Sparsely-sampled phase-insensitive two-dimensional arrays: Spatial interpolation and signal-dependent aperture
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Patrick H. Johnston↗
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Characterizing Spatiotemporal Uncertainty in Interpolated Meteorological Data
Interpolated meteorological data invariably contain errors. These errors have structure in time and space, particularly autocorrelation, which can cause the effects of errors to compound when model outputs are aggregated temporally or spatially. One way to account for this uncertainty is with a probabilistic model from which samples can be drawn that are coherent with respect to underlying spatial and temporal covariance structure. This work describes a probabilistic method for spatial interpolation of point-wise meteorological time series. Observational data from weather stations are generally sparse in space and dense in time (but sometimes missing). The method works by projecting time series onto orthogonal basis vectors and spatially interpolating each resulting component independently. Under suitable assumptions, and data transformations to better satisfy those assumptions, Gaussian process regression provides a complete description of the joint predictive distribution over a Gaussian random field. Spatiotemporally coherent realizations are generated as the sum of conditional (spatial) simulations of each orthogonal (temporal) component. Data-derived and generic orthogonal bases are considered. In addition to spatial interpolation, imputation of missing observational data is examined. The method is applied using near-surface air temperature over the Western United States and validated by comparing theoretical versus actual coverage of predictive distributions and analyzing the degree to which spatial and temporal covariance structure is reproduced. Computational considerations, relating to conditional simulation of random fields, are also addressed.
Conor T Doherty↗
Transfer of Satellite Rainfall Uncertainty from Gauged to Ungauged Regions at Regional and Seasonal Timescales
Hydrologists and other users need to know the uncertainty of the satellite rainfall data sets across the range of time/space scales over the whole domain of the data set. Here, uncertainty' refers to the general concept of the deviation' of an estimate from the reference (or ground truth) where the deviation may be defined in multiple ways. This uncertainty information can provide insight to the user on the realistic limits of utility, such as hydrologic predictability, that can be achieved with these satellite rainfall data sets. However, satellite rainfall uncertainty estimation requires ground validation (GV) precipitation data. On the other hand, satellite data will be most useful over regions that lack GV data, for example developing countries. This paper addresses the open issues for developing an appropriate uncertainty transfer scheme that can routinely estimate various uncertainty metrics across the globe by leveraging a combination of spatially-dense GV data and temporally sparse surrogate (or proxy) GV data, such as the Tropical Rainfall Measuring Mission (TRMM) Precipitation Radar and the Global Precipitation Measurement (GPM) mission Dual-Frequency Precipitation Radar. The TRMM Multi-satellite Precipitation Analysis (TMPA) products over the US spanning a record of 6 years are used as a representative example of satellite rainfall. It is shown that there exists a quantifiable spatial structure in the uncertainty of satellite data for spatial interpolation. Probabilistic analysis of sampling offered by the existing constellation of passive microwave sensors indicate that transfer of uncertainty for hydrologic applications may be effective at daily time scales or higher during the GPM era. Finally, a commonly used spatial interpolation technique (kriging), that leverages the spatial correlation of estimation uncertainty, is assessed at climatologic, seasonal, monthly and weekly timescales. It is found that the effectiveness of kriging is sensitive to the type of uncertainty metric, time scale of transfer and the density of GV data within the transfer domain. Transfer accuracy is lowest at weekly timescales with the error doubling from monthly to weekly.However, at very low GV data density (<20% of the domain), the transfer accuracy is too low to show any distinction as a function of the timescale of transfer.
