Interpolation Uncertainty Research Articles

An extended and revised Lake Suigetsu varve chronology from ∼50 to ∼10 ka BP based on detailed sediment micro-facies analyses

Lake Suigetsu (Japan) is a key site for radiocarbon (14C) calibration and palaeo-environmental reconstruction in East Asia. Here we present a description of the sediment (micro)facies, which in combination with a new approach to varve interpolation allows construction of a revised varve based chronology that extends the previous 2012 varve based chronology by ∼10 ka, back to ∼50 ka BP. Challenges in varve counting and interpolation, which were previously discussed in detail only for the Last Glacial-Interglacial Transition, are described here back to ∼50 ka BP. Furthermore, the relative merits of varve counting by μXRF scanning and by thin-section microscopy are discussed. Facies analysis reveals four facies zones, their transitions driven by both local and climatic controls. The lamination quality of the sediment is highly variable and varve interpolation reveals that in the analysed time interval, on average, only 50% of the annual cycles are represented by seasonal layers. In the remaining years seasonal layers are indistinguishable, i.e. either did not form or were not preserved. For varve interpolation an advanced version of the Varve Interpolation Program was used, which enabled the construction of the longest, purely varve dated chronology published, despite long intervals of poor lamination quality. The calculated interpolation uncertainty is +8.9% and −4.6%, which is well within expectations considering the high degree of interpolation and the length of the record.

Influence of Sampling Point Discretization on the Regional Variability of Soil Organic Carbon in the Red Soil Region, China

Research on the regional variability of soil organic carbon (SOC) has focused mostly on the influence of the number of soil sampling points and interpolation methods. Little attention has typically been paid to the influence of sampling point discretization. Based on dense soil sampling points in the red soil area of Southern China, we obtained four sample discretization levels by a resampling operation. Then, regional SOC distributions were obtained at four levels by two interpolation methods: ordinary Kriging (OK) and Kriging combined with land use information (LuK). To evaluate the influence of sample discretization on revealing SOC variability, we compared the interpolation accuracies at four discretization levels with uniformly distributed validation points. The results demonstrated that the spatial distribution patterns of SOC were roughly similar, but the contour details in some local areas were different at the various discretization levels. Moreover, the predicted mean absolute errors (MAE) and root mean square errors (RMSE) of the two Kriging methods all rose with an increase in discretization. From the lowest to the largest discretization level, the MAEs of OK and LuK rose from 4.47 and 3.02 g kg−1 to 5.46 and 3.54 g kg−1, and the RMSEs rose from 5.13 and 3.95 g kg−1 to 5.76 and 4.76 g kg−1, respectively. Though the trend of prediction errors varied with discretization levels, the interpolation accuracies of the two Kriging methods were both influenced by the sample discretization level. Furthermore, the spatial interpolation uncertainty of OK was more sensitive to the discretization level than that of the LuK method. Therefore, when the spatial distribution of SOC is predicted using Kriging methods based on the same sample quantity, the more uniformly distributed sampling points are, the more accurate the spatial prediction accuracy of SOC will be, and vice versa. The results of this study can act as a useful reference for evaluating the uncertainty of SOC spatial interpolation and making a soil sampling scheme in the red soil region of China.

Seawater pH measurements with a combination glass electrode and high ionic strength TRIS-TRIS HCl reference buffers – An uncertainty evaluation approach

The comparison of pH measurements in seawater collected at different locations or occasions, is meaningful if the same measurand (i.e. the quantity intended to be measured) is determined, if adequate measurement procedures are used, including the selection of calibrators, and if the measurement uncertainty is known. Depending on the purpose of this evaluation, the measurement uncertainty should be smaller than a defined target value. The measured pH should have a sound physical-chemical meaning to allow the adequate assessment of its impacts.In the present procedure TRIS-TRIS HCl solutions, of different molality ratios, prepared in artificial seawater with reference values estimated by primary measurements, were used to obtain proper calibrators for the pH meter used for the analysis of seawater samples.This work presents the uncertainty evaluation of pH measurements in seawater, performed by potentiometry using a combination glass electrode, from the interpolation uncertainty evaluated by the Least Squares Regression Model and by Monte Carlo Simulations of measured potentials and reference values. The uncertainty evaluation was critically assessed.The developed algorithms were implemented in a user-friendly MS-Excel file available as Electronic Supplementary Material.Seawater pH was measured with an expanded uncertainty of 0.019 enabling discriminating differences of pH of two samples larger than 0.029.

