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  • Articles  (35)
  • kriging  (23)
  • Turbulence  (12)
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  • 1
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    Pure and applied geophysics 118 (1980), S. 494-527 
    ISSN: 1420-9136
    Keywords: MST radar ; Gravity waves ; Turbulence ; Stratospheric structure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract Powerful VHF radars are capable of almost continuously monitoring the threedimensional velocity vector and the distribution of turbulence in the middle atmosphere, i.e. the stratosphere and mesosphere. Methods of radar investigations of the middle atmosphere are outlined and the basic parameters, mean and fluctuating velocities as well as reflectivity and persistency of atmospheric structures, are defined. Results of radar investigations are described which show that the tropopause level as well as a criterion on the stability of the lower stratosphere can be deduced. Besides mean wind velocities, VHF radars can measure instantaneous velocities due to acoustic gravity waves. The interaction of gravity waves with the background wind is discussed, and it is shown that cumulus convection is an effective source of gravity waves in the lower stratosphere. The vertical microstructure of the stratosphere, manifesting itself in thin stratified sheets in which temperature steps occur, is investigated by applying knowledge from investigations of the oceanic thermocline. Possible origins, like shear generation and lateral convection of the microstructure of the stratosphere, are discussed. Observations of gravity waves in the mesosphere are reviewed and their connection with turbulence structures is pointed out. Finally, some open questions which could be answered by further VHF radar investigations are summarized.
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  • 2
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    Pure and applied geophysics 120 (1982), S. 95-107 
    ISSN: 1420-9136
    Keywords: Boundary layer ; Spectral density ; Turbulence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract We have carried out a series of recordings reproducing the small scale structures of temperature and windspeed within the boundary layer. The results obtained have been processed by computer and a direct comparison made between the slope of the spectral density functions and the theoretical slopen=5/3 defining the turbulence in the region of the inertial subrange. The measurements made at various altitudes on thin atmospheric samples demonstrate the variability of the spectral density slope with altitude. The hypotheses put forward by A. M. Obukhov, R. Bolgiano, G. H. Shur, P. Misme, A. Monin or S. Panchev, for example, enable the spectral variations and certain discontinuities in the turbulence observed during the experiments, within and beyond the boundary layer, to be explained.
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  • 3
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    Pure and applied geophysics 120 (1982), S. 648-661 
    ISSN: 1420-9136
    Keywords: Atmospheric turbulence ; Entrainment ; Turbulence ; Karman constant
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract This paper extends the theory of the entity and entrainment model of turbulence to obtain a numerical value of von Karman's constant,k=0.37. The formula is, $$k = (2a^3 /A)^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} \ln \beta $$ where,a=1/12 is the entrainment constant,A=1 is the turbulent decay constant, and β is the ratio in height of the successive self-similar layers of the theory, where β is evaluated as β=e 2. These new values fork and β improve the surface roughness length estimates derived from this theory.
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  • 4
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    Pure and applied geophysics 120 (1982), S. 758-771 
    ISSN: 1420-9136
    Keywords: Diffusion ; Turbulence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract Dye plumes were generated at three depths in the seasonal thermocline between 7 and 11 m, 22 km south of Key West on 21 August 1980 and photographed at about 10 second intervals with an underwater camera system. Eleven pairs of consecutive pictures are analyzed to determine the mean current vertical shear and the width of the plumes by positioning reference points relative to the rod attached to the camera system. The relative distances of reference points are calibrated with the stereophotogrammetric method for one pair. The eddy diffusivity is calculated by use of a model of turbulent diffusion developed byTaylor (1921). Its values range from 5 to 25 cm2s−1 for the plume widths ranging from 33 to 132 cm. The Richardson number is calculated for each pair of pictures with the vertical density gradient estimated from temperature profiles. Its values are higher than the critical value of 0.25 except for one case. The diffusivity was higher by orders of magnitude than the molecular one and indicates the presence of turbulence together with billow like features of the plumes in spite of high Richardson numbers. This suggests that the billow turbulence might be caused by effects of surface gravity waves and not by the Kelvin-Helmholtz instability.