Tang, Ling↗
Efficient Implementation of an Optimal Interpolator for Large Spatial Data Sets
Interpolating scattered data points is a problem of wide ranging interest. A number of approaches for interpolation have been proposed both from theoretical domains such as computational geometry and in applications' fields such as geostatistics. Our motivation arises from geological and mining applications. In many instances data can be costly to compute and are available only at nonuniformly scattered positions. Because of the high cost of collecting measurements, high accuracy is required in the interpolants. One of the most popular interpolation methods in this field is called ordinary kriging. It is popular because it is a best linear unbiased estimator. The price for its statistical optimality is that the estimator is computationally very expensive. This is because the value of each interpolant is given by the solution of a large dense linear system. In practice, kriging problems have been solved approximately by restricting the domain to a small local neighborhood of points that lie near the query point. Determining the proper size for this neighborhood is a solved by ad hoc methods, and it has been shown that this approach leads to undesirable discontinuities in the interpolant. Recently a more principled approach to approximating kriging has been proposed based on a technique called covariance tapering. This process achieves its efficiency by replacing the large dense kriging system with a much sparser linear system. This technique has been applied to a restriction of our problem, called simple kriging, which is not unbiased for general data sets. In this paper we generalize these results by showing how to apply covariance tapering to the more general problem of ordinary kriging. Through experimentation we demonstrate the space and time efficiency and accuracy of approximating ordinary kriging through the use of covariance tapering combined with iterative methods for solving large sparse systems. We demonstrate our approach on large data sizes arising both from synthetic sources and from real applications.
Memarsadeghi, Nargess↗
Effective Interpolation of Incomplete Satellite-Derived Leaf-Area Index Time Series for the Continental United States
Many earth science modeling applications employ continuous input data fields derived from satellite data. Environmental factors, sensor limitations and algorithmic constraints lead to data products of inherently variable quality. This necessitates interpolation of one form or another in order to produce high quality input fields free of missing data. The present research tests several interpolation techniques as applied to satellite-derived leaf area index, an important quantity in many global climate and ecological models. The study evaluates and applies a variety of interpolation techniques for the Moderate Resolution Imaging Spectroradiometer (MODIS) Leaf-Area Index Product over the time period 2001-2006 for a region containing the conterminous United States. Results indicate that the accuracy of an individual interpolation technique depends upon the underlying land cover. Spatial interpolation provides better results in forested areas, while temporal interpolation performs more effectively over non-forest cover types. Combination of spatial and temporal approaches offers superior interpolative capabilities to any single method, and in fact, generation of continuous data fields requires a hybrid approach such as this.
Jasinski, Michael F.↗
Efficient Implementation of an Optimal Interpolator for Large Spatial Data Sets
Scattered data interpolation is a problem of interest in numerous areas such as electronic imaging, smooth surface modeling, and computational geometry. Our motivation arises from applications in geology and mining, which often involve large scattered data sets and a demand for high accuracy. The method of choice is ordinary kriging. This is because it is a best unbiased estimator. Unfortunately, this interpolant is computationally very expensive to compute exactly. For n scattered data points, computing the value of a single interpolant involves solving a dense linear system of size roughly n x n. This is infeasible for large n. In practice, kriging is solved approximately by local approaches that are based on considering only a relatively small'number of points that lie close to the query point. There are many problems with this local approach, however. The first is that determining the proper neighborhood size is tricky, and is usually solved by ad hoc methods such as selecting a fixed number of nearest neighbors or all the points lying within a fixed radius. Such fixed neighborhood sizes may not work well for all query points, depending on local density of the point distribution. Local methods also suffer from the problem that the resulting interpolant is not continuous. Meyer showed that while kriging produces smooth continues surfaces, it has zero order continuity along its borders. Thus, at interface boundaries where the neighborhood changes, the interpolant behaves discontinuously. Therefore, it is important to consider and solve the global system for each interpolant. However, solving such large dense systems for each query point is impractical. Recently a more principled approach to approximating kriging has been proposed based on a technique called covariance tapering. The problems arise from the fact that the covariance functions that are used in kriging have global support. Our implementations combine, utilize, and enhance a number of different approaches that have been introduced in literature for solving large linear systems for interpolation of scattered data points. For very large systems, exact methods such as Gaussian elimination are impractical since they require 0(n(exp 3)) time and 0(n(exp 2)) storage. As Billings et al. suggested, we use an iterative approach. In particular, we use the SYMMLQ method, for solving the large but sparse ordinary kriging systems that result from tapering. The main technical issue that need to be overcome in our algorithmic solution is that the points' covariance matrix for kriging should be symmetric positive definite. The goal of tapering is to obtain a sparse approximate representation of the covariance matrix while maintaining its positive definiteness. Furrer et al. used tapering to obtain a sparse linear system of the form Ax = b, where A is the tapered symmetric positive definite covariance matrix. Thus, Cholesky factorization could be used to solve their linear systems. They implemented an efficient sparse Cholesky decomposition method. They also showed if these tapers are used for a limited class of covariance models, the solution of the system converges to the solution of the original system. Matrix A in the ordinary kriging system, while symmetric, is not positive definite. Thus, their approach is not applicable to the ordinary kriging system. Therefore, we use tapering only to obtain a sparse linear system. Then, we use SYMMLQ to solve the ordinary kriging system. We show that solving large kriging systems becomes practical via tapering and iterative methods, and results in lower estimation errors compared to traditional local approaches, and significant memory savings compared to the original global system. We also developed a more efficient variant of the sparse SYMMLQ method for large ordinary kriging systems. This approach adaptively finds the correct local neighborhood for each query point in the interpolation process.