Hydrological data uncertainty and its implications

Hydrologic data are at the core of our understanding of physical hydrologic processes, our simulation models and forecasts of water resources and hazards, and our monitoring of water quantity and quality. However, hydrologic data are subject to multiple sources of uncertainty that can introduce bias and error into our analyses and decision‐making if not properly accounted for. In this article, we summarize five categories of data uncertainty: measurement uncertainty, derived data uncertainty, interpolation uncertainty, scaling uncertainty, and data management uncertainty. Hydrologic data uncertainty magnitudes are typically in the range 10–40%. To quantify data uncertainty, hydrologists should first construct a perceptual model of uncertainty that itemizes uncertainty sources. The magnitude of each source can then be estimated using replicates (repeated, nested or subsampled measurements), or information from the literature (in‐depth uncertainty results from experimental catchments, colocated gauges or method comparisons). Multiple uncertainty sources can be combined using Monte Carlo methods to determine total uncertainty. Data uncertainty analysis improves hydrologic process understanding by enabling robust hypothesis testing and identification of spatial and temporal patterns that relate to true process differences rather than data uncertainty. By quantifying uncertainty in data used for input or evaluation of hydrologic models, we can prevent parameter bias, exclude disinformative data, and enhance model performance evaluation. In water management applications, quantifying data uncertainty can lead to robust risk analysis, reduced costs, and transparent results that improve the trust of the public and water managers.This article is categorized under: Science of Water > Hydrological Processes Science of Water > Methods Engineering Water > Planning Water

An Ensemble Version of the E‐OBS Temperature and Precipitation Data Sets

AbstractWe describe the construction of a new version of the Europe‐wide E‐OBS temperature (daily minimum, mean, and maximum values) and precipitation data set. This version provides an improved estimation of interpolation uncertainty through the calculation of a 100‐member ensemble of realizations of each daily field. The data set covers the period back to 1950 and provides gridded fields at a spacing of 0.25∘ × 0.25∘ in regular latitude/longitude coordinates. As with the original E‐OBS data set, the ensemble version is based on the station series collated as part of the ECA&D initiative. Station density varies significantly over the domain, and over time, and a reliable estimation of interpolation uncertainty in the gridded fields is therefore important for users of the data set. The uncertainty quantified by the ensemble data set is more realistic than the uncertainty estimates in the original version, although uncertainty is still underestimated in data‐sparse regions. The new data set is compared against the earlier version of E‐OBS and against regional gridded data sets produced by a selection of National Meteorological Services. In terms of both climatological averages and extreme values, the new version of E‐OBS is broadly comparable to the earlier version. Nonetheless, users will notice differences between the two E‐OBS versions, especially for precipitation, which arises from the different gridding method used.

Structural and sampling uncertainty in observed UK daily precipitation extremes derived from an intercomparison of gridded data sets

This paper compares multiple gridded data sets of daily UK precipitation to evaluate structural uncertainty in our reconstructions of historical rainfall. The data sets compared reflect two different sampling strategies and three different grid interpolation methods. In order to separate the influence of sampling and interpolation uncertainties, one of the data sets (produced by the Met Office) has been recreated using the sampling strategy of stations used in the European (E‐OBS) data set. The results confirm and build upon previous studies showing that relying on a relatively sparse but homogeneous network of stations limits the ability of the resulting data set to reliably estimate extreme rainfall at the daily timescale. It is shown that gridding methods that additionally make use of reference climatological data can avoid systematic bias in both the average and extreme events even when using a relatively sparse network of observations. This is an encouraging result in terms of our potential to reliably extend such data sets further back in time where the availability of digitized data is substantially lower than the more modern era.