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  • 5
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    Boundary layer meteorology 94 (2000), S. 461-493 
    ISSN: 1573-1472
    Keywords: Radiation fog ; Large-eddy simulation ; Turbulence ; Kelvin–Helmholtz instability ; Convection ; Mixed layer
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract In order to study the three-dimensional structure of radiation fogand to obtain a basic understanding of its generation mechanism,a numerical experiment is performed with a large-eddysimulation model and compared with the observation at Cabauw in the Netherlands. After confirming that the results are insatisfactory agreement with the observations, the structure of thefog and its generation mechanism are examined in more detail. Before the fog forms, the atmosphere is stable and an inversionlayer exists almost adjacent to the ground surface. As the fog grows, however, the stratification is destabilized and a mixed layerdevelops gradually. The longwave radiative cooling near thefog top contributes to the destabilization more than thecondensational heating does. The evolution of the fog can be classified into three stagesaccording to the behaviour of turbulent kinetic energy (TKE):formation, development, and dissipation stages.The fog layer has different flow structures at each stage.During the formation stage, longitudinal rolls similar tostreaks in channel flows appear near the ground surface.The development stage is characterized by an initiation oftransverse bands due to Kelvin–Helmholtz instability anda sudden increase of TKE. During the dissipation stage, longitudinalrolls and polygonal cells due to convective instability are organized.
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  • 6
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    Boundary layer meteorology 97 (2000), S. 219-249 
    ISSN: 1573-1472
    Keywords: Conditional concentration statistics ; Mixing ; Scalar dissipation ; Turbulence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract We analyse cross-wind-integrated statistics of theconcentration field of a conserved scalar for pointand line sources in grid turbulence. In particular,using wind-tunnel measurements we calculate thecross-wind integrated probability density function(pdf) for the scalar concentration. We then use thatquantity in the exact evolution equation for the pdfto calculate the cross-wind integrated mean of therate of dissipation of scalar variance, conditional onthe scalar concentration. Much of the variation ofthese statistics with distance downstream is accountedfor by scaling with concentration, length and timescales based on the development of the mean plume.This scaling thus suggests some simple practicalparameterisations of these statistics in terms ofmean-field quantities. One of the motivations for thiswork is to find a simple parameterisation for thescalar dissipation that can be used for modellingchemical reactions in plumes. We also consider the cross-wind integral of the firstfew moments of the concentration field and show thatthe integration greatly simplifies the budgets forthese moments. Thus the first moment is just thedownstream flux of the scalar, which is constant. Thesecond moment budget provides a check on the meandissipation estimated directly from the pdf evolutionequation.
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  • 7
    ISSN: 1573-1472
    Keywords: Katabatic flow ; Spectra ; Stable boundary layer ; Turbulence ; Waves
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract Observations obtained over a glacier surface in a predominantlykatabatic flow and with a distinctwind maximum below 13-m height are presented. The data werecollected using a 13-m high profilemast and two sonic anemometers (at about 2.5-m and 10-m heights).The spectra at frequencies belowthat of the turbulence range appear to deviate considerably fromthe curves obtained by Kaimal andco-workers during the 1968 Kansas experiment. The characteristicsof these deviations are compared tothe observations of others in surface-layers disturbed by anykind of large-scale outer-layer (orinactive) turbulence. In our case the disturbances arelikely to be induced by the highmountain ridges that surround the glacier. Moreover, the deviationsobserved in the cospectra seemto result from an, as yet, unspecified interaction between theinactive outer-layer turbulenceand the local surface-layer turbulence. Near the distinctwind maximum turbulence production ceasedwhile turbulence itself did not, probably the result ofturbulence transport from other levels. Consequently, we studied thelocal similarity relations using σw instead of u* as an alternative velocity scale. Wellbelow the wind maximum, and for relatively low stability(0〈 Rig 〈0.2), the flow behaves accordingto well established local-scaling similarity relationshipsin the stable boundary layer. For higherstability (Rig 〉 0.2), and near or above the wind maximum, the boundary-layer structure conforms tothat of z-less stratification suggesting that the eddy sizeis restricted by the local stability ofthe flow. In line with this we observed that the sensibleheat fluxes relate remarkably well to thelocal flow parameters.