Memarsadeghi, Nargess↗
Improved Rainfall Data in the Philippines through Concurrent Use of GPM IMERG and Ground-Based Measurements
The availability of accurate and reliable rainfall data that are applicable to various phenomenological, climatological, and modeling studies is important, especially in the Philippines, which is considered to be highly vulnerable to natural hazards and a changing climate. The presented strategy involved constructing a dataset consisting of synoptic data, automatic rain gauge (ARG) measurements, and satellite data that are co-registered, consistent, and formatted in the same manner. Although sparse in number, the synoptic stations provide the most accurate rainfall information and were used as the baseline for creating the dataset. The ARGs that are within a distance of 1 km to the synoptic stations were used to determine the correction factors needed to make the synoptic and ARG data consistent. Subsequently, the corrected ARGs were used to make the satellite IMERG data consistent with both ARG and synoptic data. In case of the latter, only IMERG pixels with at least 10 ARGs within the relatively large footprint of the satellite sensor were used in estimating the required correction parameters derived from a combination of a power transform and linear regression correction techniques. The final results show good agreement of synoptic and corrected ARG data with correlation coefficients of 0.94 and 0.97 for the 10 day and monthly data, respectively, and improvement in the linear regression slope from 0.67 to 0.90 for 10 day data, and 0.70 to 0.94 for monthly data. In addition, the corrected ARG data agree well with the corrected IMERG data, with correlation coefficients of 0.88 and 0.93 for the 10 day and monthly data, respectively, and an improvement in slope from 0.66 to 0.87 for 10 day data, and 0.74 to 0.99 for monthly data. The merit of using a combined dataset is illustrated through comparative analyses of the IMERG data and spatially interpolated synoptic and ARG data. The results show general agreements in spatial patterns of rainfall across the datasets, especially in areas where in situ measurements are recorded. The observed discrepancy when ground data is limited emphasizes the need for satellite IMERG data to obtain the true spatial patterns of rainfall distribution.
rainfall↗
Observed Trend in Surface Wind Speed Over the Conterminous USA and CMIP5 Simulations
There has been no spatial surface wind map even over the conterminous USA due to the difficulty of spatial interpolation of wind field. As a result, the reanalysis data were often used to analyze the statistics of spatial pattern in surface wind speed. Unfortunately, no consistent trend in wind field was found among the available reanalysis data, and that obstructed the further analysis or projection of spatial pattern of wind speed. In this study, we developed the methodology to interpolate the observed wind speed data at weather stations using random forest algorithm. We produced the 1-km daily climate variables over the conterminous USA from 1979 to 2015. The validation using Ameriflux daily data showed that R2 is 0.59. Existing studies have found the negative trend over the Eastern US, and our study also showed same results. However, our new datasets also revealed the significant increasing trend over the southwest US especially from April to June. The trend in the southwestern US represented change or seasonal shift in North American Monsoon. Global analysis of CMIP5 data projected the decrease trend in mid-latitude, while increase trend in tropical region over the land. Most likely because of the low resolution in GCM, CMIP5 data failed to simulate the increase trend in the southwest US, even though it was qualitatively predicted that pole ward shift of anticyclone help the North American Monsoon.
simulations↗
An algorithm to generate input data from meteorological and space shuttle observations to validate a CH4-CO model
Objective procedures to analyze data from meteorological and space shuttle observations to validate a three dimensional model were investigated. The transport and chemistry of carbon monoxide and methane in the troposphere were studied. Four aspects were examined: (1) detailed evaluation of the variational calculus procedure, with the equation of continuity as a strong constraint, for adjustment of global tropospheric wind fields; (2) reduction of the National Meteorological Center (NMC) data tapes for data input to the OSTA-1/MAPS Experiment; (3) interpolation of the NMC Data for input to the CH4-CO model; and (4) temporal and spatial interpolation procedures of the CO measurements from the OSTA-1/MAPS Experiment to generate usable contours of the data.