A multi-objective robust optimization approach for engineering design under interval uncertainty

PurposeEngineering system design and optimization problems are usually multi-objective and constrained and have uncertainties in the inputs. These uncertainties might significantly degrade the overall performance of engineering systems and change the feasibility of the obtained solutions. This paper aims to propose a multi-objective robust optimization approach based on Kriging metamodel (K-MORO) to obtain the robust Pareto set under the interval uncertainty.Design/methodology/approachIn K-MORO, the nested optimization structure is reduced into a single loop optimization structure to ease the computational burden. Considering the interpolation uncertainty from the Kriging metamodel may affect the robustness of the Pareto optima, an objective switching and sequential updating strategy is introduced in K-MORO to determine (1) whether the robust analysis or the Kriging metamodel should be used to evaluate the robustness of design alternatives, and (2) which design alternatives are selected to improve the prediction accuracy of the Kriging metamodel during the robust optimization process.FindingsFive numerical and engineering cases are used to demonstrate the applicability of the proposed approach. The results illustrate that K-MORO is able to obtain robust Pareto frontier, while significantly reducing computational cost.Practical implicationsThe proposed approach exhibits great capability for practical engineering design optimization problems that are multi-objective and constrained and have uncertainties.Originality/valueA K-MORO approach is proposed, which can obtain the robust Pareto set under the interval uncertainty and ease the computational burden of the robust optimization process.

Comparison of the common spatial interpolation methods used to analyze potentially toxic elements surrounding mining regions

The appropriate spatial interpolation methods must be selected to analyze the spatial distributions of Potentially Toxic Elements (PTEs), which is a precondition for evaluating PTE pollution. The accuracy and effect of different spatial interpolation methods, which include inverse distance weighting interpolation (IDW) (power = 1, 2, 3), radial basis function interpolation (RBF) (basis function: thin-plate spline (TPS), spline with tension (ST), completely regularized spline (CRS), multiquadric (MQ) and inverse multiquadric (IMQ)) and ordinary kriging interpolation (OK) (semivariogram model: spherical, exponential, gaussian and linear), were compared using 166 unevenly distributed soil PTE samples (As, Pb, Cu and Zn) in the Suxian District, Chenzhou City, Hunan Province as the study subject. The reasons for the accuracy differences of the interpolation methods and the uncertainties of the interpolation results are discussed, then several suggestions for improving the interpolation accuracy are proposed, and the direction of pollution control is determined. The results of this study are as follows: (i) RBF-ST and OK (exponential) are the optimal interpolation methods for As and Cu, and the optimal interpolation method for Pb and Zn is RBF-IMQ. (ii) The interpolation uncertainty is positively correlated with the PTE concentration, and higher uncertainties are primarily distributed around mines, which is related to the strong spatial variability of PTE concentrations caused by human interference. (iii) The interpolation accuracy can be improved by increasing the sample size around the mines, introducing auxiliary variables in the case of incomplete sampling and adopting the partition prediction method. (iv) It is necessary to strengthen the prevention and control of As and Pb pollution, particularly in the central and northern areas. The results of this study can provide an effective reference for the optimization of interpolation methods and parameters for unevenly distributed soil PTE data in mining areas.

A robust optimization approach based on multi-fidelity metamodel

Multi-fidelity (MF) metamodeling approaches have recently attracted a significant amount of attention in simulation-based design optimization due to their ability to conduct trade-offs between high accuracy and low computational expenses by integrating the information from high-fidelity (HF) and low-fidelity (LF) models. While existing MF metamodel assisted design optimization approaches may yield an inferior or even infeasible solution since they generally treat the MF metamodel as the real HF model and ignore the interpolation uncertainties from the MF metamodel. This situation will be more serious in non-deterministic optimization. Hence, in this work, a MF metamodel assisted robust optimization approach is developed, in which the interpolation uncertainty of the MF metamodel and design variable uncertainty are quantified and taken into consideration. To demonstrate the effectiveness and merits of the proposed approach, two numerical examples and a long cylinder pressure vessel design optimization problem are tested. Results show that for the test cases the proposed approach can obtain a solution that is both optimal and within the feasible region even with perturbation of the uncertain variables.