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  • 8
    ISSN: 1573-1472
    Keywords: Coherent structures ; Numerical modelling ; Turbulence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract In two preceding papers, coherent structures of theatmospheric boundary layer (ABL), such as rollvortices or cells, were investigated through radar andaircraft observations collected during the TRAC-93(Turbulence Radar Aircraft Cells) experiment held inFrance in June 1993. The analysis of this experimentaldata set provided information on the spatialcharacteristics of these organisations (length scale,orientation, type ... ), their temporal and verticalevolution, and their relation with the dynamic andthermodynamic conditions of the ABL. For the thirdpaper in this series, a large eddy simulation model is used to examine the impact of thecoherent structures on the ABL vertical fluxes. Theanalysis of the simulated horizontal fields is madewith two-dimensional auto and cross-correlationsapplied on different pertinent ABL variables. Theresults emphasise a directional anisotropy of theseorganised fields throughout the ABL, much morepronounced in the heat flux fields, not only at thelength scale of organisations but also at theturbulence scales. This finding has an importantconsequence for traditional ABL flux measurementsbased on the hypothesis of isotropic and homogeneousturbulence. It can explain part of the underestimationof the surface fluxes often mentioned in theliterature. This approach makes it possible tomodify the concept of diffusion time (in chemicalmodelling) and could also lead to revised ABLparameterisations in Range Scale models.
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  • 9
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    Mathematical geology 32 (2000), S. 465-487 
    ISSN: 1573-8868
    Keywords: Bayesian inference ; elicited prior ; hierarchical models ; honesty parameter ; kriging ; optimal sequential sample design
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Spatial data are often sparse by nature. However, in many instances, information may exist in the form of “soft” data, such as expert opinion. Scientists in the field often have a good understanding of the phenomenon under study and may be able to provide valuable information on its likely behavior. It is thus useful to have a sensible mechanism that incorporates expert opinion in inference. The Bayesian paradigm suffers from an inherent subjectivity that is unacceptable to many scientists. Aside from this philosophical problem, elicitation of prior distributions is a difficult task. Moreover, an intentionally misleading expert can have substantial influence on Bayesian inference. In our experience, eliciting data is much more natural to the experts than eliciting prior distributions on the parameters of a probability model that is a purely statistical construct. In this paper we elicit data, i.e., guess values for the realization of the process, from the experts. Utilizing a hierarchical modeling framework, we combine elicited data and actual observed data for inferential purposes. A distinguishing feature of this approach is that even an intentionally misleading expert proves to be useful. Theoretical results and simulations illustrate that incorporating expert opinion via elicited data substantially improves the estimation, prediction, and design aspects of statistical inference for spatial data.
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  • 10
    ISSN: 1573-8868
    Keywords: geostatistics ; kriging ; variogram analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The Nanisivik Pb-Zn deposit at Strathcona Sound, northern Baffin Island, Canada, is an orebody that is an S-shaped lens 3000 m long, 75 m wide, and 2–20 m thick. It is an epigenetic karst channel filling. Total reserves are 6.9 million short tons of 14.12% Zn, 1.4% Pb, and 1.77 oz of Ag per ton. The mine was brought into production in 1976 at a rate of 2000 short tons per day. In the preproduction phase variogram studies based only on surface holes indicated a random distribution. For the evaluation of grades and for the determination of the confidence limits, classic statistical procedures were applied. Since the drilling density was different in different ore zones, a model was developed to test the effect of different drill hole spacings on the uncertainties in the present value in a discounted cash flow model. After the initial development and mining stage, structures could be recognized in semivariograms. Studies for a best drilling grid showed that the pattern of 25 ×20 m used in the mine could be extended to 35 ×35 m, taking into account that eight stopes are mined simultaneously. For ore reserve calculations weighing factors were calculated by kriging for six universally applicable geometric situations. Only the drill holes within a block and immediately surrounding it were taken into account. A test in a well-known zone yielded satisfactory results. A test to predict highs and lows of the grade accurately enough to optimally place the pillars failed.
