Optimal Interpolation Algorithm Research Articles

Deep learning for the super resolution of Mediterranean sea surface temperature fields

Abstract. Sea surface temperature (SST) is one of the essential variables of the Earth's climate system. Being at the air–sea interface, SST modulates heat fluxes in and out of the ocean, provides insight into several upper and interior ocean dynamical processes, and is a fundamental indicator of climate variability potentially impacting the health of marine ecosystems. Its accurate estimation and regular monitoring from space is therefore crucial. However, even if satellite infrared/microwave measurements provide much better coverage than what is achievable from in situ platforms, they cannot sense the sea surface under cloudy and rainy conditions. Large gaps are present even in merged multi-sensor satellite products, and different statistical strategies, mostly based on optimal interpolation (OI) algorithms, have thus been proposed to obtain gap-free (L4) images. These techniques, however, filter out the signals below the space–time decorrelation scales considered, significantly smoothing most of the small mesoscale and submesoscale features. Here, deep learning models, originally designed for single-image super resolution (SR), are applied to enhance the effective resolution of SST products and the accuracy of SST gradients. SR schemes include a set of computer vision techniques leveraging convolutional neural networks to retrieve high-resolution data from low-resolution images. A dilated convolutional multi-scale learning network, which includes an adaptive residual strategy and implements a channel attention mechanism, is used to reconstruct features in SST data at 1/100° spatial resolution starting from 1/16° data over the Mediterranean Sea. The application of this technique shows an improvement in the high-resolution reconstruction, capturing small-scale features and providing a root-mean-squared-difference improvement of 0.02 °C with respect to the L3 ground-truth data.

Fusion of Lagrangian drifter data and numerical model outputs for improved assessment of turbulent dispersion

Abstract. Transport and dispersion processes in the ocean are crucial, as they determine the lifetime and fate of biological and chemical quantities drifting with ocean currents. Due to the complexity of the coastal ocean environment, numerical circulation models have difficulties to accurately simulate highly turbulent flows and dispersion processes, especially in highly energetic tidal basins such as the eastern English Channel. A method of improving the results of coastal circulation modeling and tracer dispersion in the Dover Strait is proposed. Surface current velocities derived from Lagrangian drifter measurements in November 2020 and May 2021 were optimally interpolated in time and space to constrain a high-resolution coastal circulation MARS model, with careful attention given to selecting ensemble members composing the model covariance matrix. The space–time velocity covariances derived from model simulations were utilized by the optimal interpolation algorithm to determine the most likely evolution of the velocity field under constraints provided by Lagrangian observations and their error statistics. The accuracy of the velocity field reconstruction was evaluated at each time step. The results of the fusion of model outputs with surface drifter velocity measurements show a significant improvement (by ∼ 50 %) of the model capability to simulate the drift of passive tracers in the Dover Strait. Optimized velocity fields were used to quantify the absolute dispersion in the study area. The implications of these results are important, as they can be used to improve existing decision-making support tool or design new tools for monitoring the transport and dispersion in a coastal ocean environment.

Geotechnical investigation measures and techniques in construction engineering based on multi-source data fusion

Abstract Geotechnical investigation is an important aspect of construction engineering. In order to optimize this link, this paper analyzes the current status of the application of geotechnical investigation in construction engineering through the fusion of multidimensional spatial data interpolation of geotechnical geology as well as the structural design of geotechnical profiles by combining with triangular mesh search algorithm. On this basis, the parameter prediction of the fusion of multi-source parameter data in geotechnicals was carried out by determining the estimated variance and weight coefficients. In addition, the optimal interpolation algorithm was selected through the indoor test and in-situ test, and the accuracy of the modeling of the soil and the prediction of the geotechnical parameters were analyzed. The results show that the modeling width error is between [0.04,0.20], the length error is between [0.7,0.23], the planar accuracy error is around 0.2cm, the depth error is within 0.02cm, and the errors of the actual and predicted values of the rock and map parameters are within 0.1 and 0.09, respectively. Geotechnical investigation measures and techniques in construction engineering based on multi-source data fusion can optimize construction engineering.

