Many industrial design problems are characterized by a lack of an analytical expression defining the relationship between design variables and chosen quality metrics. Evaluating the quality of new designs is therefore restricted to running a predetermined process such as physical testing of prototypes. When these processes carry a high cost, choosing how to gather further data can be very challenging, whether the end goal is to accurately predict the quality of future designs or to find an optimal design. In the multi-fidelity setting, one or more approximations of a design’s performance are available at varying costs and accuracies. Surrogate modelling methods have long been applied to problems of this type, combining data from multiple sources into a model which guides further sampling. Many challenges still exist; however, the foremost among them is choosing when and how to rely on available low-fidelity sources. This tutorial-style paper presents an introduction to the field of surrogate modelling for multi-fidelity expensive black-box problems, including classical approaches and open questions in the field. An illustrative example using Australian elevation data is provided to show the potential downfalls in blindly trusting or ignoring low-fidelity sources, a question that has recently gained much interest in the community.

While finite element (FE) modeling is widely used for ultimate strength assessments of structural systems, incorporating complex distortions and imperfections into FE models remains a challenge. Conventional methods typically rely on assumptions about the periodicity of distortions through spectral or modal methods. However, these approaches are not viable under the many realistic scenarios where these assumptions are invalid. Research efforts have consistently demonstrated the ability of point cloud data, generated through laser scanning or photogrammetry-based methods, to accurately capture structural deformations at the millimeter scale. This enables the updating of numerical models to capture the exact structural configuration and initial imperfections without the need for unrealistic assumptions. This research article investigates the use of point cloud data for updating the initial distortions in a FE model of a stiffened ship deck panel, for the purposes of ultimate strength estimation. The presented approach has the additional benefit of being able to explicitly account for measurement uncertainty in the analysis. Calculations using the updated FE models are compared against ground truth test data as well as FE models updated using standard spectral methods. The results demonstrate strength estimation that is comparable to existing approaches, with the additional advantages of uncertainty quantification and applicability to a wider range of application scenarios.

Although an accurate reliability assessment is essential to build a resilient infrastructure, it usually requires time-consuming computation. To reduce the computational burden, machine learning-based surrogate models have been used extensively to predict the probability of failure for structural designs. Nevertheless, the surrogate model still needs to compute and assess a certain number of training samples to achieve sufficient prediction accuracy. This paper proposes a new surrogate method for reliability analysis called Adaptive Hyperball Kriging Reliability Analysis (AHKRA). The AHKRA method revolves around using a hyperball-based sampling region. The radius of the hyperball represents the precision of reliability analysis. It is iteratively adjusted based on the number of samples required to evaluate the probability of failure with a target coefficient of variation. AHKRA adopts samples in a hyperball instead of an n-sigma rule-based sampling region to avoid the curse of dimensionality. The application of AHKRA in ten mathematical and two practical cases verifies its accuracy, efficiency, and robustness as it outperforms previous Kriging-based methods.

In engineering design, surrogate models are often used instead of costly computer simulations. Typically, a single surrogate model is selected based on the previous experience. We observe, based on an analysis of the published literature, that fitting an ensemble of surrogates (EoS) based on cross-validation errors is more accurate but requires more computational time. In this paper, we propose a method to build an EoS that is both accurate and less computationally expensive. In the proposed method, the EoS is a weighted average surrogate of response surface models, kriging, and radial basis functions based on overall cross-validation error. We demonstrate that created EoS is accurate than individual surrogates even when fewer data points are used, so computationally efficient with relatively insensitive predictions. We demonstrate the use of an EoS using hot rod rolling as an example. Finally, we include a rule-based template which can be used for other problems with similar requirements, for example, the computational time, required accuracy, and the size of the data.

The spatial distribution of the Atterberg limits can be used to distinguish the consistency and behaviour of a soil and its engineering properties, which strongly depends on the water content of the soil and types of silts and clays in the soil. By spatial modeling, and comparing the results of ordinary kriging with the cokriging approach, this study aims to find correlations between the Atterberg limits and the selected physical soil parameters in order to examine how effective they are in generating an understanding of the dynamics of a physical soil system.

In 156 soil samples, the Atterberg limits and soil moisture conditions were determined, and auxiliary functions were selected by application of cokriging using correlation analysis and regression equations obtained by the residual maximum likelihood (REML). These techniques were evaluated by the results of the mean absolute error (MAE) and the mean squared error (MSE). Cokriging analysis was found to be more effective at estimating the liquid limit (WLL) and the plastic limit (WPL) than kriging analysis and with smaller error values. On the other hand, the kriging approach, which had smaller MAE and MSE values, was more effective at estimating the plasticity index (WPI) values than the cokriging method. Unlike the REML regression equations, the field capacity (FC) value was the more suitable parameter for the cokriging estimates. When the necessary labour and time were considered for determining the Atterberg limits, both kriging and cokriging were found to be applicable for estimation of these limits.