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  • 11
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    Mathematical geology 14 (1982), S. 475-499 
    ISSN: 1573-8868
    Keywords: kriging ; lognormal estimation ; conservation of lognormality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract A theoretical study of the general case of the estimation of regionalized variables with a lognormal distribution is presented. The results of this study are compared to those obtained assuming conservation of lognormality. The numerical significance of the different solutions is illustrated by several simple examples.
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  • 12
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    Mathematical geology 14 (1982), S. 629-644 
    ISSN: 1573-8868
    Keywords: variogram ; kriging ; interpolation ; contouring ; inverse distance weighting ; geochemical gradient
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Four variogram models for regional groundwater geochemical data are presented. These models were developed from an empirical study of the sample variograms for more than 10 elements in groundwaters from two geologic regions in the Plainview quandrangle, Texas. A procedure is given for the estimation of the variogram in the isotropic and anisotropic case. The variograms were found useful for quantifying the differences in spatial variability for elements within a geologic unit and for elements in different geologic units. Additionally, the variogram analysis enables assessment of the assumption of statistical independence of regional samples which is commonly used in many statistical procedures. The estimated variograms are used in computation of kriged estimates for the Plainview quadrangle data. The results indicate that an inverse distance weighting model was superior for prediction than simple kriging with the particular variograms used.
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  • 13
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    Mathematical geology 14 (1982), S. 87-106 
    ISSN: 1573-8868
    Keywords: coal ; geostatistics ; kriging ; regional estimation ; systematic sampling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Two-dimensional systematic sampling of small plots followed by the kriging of those plots may be employed to obtain regional estimates of coal resources and measures of the accuracy of the estimates. The use of sampling makes large savings in computation possible. Two case studies involving the estimation of coal tonnage are discussed.
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  • 14
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    Mathematical geology 15 (1983), S. 131-143 
    ISSN: 1573-8868
    Keywords: Geostatistics ; resource classification ; mineral resources ; kriging ; resource definitions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Definition of “measured” and “indicated” resources tend to be vague. Yet, the calculation of such categories of resources in a mineral deposit calls for specific technical criteria, the selection of which depends upon the method of evaluation. This paper discusses how a geostatistical methodology provides the technical criteria required to classify reasonably assured resources by levels of assurance of their existence. A preliminary discussion concludes that the reasonably assured resources of a mineral deposit should be classified locally as in the traditional approach. However, the size of local volumes should not correspond to the elementary grid unit of drill holes but if possible to the minimum unit of selection that can be supported by the chosen mining method. The “level of confidence” of the reasonably assured resources can be quantitatively represented and determined by the estimation variance, one of the most important tools of geostatistics.
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  • 15
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    Mathematical geology 15 (1983), S. 537-551 
    ISSN: 1573-8868
    Keywords: kriging ; drift ; least squares ; residuals ; water table
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Universal kriging is applied to water table data from the Souss aquifer in central Morocco. The procedure accounts for the spatial variability of the phenomenon to be mapped. With the use of measured elevations of the water table, an experimental variogram is constructed that characterizes the spatial variability of the measured water levels. Spherical and Gaussian variogram models are alternatively used to fit the experimental variogram. The models are used to develop contour maps of water table elevations and corresponding estimation variances. The estimation variances express the reliability of the kriged water table elevation maps. Universal kriging also provides a contour map of the expected elevation of the water table (drift). The differences between the expected and measured water table elevations are called residuals from the drift. Residuals from the drift are compared with residuals obtained by more traditional least-squares analysis.