Image-Like Situational Difference Mapping (ILSDM)-Based Method for Detecting Anomalies in Electromagnetic Environment

Electromagnetic environment anomaly radiation source detection (EMEARD) can provide a basis for electromagnetic (EM) equipment behavior cognition and battlefield threat assessment. Existing methods mainly focus on generating EM space situations but ignore the research on anomaly signal detection. In this paper, the connections between information geometry and performance of sensor networks for anomaly detection are explored. Nonlinear estimation methods based on information geometry can be directly used for anomaly detection. However, Riemann distance as a metric function faces the disadvantages of extensive computation and high complexity. To tackle this issue here, we propose a method based on image-like situation difference mapping, namely ILSDM. which takes the data-driven strategy to obtain the analytic expression of situation difference mapping distance function (DMDF), and anomaly detection is equivalent to the mapping reconstruction of information geometry. Firstly, the EM space situation is acquired through the distributed sensor network. In addition, to solve the perceptual blind area problem, the sensors’ data is estimated by spatial autocovariance optimal interpolation algorithm (Kriging). Secondly, we apply the Gaussian mixture model (GMM) to fit the statistical features of each sensing node into probability density function (PDF), and adjacent sensors with similar PDF are fused by the neighbor average fusion method (NAF). Then, based on the information geometry, the fused PDFs are mapped to the Riemannian manifold space as a series of coordinates, and we further propose to use image-like situation (ILS) to represented the EM space situation. Finally, DMDF measures the information distance between different ILSs to achieve anomaly detection. Numerical simulations and real experiments were conducted to prove the superiority and feasibility of our method for anomaly detection in EM space.

An adaptive infrared image denoising method based on two-dimensional empirical mode decomposition for distribution network inspection UAV

An adaptive denoising method based on 2D empirical mode decomposition (EMD) is proposed to improve the infrared image quality of inspection UNMANNED aerial vehicles (UAVs) and provide guarantee for improving the inspection level of distribution network. Through rapid adaptive two-dimensional empirical mode decomposition algorithm decomposition of a UAV collected for distribution network inspection original noise of infrared image, get more than the IMF component and the residual amount, a forecast noise dominated the IMF component parameters such as threshold value and the variance of noise, using the estimated parameters in combination with the optimal linear interpolation algorithm of noise threshold function of leading the IMF component implementation of threshold denoising. After the denoised IMF component is obtained, the denoised infrared image is obtained after reconstruction with the signal-dominated IMF component, and the adaptive denoising of the infrared image of the distribution network inspection UAV is realized. The experimental results show that the method in this paper can maintain the details of the image, improve the definition, significantly improve the visual effect, the overall denoising performance is stable and feasible, and ensure the quality inspection UAV to collect infrared images.

Optimal linear interpolation algorithm with fixed delay

One of the most important characteristics of satellite navigation receiving equipment is the accuracy of navigation definitions. To improve this characteristic, the paper considers the possibility of using optimal interpolation algorithms with a fixed delay. A new interpolation algorithm with a fixed delay in discrete time was deduced. The use of the proposed interpolation algorithm instead of filtering made it possible to reduce the standard deviation by 2 times with a delay in the output of the result by 10 cycles. The advantage of the proposed algorithm is the reduction in computational complexity compared to the optimal two-way algorithm. The proposed algorithm requires 2 matrix accesses per cycle of operation, subject to the use of additional memory resources, to save pre-calculated elements.

Assessing Sea Surface Temperatures Estimated from Fused Infrared and Microwave Data

Sea surface temperature (SST), a critical parameter of the global ocean–atmosphere system, is an essential element in the study and in the application of marine science. Satellite–infrared observations currently represent the only available method for continuous, large-scale observation of SST. Although passive microwave observations are not blocked by clouds, allowing for data collection in all weather conditions, this technological tool is characterized by low spatial resolution. Conversely, infrared observations offer high resolution but are susceptible to cloud obscuration. Accordingly, a technique that effectively fuses microwave and infrared satellite observations into a high-resolution SST field with global coverage close to the actual distribution is of practical significance. This paper describes fusing MODIS infrared remote sensing and AMSR-2 microwave remote sensing SST data with an optimal interpolation (OI) approach to produce a high-resolution SST data. The study chose the coastal Kuroshio region of China to establish an appropriate scale for examining the spatial structure of SST and attaining a more realistic picture of SST observations and impacts. The included discussion of the sources of error in the fusion process provides a reference for improving the accuracy of fused marine remote sensing data. The study also compared the fused SST results and the current international mainstream multi-temporal resolution of the three using the OI algorithm. We compared the fusion product with ARGO data with and without typhoon impact to explore and practice the OI in SST fusion when evaluating the accuracy of different data in the case of external disturbance being present. The research results have great significance for improving regional SST forecast accuracy while ensuring the applicability of various approaches to fusing SST data by incorporating the influence of typhoons in the offshore region of the East China Sea (ECS). Implications for the future development of SST fusion data are also included in the discussion.