An intensive field survey of an eastern Nebraska corn and soybean field was conducted to characterize the spatial structure and temporal stability of broadleaf weed seedling populations over two growing seasons. Anisotropy, the effect of direction on the relationship between observations, is present in the semivariogram for the velvetleaf and common sunflower populations in 1992 and 1993. The directional trends in aggregation are visible in kriged maps as elliptical shapes oriented east to west across the study area. In addition, there are two distinct zones of aggregation from north to south. These two distinct areas of aggregation are reflected as a ‘plateau’ in the north-south semivariogram. The distance over which this plateau extends indicates that the shape or size of the patch is contracting in the north-south direction (perpendicular to the crop row). The slope of the semivariogram in the east-west direction (aligned with the crop row) remains consistent from 1992 to 1993 suggesting that the shape of the patch is not changing. For sunflower populations, the slope of the north-south empirical semivariogram changes at 20 m, similar to the velvetleaf population semivariograms. This change, however, is reflected as a downward trend in the empirical semivariogram. The distance over which this trend occurs increases from 1992 to 1993 suggesting that seedling patch size was smaller in 1993 compared to 1992. Weed seedling establishment resulting from seed dispersal, differential seed and seedling mortality, or emergence may have resulted in the observed patch dynamics.

The nonuniform spatial distribution of weeds complicates sampling, modeling, and management of weed populations. Principles of a rational approach to analysis of weed spatial distribution, combining classical and spatial statistics, are presented using data for cumulative emergence of common lambsquarters in no-tillage soybean fields in 1990 and 1993. Classical statistics, dispersion indices, mean/variance relationships, and frequency histograms confirmed that raw and loge-transformed data were not normally distributed, that populations were aggregated, and that large-scale trends in population means violated assumptions of spatial statistics. Detrending was accomplished by median polishing loge-transformed data and confirmed by evaluation of standardized residuals and frequency histograms. Detrended residuals were used to construct omni-directional and uni-directional semivariograms to describe the spatial structure of the populations. A spherical model fit to the data was verified by cross validation. Semivariograms showed that common lambsquarters density was spatially autocorrelated at distances to 16 m, with more than 30% of the variance in density due to distance between field locations. Comparisons of kriged estimates and their standard deviations with and without detrending indicated that estimates using detrended data were more appropriate and more precise. Kriged estimates of common lambsquarters density were used to draw contour maps of the populations.

Precision agriculture (PA) requires reasonably homogeneous areas for site-specific management. This work explores the applicability of digital terrain classes obtained from a digital elevation model derived from UAV-acquired images, to define management units in in a relative flat area of about 6 ha. Elevation, together with other terrain variables such as: slope degree, profile curvature, plan curvature, topographic wetness index, sediment transport index, were clustered using the Fuzzy Kohonen Clustering Network (FKCN). Four terrain classes were obtained. The result was compared with a map produced by a classification of soil properties previously interpolated by ordinary kriging. The results suggest that areas for site-specific management can be defined from terrain classes based on environmental covariates, saving time and cost in comparison with interpolation of soil variables.

Current limit state surrogate modeling methods for system reliability analysis usually build surrogate models for failure modes individually or build composite limit states. In practical engineering applications, multiple system responses may be obtained from a single setting of inputs. In such cases, building surrogate models individually will ignore the correlation between different system responses and building composite limit states may be computationally expensive because the nonlinearity of composite limit state is usually higher than individual limit states. This paper proposes a new efficient Kriging surrogate modeling approach for system reliability analysis by constructing composite Kriging surrogates through selection of Kriging surrogates constructed individually and Kriging surrogates built based on singular value decomposition. The resulting composite surrogate model will combine the advantages of both types of Kriging surrogate models and thus reduce the number of required training points. A new stopping criterion and a new surrogate model refinement strategy are proposed to further improve the efficiency of this approach. The surrogate models are refined adaptively with high accuracy near the active failure boundary until the proposed new stopping criterion is satisfied. Three numerical examples including a series, a parallel, and a combined system are used to demonstrate the effectiveness of the proposed method.

