse Repetitive Frequency (PRF) signal, the range ambiguity occurs. The clutter of short-range clutter has serious range dependence problem for non-Side Looking Airborne Radar (non-SLAR). As a result, the clutter plus noise covariance matrix can not be estimated correctly, and the performance of clutter suppression obtained by Space-Time Adaptive Processing (STAP) degrades greatly. The uniform linear array has not elevation degrees; therefore, the short-range clutter can not be suppressed directly. A short-range clutter suppression method is proposed. The method first estimate the elevation angles of the ambiguous short-range gate, then eliminates short-range clutter by space time interpolation and adds moving target protection in the procedure. This method can suppress the short-range clutter well. Simulation results show the validity of the method.

An ensemble approach to space–time interpolation

The availability of spatial data on an unprecedented scale as well as advancements in analytical and visualization techniques gives researchers the opportunity to study complex problems over large urban and regional areas. Nevertheless, few individual data sets exist that provide both the requisite spatial and/or temporal observational frequency to truly facilitate detailed investigations. Some data are collected frequently over time but only at a few geographic locations (e.g., weather stations). Similarly, other data are collected with a high level of spatial resolution but not at regular or frequent time intervals (e.g., satellite data). The purpose of this article is to present an interpolation approach that leverages the relative temporal richness of one data set with the relative spatial richness of another to fill in the gaps. Because different interpolation techniques are more appropriate than others for specific types of data, we propose a space–time interpolation approach whereby two interpolation methods – one for the temporal and one for the spatial dimension – are used in tandem to increase the accuracy results. We call our ensemble approach the space–time interpolation environment (STIE). The primary steps within this environment include a spatial interpolation processor, a temporal interpolation processor, and a calibration processor, which enforces phenomenon-related behavioral constraints. The specific interpolation techniques used within the STIE can be chosen on the basis of suitability for the data and application at hand. In this article, we first describe STIE conceptually including the data input requirements, output structure, details of the primary steps, and the mechanism for coordinating the data within those steps. We then describe a case study focusing on urban land cover in Phoenix, Arizona, using our working implementation. Our empirical results show that our approach increased the accuracy for estimating urban land cover better than a single interpolation technique.