Pattern Reconstruction of 3-D Modular Antennas by Means of a Non-Redundant Near-Field Spherical Scan

A very flexible source model is proposed here to reduce the volumetric redundancy when considering the pattern reconstruction of three-dimensional modular antennas by means of a near-field spherical scan using a non-redundant sampling representation. Since this last facet is based on the appropriate choice of antenna model for the evaluation of the optimal parameters to be used, the proposed geometry guaranteed the minimum number of needed samples and then a significant time saved for data acquisition on the near-field spherical grid. Then, an optimal interpolation algorithm used these non-redundant samples for an accurate evaluation of the near-field data that were usable in the classical near-field to far-field transformation. The reliability and accuracy of the reconstruction process were proven by means of numerical tests. These last showed a remarkable reduction (about 53%) in needed near-field samples as compared to those required by the classical near-field to far-field transformation and this was achieved without any loss in accuracy.

Including parameterized error covariance in local ensemble solvers: Experiments in a 1D model with balance constraints

AbstractLack of efficient ways to include parameterized error covariance in ensemble‐based local volume solvers (e.g. the local ensemble‐transform Kalman filter – the LETKF) remains an outstanding problem in data assimilation. Here, we describe two new algorithms: GETKF‐OI and LETKF‐OI. These algorithms are similar to the traditional optimal interpolation (OI) algorithm in that they use parameterized error covariance to update each of the local volume solutions. However, unlike the traditional OI that scales poorly as the number of observations increases, the new algorithms achieve linear scalability by using either the observational‐space localization strategy of the traditional LETKF algorithm or the modulated ensembles of the gain‐form (GETKF) algorithm. In our testing with a simple one‐dimensional univariate system, we find that the GETKF‐OI algorithm can recover the exact solution within the truncation bounds of the modulated ensemble and the LETKF‐OI algorithm achieves a close approximation to the exact solution. We also demonstrate how to extend GETKF‐OI algorithm to a toy multivariate system with balance constraints.

Retrieving Mediterranean Sea Surface Salinity Distribution and Interannual Trends from Multi-Sensor Satellite and In Situ Data

Sea surface salinity (SSS) is one of the Essential Climate Variables (ECVs), defined by the Global Climate Observing System (GCOS). Salinity is modified by river discharge, land run-off, precipitation, and evaporation, and it is advected by oceanic currents. In turn, ocean circulation, the water cycle, and biogeochemistry are deeply impacted by salinity variations. The Mediterranean Sea represents a hot spot for the variability of salinity. Despite the ever-increasing number of moorings and floating buoys, in situ SSS estimates have low coverage, hindering the monitoring of SSS patterns. Conversely, satellite sensors provide SSS surface data at high spatial and temporal resolution, complementing the sparseness of in situ datasets. Here, we describe a multidimensional optimal interpolation algorithm, specifically configured to provide a new daily SSS dataset at 1/16° grid resolution, covering the entire Mediterranean Sea (Med L4 SSS). The main improvements in this regional algorithm are: the ingestion of satellite SSS estimates from multiple satellite missions (NASA’s Soil Moisture Active Passive (SMAP), ESA’s Soil Moisture and Ocean Salinity (SMOS) satellites), and a new background (first guess), specifically built to improve coastal reconstructions. The multi-sensor Med L4 SSS fields have been validated against independent in situ SSS samples, collected between 2010–2020. They have also been compared with global weekly Copernicus Marine Environment Monitoring Service (CMEMS) and Barcelona Expert Centre (BEC) regional products, showing an improved performance. Power spectral density analyses demonstrated that the Med L4 SSS field achieves the highest effective spatial resolution, among all the datasets analysed. Even if the time series is relatively short, a clear interannual trend is found, leading to a marked salinification, mostly occurring in the Eastern Mediterranean Sea.