A kriging metamodel-assisted robust optimization method based on a reverse model

ABSTRACTThe goal of robust optimization methods is to obtain a solution that is both optimum and relatively insensitive to uncertainty factors. Most existing robust optimization approaches use outer–inner nested optimization structures where a large amount of computational effort is required because the robustness of each candidate solution delivered from the outer level should be evaluated in the inner level. In this article, a kriging metamodel-assisted robust optimization method based on a reverse model (K-RMRO) is first proposed, in which the nested optimization structure is reduced into a single-loop optimization structure to ease the computational burden. Ignoring the interpolation uncertainties from kriging, K-RMRO may yield non-robust optima. Hence, an improved kriging-assisted robust optimization method based on a reverse model (IK-RMRO) is presented to take the interpolation uncertainty of kriging metamodel into consideration. In IK-RMRO, an objective switching criterion is introduced to determine whether the inner level robust optimization or the kriging metamodel replacement should be used to evaluate the robustness of design alternatives. The proposed criterion is developed according to whether or not the robust status of the individual can be changed because of the interpolation uncertainties from the kriging metamodel. Numerical and engineering cases are used to demonstrate the applicability and efficiency of the proposed approach.

Hydrodynamic Modeling of the St. Lawrence Fluvial Estuary. I: Model Setup, Calibration, and Validation

In this study, a high-resolution, two-dimensional (2D), time-dependent hydrodynamic model of the St. Lawrence fluvial estuary was developed with the objective of documenting the tidal hydrodynamics of this complex yet poorly understood region. The hydrodynamic model solves the shallow-water equations over a finite-element–discretized domain, with an average spatial resolution of 50 m, and includes a drying–wetting component for the treatment of shallow intertidal areas. The numerical terrain model is composed of high-density topographic data and detailed friction fields associated with bottom substrate and macrophytes. Calibration and validation were carried out using recently acquired data for water level and velocity. Results show very good accuracy in water levels, with prediction skills higher than 0.99 at all stations (where a skill of 1 means perfect agreement between model and observations in terms of their relative average error) and root-mean-square errors (RMSEs) less than 5% of local tidal ranges downstream; at upstream stations where tidal ranges are significantly reduced, RMSEs were lower than 6 cm. Discharges were reproduced with similarly good accuracy, with errors lower than 6% of the maximum observed discharges at 11 of the 13 surveyed transects; the two remaining sections are subject to larger interpolation and bathymetric uncertainties. In this paper, critical aspects of model development are discussed, including the 2D approximation, temporal and spatial resolution, bathymetric uncertainty, error in the boundary conditions, and calibration under nonstationary conditions. This work is the first part of a two-part investigation serving as a methodological framework for model setup, calibration, and validation in large tidal rivers.

Aerial observations of sea ice and melt ponds near the North Pole during CHINARE2010

An aerial photography has been used to provide validation data on sea ice near the North Pole where most polar orbiting satellites cannot cover. This kind of data can also be used as a supplement for missing data and for reducing the uncertainty of data interpolation. The aerial photos are analyzed near the North Pole collected during the Chinese national arctic research expedition in the summer of 2010 (CHINARE2010). The result shows that the average fraction of open water increases from the ice camp at approximately 87°N to the North Pole, resulting in the decrease in the sea ice. The average sea ice concentration is only 62.0% for the two flights (16 and 19 August 2010). The average albedo (0.42) estimated from the area ratios among snow-covered ice, melt pond and water is slightly lower than the 0.49 of HOTRAX 2005. The data on 19 August 2010 shows that the albedo decreases from the ice camp at approximately 87°N to the North Pole, primarily due to the decrease in the fraction of snow-covered ice and the increase in fractions of melt-pond and open-water. The ice concentration from the aerial photos and AMSR-E (The Advanced Microwave Scanning Radiometer-Earth Observing System) images at 87.0°–87.5°N exhibits similar spatial patterns, although the AMSR-E concentration is approximately 18.0% (on average) higher than aerial photos. This can be attributed to the 6.25 km resolution of AMSR-E, which cannot separate melt ponds/submerged ice from ice and cannot detect the small leads between floes. Thus, the aerial photos would play an important role in providing high-resolution independent estimates of the ice concentration and the fraction of melt pond cover to validate and/or supplement space-borne remote sensing products near the North Pole.