Landscape-induced differences in soil variability and other parameters have potential effects on herbicide sorption, persistence, degradation, and, ultimately, efficacy. This study examined the spatial variability of herbicide efficacy across an eastern South Dakota field in continuous corn. Atrazine and alachlor had been applied for the previous 10 yr. The spatial variability observed in weed control was compared with herbicide sorption (Kd), mineralization rate, and first-order half-life (t 1/2), and field herbicide dissipation rates (DT50). Spatial structure was present in atrazine mineralization, weed biomass, and corn biomass data. The amount of atrazine and alachlor sorbed to soil collected from the summit position of the field was 10 and 20% less, respectively, than the amounts sorbed to backslope or toeslope soils. Generally, both herbicides had faster mineralization rates and shorter t 1/2 in summit than in backslope or toeslope soils. Weed biomass was correlated positively with elevation and total amount of atrazine mineralized, whereas corn biomass was correlated negatively with these parameters. These findings suggest that weed control can be improved by accounting for the landscape positional effects on differential herbicide mineralization and dissipation in fields.

Geostatistical techniques were used to describe and map the spatial distribution of crenate broomrape populations parasitizing broad bean over 6 yr (from 1985 to 1990). In the first year, the spatial distribution was random, but from 1986 to 1989, crenate broomrape populations were clearly aggregated. The crenate broomrape infection severity (IS: number of emerged broomrape m−2) increased every year, from an average of 0.45 in 1985 to 29.4 in 1989, with a slight decrease the following year (IS = 27.4). Spherical functions provided the best fit because the cross-validation criteria were accomplished in all study cases. Kriged estimates were used to draw contour maps of the populations. About 34.3, 43.3, and 74.3% of the field plot surface exhibited an IS ≥ 1 (economic threshold) in 1985, 1986, and 1987, respectively, and nearly 100% of the area exceeded the economic threshold from 1988 to 1990; 1985 and 1986 were key years for control of the parasitic weed population. The percentage of infested area at different IS intervals in each year's map obtained by kriging was used to estimate the percentage of yield losses in each infested area (YA) with the equation: YA = A ∗ Ymax ∗ (1 − IS ∗ 0.124), where A is the infested area at a given IS interval and Ymax is the expected broomrape-free broad bean yield. Yield losses under different IS intervals were compared with yield loss attributable to a uniform distribution of crenate broomrape. Results showed that yield loss assuming a uniform distribution of crenate broomrape was clearly overestimated, which is important to avoid overuse of herbicides.

Geostatistical techniques were used to describe and map weed spatial distribution in two sunflower fields in Cabello and Monclova, southern Spain. Data from the study were used to design intermittent spraying strategies. Weed species, overall infestation severity (IS) index, and spatial distribution varied considerably between the two sites. Weed species displayed differences in spatial dependence regardless of IS. The IS mapping of each single weed and of the overall infestation was achieved by kriging, and site-specific application maps were then drawn based on the multi-species weed map and the estimated economic threshold (ET). Herbicide treatment was assumed to be needed for an overall IS score of 2 or 3, and the infested “area exceeding the economic threshold” was determined. The overall weed-infested area varied considerably between locations. About 99 and 38% of the total area was moderately infested (IS ≥ 2) at Monclova and Cabello, respectively. Therefore, if a given herbicide were applied just to the areas exceeding the ET, a significant herbicide saving would be realized in Cabello but not in Monclova. A multi-species spatial analysis provides an opportunity to make site-specific management recommendations from a map of the distribution of IS of the total infestation. Furthermore, only in fields with hard-to-control weed species (e.g., nodding broomrape and corn caraway) would site-specific herbicide application maps developed from total weed infestations need to be complemented with targeted site-specific herbicide treatments to prevent further spread of these species, although their IS might be low.

Recent interest in describing the spatial distribution of weeds and studying their association with site properties has increased the use of interpolation to estimate weed seedling density from spatially referenced data. In addition, farmers and consultants adopting elements of site-specific farming practices are using interpolation methods for mapping weed densities as well as soil properties. This study was conducted to compare the performance of four interpolation methods, namely inverse-distance weighting (IDW), ordinary point kriging (OPK), minimum surface curvature (MC), and multiquadric radial basis function (MUL), with respect to their ability to map weed-seedling densities. These methods were evaluated on data from four weed species, velvetleaf, hemp dogbane, common sunflower, and foxtail species, of contrasting biology and infestation levels in corn and soybean production fields in Nebraska. Mean absolute difference (MAD) and root mean square (RMS) between the observed point sample data and the estimated weed seedling density surfaces were used to evaluate the performance of the interpolation methods. Four neighborhood search types were compared within each interpolation method, and Search3 (12 to 16 neighboring locations) generated an interpolated surface with the smallest MAD and RMS indicating the highest precision. IDW with a power coefficient of p = 4 gave the smallest MAD and RMS, as did a test with an elliptical search and no anisotropy. The level of precision of all four interpolation methods was very poor for weed species with low infestation levels (> 75% of field weed-free; MAD ranged from 100 to 187% of the observed mean density), whereas precision was improved for weed species with high infestation levels (< 25% of field weed-free; MAD ranged from 45 to 85%). No single interpolator appears to be more precise than another. Implications of this study indicate that grid sample spacing and quadrat size are more important than the specific interpolation method chosen.