A note on optimal Hermite interpolation in Sobolev spaces

This paper investigates the optimal Hermite interpolation of Sobolev spaces W_{infty }^{n}[a,b], nin mathbb{N} in space L_{infty }[a,b] and weighted spaces L_{p,omega }[a,b], 1le p< infty with ω a continuous-integrable weight function in (a,b) when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in L_{infty } (or L_{p,omega }[a,b], 1le p<infty ) are optimal for W_{infty }^{n}[a,b] in L_{infty }[a,b] (or L_{p,omega }[a,b], 1le p<infty ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.

MODELING OF ASYMPTOTICALLY OPTIMAL PIECEWISE LINEAR INTERPOLATION OF PLANE PARAMETRIC CURVES

Context. Piecewise linear approximation of curves has a large number of applications in computer algorithms, as the reconstruction of objects of complex shapes on monitors, CNC machines and 3D printers. In many cases, it is required to have the smallest number of segments for a given accuracy.&#x0D; Objective. The objective of this paper is to improve the method of asymptotically optimal piecewise linear interpolation of plane parametric curves. This improvement is based to research influence of the method parameters and algorithms to distributions of approximation errors.&#x0D; Method. An asymptotically optimal method of curves interpolation is satisfied to the condition of minimum number of approximation units. Algorithms for obtaining the values of the sequence of approximation nodes are suggested. This algorithm is based on numerical integration of the nodes regulator function with linear and spline interpolation of its values. The method of estimating the results of the curve approximation based on statistical processing of line segments sequence of relative errors is substantiated. Modeling of real curves approximation is carried out and influence of the sampling degree of integral function – the nodes regulator on distribution parameters of errors is studied. The influence is depending on a method of integral function interpolation.&#x0D; Results. Research allows to define necessary the number of discretization nodes of the integral function in practical applications. There have been established that with enough sampling points the variance of the error’s distribution stabilizes and further increasing this number does not significantly increase the accuracy of the curve approximation. In the case of spline interpolation of the integral function, the values of the distribution parameters stabilized much faster, which allows to reduce the number of initial sampling nodes by 5–6 times having similar accuracy.&#x0D; Conclusions. Modelling of convex planar parametric curves reconstruction by an asymptotically optimal linear interpolation algorithm showed acceptable results without exceeding the maximum errors limit in cases of a sufficient discretization of the integral function. The prospect of further research is to reduce the computational complexity when calculating the values of the integral distribution function by numerical methods, and to use discrete analogues of derivatives in the expression of this function.

New Gridded Product for the Total Columnar Atmospheric Water Vapor over Ocean Surface Constructed from Microwave Radiometer Satellite Data

Based on the optimal interpolation (OI) algorithm, a daily fusion product of high-resolution global ocean columnar atmospheric water vapor with a resolution of 0.25° was generated in this study from multisource remote sensing observations. The product covers the period from 2003 to 2018, and the data represent a fusion of microwave radiometer observations, including those from the Special Sensor Microwave Imager Sounder (SSMIS), WindSat, Advanced Microwave Scanning Radiometer for Earth Observing System sensor (AMSR-E), Advanced Microwave Scanning Radiometer 2 (AMSR2), and HY-2A microwave radiometer (MR). The accuracy of this water vapor fusion product was validated using radiosonde water vapor observations. The comparative results show that the overall mean deviation (Bias) is smaller than 0.6 mm; the root mean square error (RMSE) and standard deviation (SD) are better than 3 mm, and the mean absolute deviation (MAD) and correlation coefficient (R) are better than 2 mm and 0.98, respectively.