High-resolution daily gridded data sets of air temperature and wind speed for Europe

Abstract. New high-resolution data sets for near-surface daily air temperature (minimum, maximum and mean) and daily mean wind speed for Europe (the CORDEX domain) are provided for the period 2001–2010 for the purpose of regional model validation in the framework of DecReg, a sub-project of the German MiKlip project, which aims to develop decadal climate predictions. The main input data sources are SYNOP observations, partly supplemented by station data from the ECA&D data set (http://www.ecad.eu). These data are quality tested to eliminate erroneous data. By spatial interpolation of these station observations, grid data in a resolution of 0.044° (≈ 5km) on a rotated grid with virtual North Pole at 39.25° N, 162° W are derived. For temperature interpolation a modified version of a regression kriging method developed by Krähenmann et al.(2011) is used. At first, predictor fields of altitude, continentality and zonal mean temperature are used for a regression applied to monthly station data. The residuals of the monthly regression and the deviations of the daily data from the monthly averages are interpolated using simple kriging in a second and third step. For wind speed a new method based on the concept used for temperature was developed, involving predictor fields of exposure, roughness length, coastal distance and ERA-Interim reanalysis wind speed at 850 hPa. Interpolation uncertainty is estimated by means of the kriging variance and regression uncertainties. Furthermore, to assess the quality of the final daily grid data, cross validation is performed. Variance explained by the regression ranges from 70 to 90 % for monthly temperature and from 50 to 60 % for monthly wind speed. The resulting RMSE for the final daily grid data amounts to 1–2 K and 1–1.5 ms−1 (depending on season and parameter) for daily temperature parameters and daily mean wind speed, respectively. The data sets presented in this article are published at doi:10.5676/DWD_CDC/DECREG0110v2.

Geostatistically based optimization of a rainfall monitoring network extension: case of the climatically heterogeneous Tunisia

Rainfall data are an essential input for many simulation models. In fact, these latter have a decisive role in the development and application of rational water policies. Since the accuracy of the simulation depends strongly on the available data, the task of optimizing the monitoring network is of great importance. In this paper, an application is presented aiming at the evaluation of a precipitation monitoring network by predicting monthly, seasonal, and interannual average rainfall. The method given here is based on the theory of the regionalized variables using the well-known geostatistical variance reduction method. The procedure, which involves different analysis methods of the available data, such as estimation of the interpolation uncertainty and data cross validation, is applied to a case study data set in Tunisia in order to demonstrate the potential for improvement of the observation network quality. Root mean square error values are the criteria for evaluating rainfall estimation, and network performance is discussed based on kriging variance reduction. Based on this study, it was concluded that some sites should be dropped to eliminate redundancy and some others need to be added to the existing network, essentially in the center and the south, to have a more informative network.

Effects of uncertainties in hydrological modelling. A case study of a mountainous catchment in Southern Norway

Summary In this study, we explore the effect of uncertainty and poor observation quality on hydrological model calibration and predictions. The Osali catchment in Western Norway was selected as case study and an elevation distributed HBV-model was used. We systematically evaluated the effect of accounting for uncertainty in parameters, precipitation input, temperature input and streamflow observations. For precipitation and temperature we accounted for the interpolation uncertainty, and for streamflow we accounted for rating curve uncertainty. Further, the effects of poorer quality of precipitation input and streamflow observations were explored. Less information about precipitation was obtained by excluding the nearest precipitation station from the analysis, while reduced information about the streamflow was obtained by omitting the highest and lowest streamflow observations when estimating the rating curve. The results showed that including uncertainty in the precipitation and temperature inputs has a negligible effect on the posterior distribution of parameters and for the Nash–Sutcliffe (NS) efficiency for the predicted flows, while the reliability and the continuous rank probability score (CRPS) improves. Less information in precipitation input resulted in a shift in the water balance parameter Pcorr, a model producing smoother streamflow predictions, giving poorer NS and CRPS, but higher reliability. The effect of calibrating the hydrological model using streamflow observations based on different rating curves is mainly seen as variability in the water balance parameter Pcorr. When evaluating predictions, the best evaluation scores were not achieved for the rating curve used for calibration, but for rating curves giving smoother streamflow observations. Less information in streamflow influenced the water balance parameter Pcorr, and increased the spread in evaluation scores by giving both better and worse scores.