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  • 16
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    Mathematical geology 16 (1984), S. 3-18 
    ISSN: 1573-8868
    Keywords: Geostatistics ; kriging ; robust estimation ; time series
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Geological data frequently have a heavy-tailed normal-in-the-middle distribution, which gives rise to grade distributions that appear to be normal except for the occurrence of a few outliers. This same situation also applies to log-transformed data to which lognormal kriging is to be applied. For such data, linear kriging is nonrobust in that (1)kriged estimates tend to infinity as the outliers do, and (2)it is also not minimum mean squared error. The more general nonlinear method of disjunctive kriging is even more nonrobust, computationally more laborious, and in the end need not produce better practical answers. We propose a robust kriging method for such nearly normal data based on linear kriging of an editing of the data. It is little more laborious than conventional linear kriging and, used in conjunction with a robust estimator of the variogram, provides good protection against the effects of data outliers. The method is also applicable to time series analysis.
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  • 17
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    Mathematical geology 16 (1984), S. 601-615 
    ISSN: 1573-8868
    Keywords: kriging ; splines ; interpolation ; smoothing ; prediction ; Green's function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Let scalar measurements at distinct points x1, ⋯, xn be y1, ⋯, yn.We may look for a smooth function f(x)that goes through or near the points (xi, yi).Kriging assumes f(x)is a random function with known (possibly estimable) covariance function (in the simplest case). Splines assume a definition of the smoothness of a nonrandom function f(x).An elementary explanation is given of the fact that spline approximations are special cases of the solution of a kriging problem.
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  • 18
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    Mathematical geology 15 (1983), S. 687-699 
    ISSN: 1573-8868
    Keywords: cross validation ; kriging ; moving neighborhood ; unique neighborhood
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Cross validation is an appropriate tool for testing interpolation methods: it consists of leaving out one data point at a time, and determining how well this point can be estimated from the other data. Cross validation is often used for testing “moving neighborhood” kriging models; in this case, each unknown value is predicted from a small number of surrounding data. In “unique neighborhood” kriging algorithms, each estimation uses all the available data; as a result, cross validation would spend much computer time. For instance, with ndata points it would cost at least the resolution of nsystems of n × nlinear equations (each with a different matrix).Here, we present a much faster method for cross validation in a unique neighborhood. Instead of solving nsystems n × n,it only requires the inversion of one n × nmatrix. We also generalized this method to leaving out several points instead of one.
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  • 19
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    Mathematical geology 16 (1984), S. 19-35 
    ISSN: 1573-8868
    Keywords: groundwater ; geostatistics ; cokriging ; kriging ; transmissivity ; specific capacity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract This paper presents a new application of the cokriging technique for constructing maps of aquifer transmissivity from field measurements of transmissivity and specific capacity. The technique is illustrated using data from Yolo Basin, California. Cokriging is well-suited for estimating undersampled variables. To improve the accuracy of the estimation, cokriging considers the spatial auto-correlation of the variable to be estimated and the spatial cross-correlation between the variable to be estimated and other, better-sampled variables. Consequently, in regions that lack data of the variable to be estimated, accurate estimation can still be made on the basis of auto- and cross-correlation. In addition, estimation variances can be obtained with a little additional computation effort.
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  • 20
    ISSN: 1573-8868
    Keywords: kriging ; cokriging ; hydrology ; groundwater
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract A geostatistically based approach is developed for the identification of aquifer transmissivities in Yolo Basin, California. The approach combines weighted least-squares with universal kriging and cokriging techniques in an overall scheme that (1)considers a prioriknown information on aquifer transmissivity and specific capacities of wells, (2)considers uncertainties in water level and transmissivity data, and (3)estimates the reliability of the generated transmissivity values. Minimization of a global least-squares function that incorporates calibration and plausibility criteria leads to a transmissivity map that shows a good agreement with pumping-test results.