Reconstruction of arctic SST data and generation of multi-source satellite fusion products with high temporal and spatial resolutions

ABSTRACT The use of remote sensing data, whether for change detection, spatial and temporal analysis, or numerical simulation of water bodies, requires a complete data set. Here, high temporal and spatial resolution sea surface temperature (SST) data for the Arctic are produced based on optimal interpolation and data reconstruction algorithms with a temporal resolution of 12 h and a spatial resolution of 9 km. The accuracy of the reconstructed data sets is verified using Argo measurements to demonstrate the validity and reliability of the data interpolation empirical orthogonal functions method. The root mean square error of the fusion data (a temporal resolution of 12 h) is relatively stable, ranging from 0.49°C to 0.62°C. In comparison with remote sensing systems (RSS) MW-IR SST and Advanced Very High Resolution Radiometer (AVHRR) OISST, the accuracy of the fusion data (a temporal resolution of 24 h) is in the range of 0.48–0.65°C, while that of the MW-IR SST is in the range of 0.48–0.60°C and AVHRR OISST is in the range of 0.66–0.79°C. The fusion data is more accurate than those of AVHRR OISST, and there is no obvious difference with RSS MW-IR SST.

Sample numbers and optimal Lagrange interpolation in Sobolev spaces

This paper investigates the optimal recovery of Sobolev spaces W∞r[−1,1],r∈ℕ in space L∞[−1,1] and weighted spaces Lp,ω[−1,1],1≤p<∞ with ω a continuous integrable weight function in (−1,1). We obtain the values of the sampling numbers of W∞r[−1,1] in L∞[−1,1] and Lp,ω[−1,1],1≤p<∞. We prove that the Lagrange interpolation algorithms based on the Chebyshev nodes of the first kind are optimal for p=∞. Meanwhile, we prove that the Lagrange interpolation algorithms based on the zeros of polynomial of degree r with the leading coefficient 1 of the least deviation from zero in Lp,ω[−1,1] are optimal for 1≤p<∞. We also give the optimal Lagrange interpolation algorithms when we ask the endpoints to be included in the nodes.

Estimation of Stochastic Processes with Random Structure and Markov Switches in Discreet Time (Review)

The review of algorithms of estimation of stochastic processes with random structure and Markov switch obtained on a basis of mathematic tool of mixed Markov processes in discrete time is represented. It is shown that Markov property including continuous valued process with random structure in discrete time and Markov chain controlling its structure modification. There are considered recurrent optimal and quasi-optimal filtering algorithms describing the evolution of a posteriori probability density of the mixed process. Adaptive filters belong to devices class with feedbacks between channels. There is represented Bayesian decision rule for definition of the estimations of discrete and continuous components which are mutually coupled. There are considered recurrent optimal interpolation algorithms: at the fixed point, in the fixed interval and with constant delay and their analysis is carried out. There are represented examples of application of considered estimation algorithms for solution of applicable problems. There are considered two-step algorithms of mutual filtering and segmentation of textural images allowing to preserve computational advantages of single-dimension algorithms of estimation of the processes with random structure and adequate to digital devices with parallel architecture.

An Adaptive Optimal Interpolation Based on Analog Forecasting: Application to SSH in the Gulf of Mexico

AbstractBecause of the irregular sampling pattern of raw altimeter data, many oceanographic applications rely on information from sea surface height (SSH) products gridded on regular grids where gaps have been filled with interpolation. Today, the operational SSH products are created using the simple, but robust, optimal interpolation (OI) method. If well tuned, the OI becomes computationally cheap and provides accurate results at low resolution. However, OI is not adapted to produce high-resolution and high-frequency maps of SSH. To improve the interpolation of SSH satellite observations, a data-driven approach (i.e., constructing a dynamical forecast model from the data) was recently proposed: analog data assimilation (AnDA). AnDA adaptively chooses analog situations from a catalog of SSH scenes—originating from numerical simulations or a large database of observations—which allow the temporal propagation of physical features at different scales, while each observation is assimilated. In this article, we review the AnDA and OI algorithms and compare their skills in numerical experiments. The experiments are observing system simulation experiments (OSSE) on the Lorenz-63 system and on an SSH reconstruction problem in the Gulf of Mexico. The results show that AnDA, with no necessary tuning, produces comparable reconstructions as does OI with tuned parameters. Moreover, AnDA manages to reconstruct the signals at higher frequencies than OI. Finally, an important additional feature for any interpolation method is to be able to assess the quality of its reconstruction. This study shows that the standard deviation estimated by AnDA is flow dependent, hence more informative on the reconstruction quality, than the one estimated by OI.