Peters, L. K.↗
The Spatial Distribution of Forest Biomass in the Brazilian Amazon: A Comparison of Estimates
The amount of carbon released to the atmosphere as a result of deforestation is determined, in part, by the amount of carbon held in the biomass of the forests converted to other uses. Uncertainty in forest biomass is responsible for much of the uncertainty in current estimates of the flux of carbon from land-use change. We compared several estimates of forest biomass for the Brazilian Amazon, based on spatial interpolations of direct measurements, relationships to climatic variables, and remote sensing data. We asked three questions. First, do the methods yield similar estimates? Second, do they yield similar spatial patterns of distribution of biomass? And, third, what factors need most attention if we are to predict more accurately the distribution of forest biomass over large areas? Amazonian forests (including dead and below-ground biomass) vary by more than a factor of two, from a low of 39 PgC to a high of 93 PgC. Furthermore, the estimates disagree as to the regions of high and low biomass. The lack of agreement among estimates confirms the need for reliable determination of aboveground biomass over large areas. Potential methods include direct measurement of biomass through forest inventories with improved allometric regression equations, dynamic modeling of forest recovery following observed stand-replacing disturbances (the approach used in this research), and estimation of aboveground biomass from airborne or satellite-based instruments sensitive to the vertical structure plant canopies.
Houghton, R. A.↗
An Automated Approach to Very High Order Aeroacoustic Computations in Complex Geometries
Computational aeroacoustics requires efficient, high-resolution simulation tools. And for smooth problems, this is best accomplished with very high order in space and time methods on small stencils. But the complexity of highly accurate numerical methods can inhibit their practical application, especially in irregular geometries. This complexity is reduced by using a special form of Hermite divided-difference spatial interpolation on Cartesian grids, and a Cauchy-Kowalewslci recursion procedure for time advancement. In addition, a stencil constraint tree reduces the complexity of interpolating grid points that are located near wall boundaries. These procedures are used to automatically develop and implement very high order methods (>15) for solving the linearized Euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point. The accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries.
Dyson, Rodger W.↗
Automated Approach to Very High-Order Aeroacoustic Computations
Computational aeroacoustics requires efficient, high-resolution simulation tools. For smooth problems, this is best accomplished with very high-order in space and time methods on small stencils. However, the complexity of highly accurate numerical methods can inhibit their practical application, especially in irregular geometries. This complexity is reduced by using a special form of Hermite divided-difference spatial interpolation on Cartesian grids, and a Cauchy-Kowalewski recursion procedure for time advancement. In addition, a stencil constraint tree reduces the complexity of interpolating grid points that am located near wall boundaries. These procedures are used to develop automatically and to implement very high-order methods (> 15) for solving the linearized Euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point. The accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries.
Dyson, Rodger W.↗
Digital image processing of Seabeam bathymetric data for structural studies of seamounts near the East Pacific Rise
The problem of displaying information on the seafloor morphology is attacked by utilizing digital image processing techniques to generate images for Seabeam data covering three young seamounts on the eastern flank of the East Pacific Rise. Errors in locations between crossing tracks are corrected by interactively identifying features and translating tracks relative to a control track. Spatial interpolation techniques using moving averages are used to interpolate between gridded depth values to produce images in shaded relief and color-coded forms. The digitally processed images clarify the structural control on seamount growth and clearly show the lateral extent of volcanic materials, including the distribution and fault control of subsidiary volcanic constructional features. The image presentations also clearly show artifacts related to both residual navigational errors and to depth or location differences that depend on ship heading relative to slope orientation in regions with steep slopes.
Edwards, M. H.↗
Sparsely Sampled Phase-Insensitive Ultrasonic Transducer Arrays
Three methods of interpretation of outputs from sparsely sampled two-dimensional array of receiving ultrasonic transducers used in transmission experiments investigated. Methods are: description of sampled beam in terms of first few spatial moments of sampled distribution of energy; use of signal-dependent cutoff to limit extent of effective receiver aperture; and use of spatial interpolation to increase apparent density of sampling during computation. Methods reduce errors in computations of shapes of ultrasonic beams.