Combining high resolution vertical gradients and sequence stratigraphy to delineate hydrogeologic units for a contaminated sedimentary rock aquifer system

Summary Hydrogeologic units (HGUs), representing subsurface contrasts in hydraulic conductivity, form the basis for all conceptual and numerical models of groundwater flow. However, conventionally, delineation of these units relies heavily on data sets indirect with respect to hydraulic properties. Here, we use the spatial and temporal characteristics of the vertical component of hydraulic gradient (i.e., vertical gradient) as the primary line of evidence for delineating HGUs for Cambrian–Ordovician sedimentary rocks at a site in Dane County, Wisconsin. The site includes a 16 km2 area encompassing a 3 km long mixed organic contaminants plume. The vertical gradients are derived from hydraulic head profiles obtained using high resolution Westbay multilevel systems installed at 7 locations along two, orthogonal 4 km long cross-sections and monitoring to depths between 90 and 146 m with an average of 3–4 monitoring zones per 10 m. These vertical gradient cross-sections reveal 11 laterally extensive HGUs with contrasting vertical hydraulic conductivity (Kv). The position and thickness of the Kv contrasts are consistently associated with sequence stratigraphic features (maximum flooding intervals and sequence boundaries) distinguished at the site using cores and borehole geophysical logs. The same sequence stratigraphic features are also traceable across much of the Cambrian–Ordovician aquifer system of the Midwest US. The vertical gradients and sequence stratigraphy were arrived at independently and when combined provide a hydraulically calibrated sequence stratigraphic framework for the site. This framework provides increased confidence in the precise delineation and description of the nature of HGU contacts in each borehole, reduced uncertainty in interpolation of the HGUs between boreholes, and some capability to predict HGU boundaries and thickness in offsite areas where high resolution hydraulic data sets are not available. Consequently, this HGU conceptual model will serve as a better basis for numerical simulations of groundwater flow and contaminant transport needed for continued risk assessment and to evaluate potential remedial technologies for the site. More broadly, the association between Kv contrasts and sequence stratigraphy demonstrated at this site provides valuable insight into the relationship between geology and hydraulic contrasts that can be transferred to other areas of the upper Midwest US and perhaps elsewhere to strata deposited in similar settings.

Mapping Atmospheric Moisture Climatologies across the Conterminous United States

Spatial climate datasets of 1981–2010 long-term mean monthly average dew point and minimum and maximum vapor pressure deficit were developed for the conterminous United States at 30-arcsec (~800m) resolution. Interpolation of long-term averages (twelve monthly values per variable) was performed using PRISM (Parameter-elevation Relationships on Independent Slopes Model). Surface stations available for analysis numbered only 4,000 for dew point and 3,500 for vapor pressure deficit, compared to 16,000 for previously-developed grids of 1981–2010 long-term mean monthly minimum and maximum temperature. Therefore, a form of Climatologically-Aided Interpolation (CAI) was used, in which the 1981–2010 temperature grids were used as predictor grids. For each grid cell, PRISM calculated a local regression function between the interpolated climate variable and the predictor grid. Nearby stations entering the regression were assigned weights based on the physiographic similarity of the station to the grid cell that included the effects of distance, elevation, coastal proximity, vertical atmospheric layer, and topographic position. Interpolation uncertainties were estimated using cross-validation exercises. Given that CAI interpolation was used, a new method was developed to allow uncertainties in predictor grids to be accounted for in estimating the total interpolation error. Local land use/land cover properties had noticeable effects on the spatial patterns of atmospheric moisture content and deficit. An example of this was relatively high dew points and low vapor pressure deficits at stations located in or near irrigated fields. The new grids, in combination with existing temperature grids, enable the user to derive a full suite of atmospheric moisture variables, such as minimum and maximum relative humidity, vapor pressure, and dew point depression, with accompanying assumptions. All of these grids are available online at http://prism.oregonstate.edu, and include 800-m and 4-km resolution data, images, metadata, pedigree information, and station inventory files.