Weed maps are typically produced from data sampled at discrete intervals on a regular grid. Errors are expected to occur as data are sampled at increasingly coarse scales. To demonstrate the potential effect of sampling strategy on the quality of weed maps, we analyzed a data set comprising the counts of capeweed in 225,000 quadrats completely covering a 0.9-ha area. The data were subsampled at different grid spacings, quadrat sizes, and starting points and were then used to produce maps by kriging. Spacings of 10 m were found to overestimate the geostatistical range by 100% and missed details apparently resulting from the spraying equipment. Some evidence was found supporting the rule of thumb that surveys should be conducted at a spacing of about half the scale of interest. Quadrat size had less effect than spacing on the map quality. At wider spacings the starting position of the sample grid had a considerable effect on the qualities of the maps but not on the estimated geostatistical range. Continued use of arbitrary survey designs is likely to miss the information of interest to biologists and may possibly produce maps inappropriate to spray application technology.

The size, location, and variation in time of weed patches within an arable field were analyzed with the ultimate goal of simplifying weed mapping. Annual and perennial weeds were sampled yearly from 1993 to 1997 at 410 permanent grid points in a 1.3-ha no-till field sown to row crops each year. Geostatistical techniques were used to examine the data as follows: (1) spatial structure within years; (2) relationships of spatial structure to literature-derived population parameters, such as seed production and seed longevity; and (3) stability of weed patches across years. Within years, densities were more variable across crop rows and patches were elongated along rows. Aggregation of seedlings into patches was strongest for annuals and, more generally, for species whose seeds were dispersed by combine harvesting. Patches were most persistent for perennials and, more generally, for species whose seeds dispersed prior to expected dates of combine harvesting. For the most abundant weed in the field, the annual, Setaria viridis, locations of patches in the current year could be used to predict patch locations in the following year, but not thereafter.

Weeds generally occur in patches in production fields. Are these patches spatially and temporally stable? Do management recommendations change on the basis of these data? The population density and location of annual grass weeds and common ragweed were examined in a 65-ha corn/soybean production field from 1995 to 2004. Yearly treatment recommendations were developed from field means, medians, and kriging grid cell densities, using the hyperbolic yield loss (YL) equation and published incremental YL values (I), maximum YL values (A), and YL limits of 5, 10, or 15%. Mean plant densities ranged from 12 to 131 annual grasses m−2 and < 1 to 37 common ragweed m−2. Median weed densities ranged from 0 to 40 annual grasses m−2 and were 0 for common ragweed. The grass I values used to estimate corn YL were 0.1 and 2% and treatment was recommended in only 1 yr when the high I value and either the mean or median density was used. The grass I values used for soybean were 0.7 and 10% and estimated YL was over 10% all years, regardless of I value. The common ragweed I values were 4.5 and 6% for corn and 5.1 and 15.6% for soybean. On the basis of mean densities, fieldwide treatment would have been recommended in 6 of 9 yr but in no years when the median density was used. Recommendations on the basis of grid cell weed density and kriging ranged from > 80% of the field treated for grass weeds in 3 of 4 yr in soybean to < 20% of the field treated for common ragweed in 2002 and 2004 (corn). Grass patches were more stable in time, space, and density than common ragweed patches. Population densities and spatial distribution generally were variable enough so that site-specific information within this field would improve weed management decisions.

Targeting weed patches for site-specific herbicide applications potentially represents cost savings for operators, reduction in environmental herbicide effects, and increased efficiency of weed control. An experiment was initiated in a no-till corn field in Ontario, Canada, in 1998 and was continued in rotation with no-till soybeans in 1999. Weeds were intensively scouted, and distribution maps of the most common weeds (field horsetail, spiny sowthistle, dandelion, and common lambsquarters) were generated for both years. A prescription map for each plot was made using the weed density maps. Treatment decisions were based on a weed threshold value of 1 shoot m−2. Four herbicide treatments were compared: a conventional broadcast, a site-specific application targeting weed patches only, and two combinations of broadcast and site-specific applications. Treatments were applied using a direct-injection sprayer. Efficacy of weed control and yield were compared among treatments. In 1998 and 1999 there were no differences in the level of weed control or yield among treatments. The average percent area sprayed was reduced as much as 26% in the site-specific treatment in 1998 and up to 59% in the site-specific and broadcast combination treatments in 1999. For those species present in the field, patches ranged from highly aggregated to completely random, and patch stability ranged from very stable to very unstable over the 2 yr.