Patrick H. Johnston↗
Monitoring global snow cover
A snow model that supports the daily, operational analysis of global snow depth and age has been developed. It provides improved spatial interpolation of surface reports by incorporating digital elevation data, and by the application of regionalized variables (kriging) through the use of a global snow depth climatology. Where surface observations are inadequate, the model applies satellite remote sensing. Techniques for extrapolation into data-void mountain areas and a procedure to compute snow melt are also contained in the model.
Armstrong, Richard↗
A catchment scale water balance model for FIFE
A catchment scale water balance model is presented and used to predict evaporation from the King's Creek catchment at the First ISLSCP Field Experiment site on the Konza Prairie, Kansas. The model incorporates spatial variability in topography, soils, and precipitation to compute the land surface hydrologic fluxes. A network of 20 rain gages was employed to measure rainfall across the catchment in the summer of 1987. These data were spatially interpolated and used to drive the model during storm periods. During interstorm periods the model was driven by the estimated potential evaporation, which was calculated using net radiation data collected at site 2. Model-computed evaporation is compared to that observed, both at site 2 (grid location 1916-BRS) and the catchment scale, for the simulation period from June 1 to October 9, 1987.
Famiglietti, J. S.↗
Development and Evaluation of High-Resolution Climate Simulations Over the Mountainous Northeastern United States
The mountain regions of the northeastern United States are a critical socioeconomic resource for Vermont, New York State, New Hampshire, Maine, and southern Quebec. While global climate models (GCMs) are important tools for climate change risk assessment at regional scales, even the increased spatial resolution of statistically downscaled GCMs (commonly approximately 1/ 8 deg) is not sufficient for hydrologic, ecologic, and land-use modeling of small watersheds within the mountainous Northeast. To address this limitation, an ensemble of topographically downscaled, high-resolution (30"), daily 2-m maximum air temperature; 2-m minimum air temperature; and precipitation simulations are developed for the mountainous Northeast by applying an additional level of downscaling to intermediately downscaled (1/ 8 deg) data using high-resolution topography and station observations. First, observed relationships between 2-m air temperature and elevation and between precipitation and elevation are derived. Then, these relationships are combined with spatial interpolation to enhance the resolution of intermediately downscaled GCM simulations. The resulting topographically downscaled dataset is analyzed for its ability to reproduce station observations. Topographic downscaling adds value to intermediately downscaled maximum and minimum 2-m air temperature at high-elevation stations, as well as moderately improves domain-averaged maximum and minimum 2-m air temperature. Topographic downscaling also improves mean precipitation but not daily probability distributions of precipitation. Overall, the utility of topographic downscaling is dependent on the initial bias of the intermediately downscaled product and the magnitude of the elevation adjustment. As the initial bias or elevation adjustment increases, more value is added to the topographically downscaled product.
climatology↗
High Resolution Image Reconstruction from Projection of Low Resolution Images DIffering in Subpixel Shifts
In this paper, we demonstrate a simple algorithm that projects low resolution (LR) images differing in subpixel shifts on a high resolution (HR) also called super resolution (SR) grid. The algorithm is very effective in accuracy as well as time efficiency. A number of spatial interpolation techniques using nearest neighbor, inverse-distance weighted averages, Radial Basis Functions (RBF) etc. used in projection yield comparable results. For best accuracy of reconstructing SR image by a factor of two requires four LR images differing in four independent subpixel shifts. The algorithm has two steps: i) registration of low resolution images and (ii) shifting the low resolution images to align with reference image and projecting them on high resolution grid based on the shifts of each low resolution image using different interpolation techniques. Experiments are conducted by simulating low resolution images by subpixel shifts and subsampling of original high resolution image and the reconstructing the high resolution images from the simulated low resolution images. The results of accuracy of reconstruction are compared by using mean squared error measure between original high resolution image and reconstructed image. The algorithm was tested on remote sensing images and found to outperform previously proposed techniques such as Iterative Back Projection algorithm (IBP), Maximum Likelihood (ML), and Maximum a posterior (MAP) algorithms. The algorithm is robust and is not overly sensitive to the registration inaccuracies.
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