Stochastic Polynomial Interpolation for Uncertainty Quantification With Computer Experiments

Multivariate polynomials are increasingly being used to construct emulators of computer models for uncertainty quantification. For deterministic computer codes, interpolating polynomial metamodels should be used instead of noninterpolating ones for logical consistency and prediction accuracy. However, available methods for constructing interpolating polynomials only provide point predictions. There is no known method that can provide probabilistic statements about the interpolation error. Furthermore, there are few alternatives to grid designs and sparse grids for constructing multivariate interpolating polynomials. A significant disadvantage of these designs is the large gaps between allowable design sizes. This article proposes a stochastic interpolating polynomial (SIP) that seeks to overcome the problems discussed above. A Bayesian approach in which interpolation uncertainty is quantified probabilistically through the posterior distribution of the output is employed. This allows assessment of the effect of interpolation uncertainty on estimation of quantities of interest based on the metamodel. A class of transformed space-filling design and a sequential design approach are proposed to efficiently construct the SIP with any desired number of runs. Simulations demonstrate that the SIP can outperform Gaussian process (GP) emulators. This article has supplementary material online.

Measurement and interpolation uncertainties in rainfall maps from cellular communication networks

Abstract. Accurate measurements of rainfall are important in many hydrological and meteorological applications, for instance, flash-flood early-warning systems, hydraulic structures design, irrigation, weather forecasting, and climate modelling. Whenever possible, link networks measure and store the received power of the electromagnetic signal at regular intervals. The decrease in power can be converted to rainfall intensity, and is largely due to the attenuation by raindrops along the link paths. Such an alternative technique fulfils the continuous effort to obtain measurements of rainfall in time and space at higher resolutions, especially in places where traditional rain gauge networks are scarce or poorly maintained. Rainfall maps from microwave link networks have recently been introduced at country-wide scales. Despite their potential in rainfall estimation at high spatiotemporal resolutions, the uncertainties present in rainfall maps from link networks are not yet fully comprehended. The aim of this work is to identify and quantify the sources of uncertainty present in interpolated rainfall maps from link rainfall depths. In order to disentangle these sources of uncertainty, we classified them into two categories: (1) those associated with the individual microwave link measurements, i.e. the errors involved in link rainfall retrievals, such as wet antenna attenuation, sampling interval of measurements, wet/dry period classification, dry weather baseline attenuation, quantization of the received power, drop size distribution (DSD), and multi-path propagation; and (2) those associated with mapping, i.e. the combined effect of the interpolation methodology and the spatial density of link measurements. We computed ~ 3500 rainfall maps from real and simulated link rainfall depths for 12 days for the land surface of the Netherlands. Simulated link rainfall depths refer to path-averaged rainfall depths obtained from radar data. The ~ 3500 real and simulated rainfall maps were compared against quality-controlled gauge-adjusted radar rainfall fields (assumed to be the ground truth). Thus, we were able to not only identify and quantify the sources of uncertainty in such rainfall maps, but also test the actual and optimal performance of one commercial microwave network from one of the cellular providers in the Netherlands. Errors in microwave link measurements were found to be the source that contributes most to the overall uncertainty.

Calendar time interpolation of amplitude maps from 4D seismic data

ABSTRACTA calendar time interpolation method for 2D seismic amplitude maps, done in two steps, is presented. The contour interpolation part is formulated as a quadratic programming problem, whereas the amplitude value interpolation is based on a conditional probability formulation. The method is applied on field data from the Sleipner CO2 storage project. The output is a continuous image (movie) of the CO2 plume. Besides visualization, the output can be used to better couple 4D seismic to other types of data acquired. The interpolation uncertainty increases with the time gap between consecutive seismic surveys and is estimated by leaving a survey out (blind test). Errors from such tests can be used to identify problems in understanding the flow and possibly improve the interpolation scheme for a given case. Field‐life cost of various acquisition systems and repeat frequencies are linked to the time‐lapse interpolation errors. The error in interpolated amplitudes increased by 3%‐4% per year of interpolation gap for the Sleipner case. Interpolation can never fully replace measurements.