Ocean Eddies in the Mediterranean Sea From Satellite Altimetry: Sensitivity to Satellite Track Location

Mesoscale eddies are a key oceanic feature relevant to the transport of water properties and biological material. These structures, through their surface signature, have been characterized and widely investigated using sea level anomaly (SLA) maps retrieved from satellite altimetry observations. Gridded SLA fields are routinely computed combining SLA data along satellite tracks with an optimal interpolation algorithm using a selected set of parameters. These are crucial because they define and constrain the temporal and spatial scales of the structures resolved. Here, we investigate the impact of the choice of the interpolation parameters in global and in the Mediterranean SLA products. Our findings demonstrate that the number of eddies detected in SLA maps over the Mediterranean Sea regional products is significantly larger along the satellite tracks than between the tracks, irrespective of the number of satellites used to produce the gridded maps, of the time period considered and of the eddy size. We also show that this is not the case of the global SLA product. We attribute the anomalously high number of eddies detected along the tracks in the regional products to the correlation scales of the optimal interpolation algorithm, which are different from those in the global product due to a smaller Rossby radius of the Mediterranean structures. Among these, we find that the time correlation scale is more restrictive than the spatial one.

Copernicus Imaging Microwave Radiometer (CIMR) Benefits for the Copernicus Level 4 Sea-Surface Salinity Processing Chain

We present a study on the potential of the Copernicus Imaging Microwave Radiometer (CIMR) mission for the global monitoring of Sea-Surface Salinity (SSS) using Level-4 (gap-free) analysis processing. Space-based SSS are currently provided by the Soil Moisture and Ocean Salinity (SMOS) and Soil Moisture Active Passive (SMAP) satellites. However, there are no planned missions to guarantee continuity in the remote SSS measurements for the near future. The CIMR mission is in a preparatory phase with an expected launch in 2026. CIMR is focused on the provision of global coverage, high resolution sea-surface temperature (SST), SSS and sea-ice concentration observations. In this paper, we evaluate the mission impact within the Copernicus Marine Environment Monitoring Service (CMEMS) SSS processing chain. The CMEMS SSS operational products are based on a combination of in situ and satellite (SMOS) SSS and high-resolution SST information through a multivariate optimal interpolation. We demonstrate the potential of CIMR within the CMEMS SSS operational production after the SMOS era. For this purpose, we implemented an Observing System Simulation Experiment (OSSE) based on the CMEMS MERCATOR global operational model. The MERCATOR SSSs were used to generate synthetic in situ and CIMR SSS and, at the same time, they provided a reference gap-free SSS field. Using the optimal interpolation algorithm, we demonstrated that the combined use of in situ and CIMR observations improves the global SSS retrieval compared to a processing where only in situ observations are ingested. The improvements are observed in the 60% and 70% of the global ocean surface for the reconstruction of the SSS and of the SSS spatial gradients, respectively. Moreover, the study highlights the CIMR-based salinity patterns are more accurate both in the open ocean and in coastal areas. We conclude that CIMR can guarantee continuity for accurate monitoring of the ocean surface salinity from space.

Tsunami Data Assimilation Without a Dense Observation Network

AbstractThe tsunami data assimilation method enables tsunami forecasting directly from observations, without the need of estimating tsunami sources. However, it requires a dense observation network to produce desirable results. Here we propose a modified method of tsunami data assimilation for regions with a sparse observation network. The method utilizes interpolated waveforms at virtual stations. The tsunami waveforms at the virtual stations between two existing observation stations are estimated by shifting arrival times with the linear interpolation of observed arrival times and by correcting the amplitudes for their water depths. In our new data assimilation approach, we employ the Optimal Interpolation algorithm to both the real observations and virtual stations, in order to construct a complete wavefront of tsunami propagation. The application to the 2004 Sumatra‐Andaman earthquake and the 2009 Dusky Sound, New Zealand, earthquake reveals that addition of virtual stations greatly helps improve the tsunami forecasting accuracy.