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  • 21
    ISSN: 1573-8868
    Keywords: Spatial variability ; nonintrinsic ; nonstationary ; drift ; kriging ; residuals ; groundwater levels
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract A method for obtaining pointwise or spatially averaged estimates of a nonintrinsic function is introduced based on residual kriging. The method relies on a stepwise iterative regression process for simultaneously estimating the global drift and residual semivariogram. Estimates of the function are then obtained by solving a modified set of simple kriging equations written for the residuals. The modification consists of replacing the true variogram in the kriging equations by the variogram of the residual estimates as obtained from the iterative regression process. The method is illustrated by considering groundwater levels in an Arizona aquifer. The results are compared with those obtained for the aquifer by the generalized covariance package BLUEPACK-3D.
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  • 22
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    Mathematical geology 32 (2000), S. 815-850 
    ISSN: 1573-8868
    Keywords: sequential simulation ; kriging ; local vs. global accuracy ; bootstrap ; entropy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Geostatistical simulations are globally accurate in the sense that they reproduce global statistics such as variograms and histograms. Kriging is locally accurate in the minimum local error variance sense. Building on the concept of direct sequential simulation, we propose a fast simulation method that can share these opposing objectives. It is shown that the multiple-point entropy of the resulting simulation is related to the univariate entropy of the local conditional distributions used to draw simulated values. Adding local accuracy to conditional simulations does not detract much from variogram reproduction and can be used to increase multiple-point entropy. The methods developed are illustrated using a case study.
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  • 23
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    Mathematical geology 32 (2000), S. 787-813 
    ISSN: 1573-8868
    Keywords: local accuracy ; conditional bias ; spectral amplitudes ; kriging ; stochastic simulation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The postprocessing algorithm introduced by Yao for imposing the spectral amplitudes of a target covariance model is shown to be efficient in correcting the smoothing effect of estimation maps, whether obtained by kriging or any other interpolation technique. As opposed to stochastic simulation, Yao's algorithm yields a unique map starting from an original, typically smooth, estimation map. Most importantly it is shown that reproduction of a covariance/semivariogram model (global accuracy) is necessarily obtained at the cost of local accuracy reduction and increase in conditional bias. When working on one location at a time, kriging remains the most accurate (in the least squared error sense) estimator. However, kriging estimates should only be listed, not mapped, since they do not reflect the correct (target) spatial autocorrelation. This mismatch in spatial autocorrelation can be corrected via stochastic simulation, or can be imposed a posteriori via Yao's algorithm.
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  • 24
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    Mathematical geology 12 (1980), S. 115-125 
    ISSN: 1573-8868
    Keywords: Geostatistics ; kriging ; robust estimation ; variogram
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract It is a matter of common experience that ore values often do not follow the normal (or lognormal) distributions assumed for them, but, instead, follow some other heavier-tailed distribution. In this paper we discuss the robust estimation of the variogram when the distribution is normal-like in the central region but heavier than normal in the tails. It is shown that the use of a fourth-root transformation with or without the use of M-estimation yields stable robust estimates of the variogram.
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    Mathematical geology 13 (1981), S. 415-428 
    ISSN: 1573-8868
    Keywords: geostatistics ; hydrologic transport ; kriging ; radioactive waste disposal
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract A significant part of the work of evaluating a geologic formation as a potential repository for hazardous wastes is the modeling of contaminant transport in the surrounding media in the event the repository is breached. The transport equations that are commonly used are deterministic functions. However, because the data can vary within the area being considered, there is a degree of uncertainty associated with the results obtained from the contaminant transport models. There are several ways to incorporate uncertainties into the transport equations, but they assume that distributions and parameters such as variances and covariances are known. This paper discusses the application of geostatistical spatial estimation techniques to estimate quantities used in transport modeling. The techniques are illustrated on data from an electric analog simulation of a two-dimensional ground water system. Geostatistical methods were used to estimate potential and hydraulic conductivity surfaces from data generated from the simulation of the ground water system. Although the two surfaces were highly dependent through Darcy's Law, they were estimated independently. Independent verification of the two surfaces showed that they approximately satisfied the required conservation of mass condition that: ∇ ⋅ v = 0.
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  • 26
    ISSN: 1573-8868
    Keywords: geostatistics ; kriging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Parallel variogram analyses, block kriging, and follow-up studies were effected for the lead content of part of the Prieska copper-zinc ore body and for the gold content of the highly variable Breef in a section of the Loraine gold mine, based first on untransformed values and second on logarithmically transformed values using the lognormal-de Wijsian model. For both models the effect was also analyzed of using the population mean or ignoring it. Practical follow-up comparisons confirm theoretical considerations and show that on these mines conditional biases can be eliminated conveniently by kriging with mean; also that the lognormal-de Wijsian model with mean gives the best results.
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    Mathematical geology 15 (1983), S. 245-257 
    ISSN: 1573-8868
    Keywords: kriging ; splines ; measurement errors ; generalized covariance ; structural identification
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract When drawing a contour map from a set of irregularly spaced data points, two methods are often used: The first corresponds to a rather aesthetic criterion and consists of obtaining contour lines which will be as “smooth”as possible and will honor the data points. This generally is the objective of the draftsman, and it can be automatically performed by the method of spline interpolation. The other method, used in kriging, is to compute the Best Linear Unbiased Estimator (B.L.U.E.),that is, to obtain a map as accurate as possible. Is it possible, in practice, to predict whether the aesthetic map will also be accurate? In this paper, we first examine the theoretical point of view: Spline interpolation is equivalent to kriging with a given (generalized)covariance. We then take an example to show how this question can be answered in practice: by testing how well the spline covariance is suited to the data.
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    Mathematical geology 15 (1983), S. 25-45 
    ISSN: 1573-8868
    Keywords: Discriminant analysis ; autocorrelation ; kriging ; multiple regression ; uranium ; mineral resources
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Geological and geophysical data digitized by Fabbri (1981)are augmented by eleven lakesediment geochemical variables in a quantitative study of the Kasmere Lake—Whiskey Jack Lake area, NW Manitoba. Each of 33 geological map units (Archean and Aphebian igneous and metamorphic rocks)have a fairly distinct multivariate geochemical and geophysical signal, as shown by discriminant analysis which is able to “recover” the geological map from the geophysical and geochemical data. Autocorrelation analysis of the geochemical variables, after removing a quadratic trend, indicate that copper, nickel, uranium, and cobait have a zone of influence from 15 to 30 km or more, with a marked anisotropy parallel to geological, geophysical, and structural trends which run NE-SW, parallel to the Wollaston fold belt. The observed uranium autocorrelation coefficients are modeled as a two-dimensional exponential function with elliptical contours. A trend-signal-noise model is applied to the lake-sediment uranium using kriging on residuals from the quadratic trend. A map comparing the resulting lake-sediment uranium signal with radiometric uranium/thorium patterns indicates considerable overlap and a generally good prediction of known uranium occurrences. In areas with a strong lake-sediment uranium signal, Aphebian metasediments have a somewhat different multivariate geochemical character from Archean and Hudsonian igneous and metamorphic rocks, suggesting a method of removing areas from the signal underlain by granites that may be economically uninteresting. Logistic and stepwise regression experiments based on a control area which includes most of the presently reported occurrences are used to predict areas favorable for uranium exploration.
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    Mathematical geology 16 (1984), S. 249-265 
    ISSN: 1573-8868
    Keywords: kriging ; moving neighborhood ; global neighborhood ; geostatistics ; automatic contouring
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The kriging estimator is usually computed in a moving neighborhood; only the data near the point to be estimated are used. This moving neighborhood approach creates discontinuities in mapping applications. An alternative approach is presented here, whereby all points are estimated using all the available data. To solve the resulting large linear system the kriging estimator is expressed in terms of the inverse of the covariance matrix. The covariance matrix has the advantage of being positive definite and the size of system which can be solved without encountering numerical instability is substantially increased. Because the kriging matrix does not change, the estimator can be written in terms of scalar products, thus avoiding the more time-consuming matrix multiplications of the standard approach. In the particular case of a covariance which is zero for distances greater than a fixed value (the range), the resulting banded structure of the covariance matrix is shown to lead to substantial computational savings in both run time and storage space. In this case the calculation time for the kriging variance is also substantially reduced. The present method is extended to the nonstationary case.
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    Mathematical geology 16 (1984), S. 283-301 
    ISSN: 1573-8868
    Keywords: Bayesian classifiers ; classification ; cluster analysis ; cluster validity ; fuzzyc means ; fuzzyc varieties ; fuzzy covariances ; geochemical prospecting ; glacial till ; kriging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract To evaluate the provenance of glacial till, the trace element content of magnetite was used. Magnetite was present in all known rock types and all till samples in the area investigated. By using fuzzy-set theory it was possible to group samples of magnetite taken from bedrock into relatively homogeneous and geologically meaningful groups and also, by fuzzy classification, to relate the till samples to the rocks in such a way that the relative contribution of each rock type to the till is estimated. Each rock and till sample is assigned a membership value between 0 and 1 for each rock type. The membership values, for a certain rock type in the till, are then interpolated by kriging onto maps. Magnetites from skarns associated with sulfide ores especially are rather distinct, and so a map of such membership values for till unveils all known ore deposits some 1–5 km “downstream”in the general direction of the ice flow. Other anomalies show up which cannot be related to hitherto known ores or mineralizations.
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    Mathematical geology 16 (1984), S. 809-822 
    ISSN: 1573-8868
    Keywords: variogram ; robustness ; kriging ; conditioning number
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Current ideas of robustness in geostatistics concentrate upon estimation of the experimental variogram. However, predictive algorithms can be very sensitive to small perturbations in data or in the variogram model as well. To quantify this notion of robustness, nearness of variogram models is defined. Closeness of two variogram models is reflected in the sensitivity of their corresponding kriging estimators. The condition number of kriging matrices is shown to play a central role. Various examples are given. The ideas are used to analyze more complex universal kriging systems.
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    Boundary layer meteorology 94 (2000), S. 139-163 
    ISSN: 1573-1472
    Keywords: Turbulence ; Coherent eddies ; Plant canopy ; Plane mixing layer ; Canopy shear scale ; Atmospheric stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract An analogy has been established between a plane mixing layer and the atmospheric flow near the top of a vegetation canopy. It is based on a common feature, a strong inflection in the mean velocity profile, responsible for hydrodynamical instabilities that set the pattern for the coherent eddies and determine the turbulence length scales. In an earlier study, this analogy was tested using a small data set from thirteen experiments, all in near-neutral conditions. It provided a good prediction of the streamwise spacing Λw of the dominant canopy eddies (evaluated from time series of vertical velocity) that appears to depend on a shear length scale Ls = U(h)/U'(h), where h is canopy height, U is mean velocity and U' the vertical gradient dU/dz. The present analysis utilizes an extensive data set of approximately 700 thirty-minute runs, from six experiments on two forest sites and a maize crop, with a large range of stability conditions. Λw was estimated for each run using the wavelet transform as an objective, automated detection method. First, the variations of Λw and Ls with atmospheric stability are discussed. Neutral and unstable values exhibit a large scatter whereas in stable conditions both variables decrease with increasing stability. It is subsequently found that Λw is directly related to Ls, in a way close to the neutral prediction Λw /h = 8.1Ls/h.The Strouhal number Str = Ls /Λw is then shown to vary with atmospheric stability, weakly in unstable conditions, more significantly in stable conditions. Altogether these results suggest that, to some extent, the plane mixing-layer analogy can be extended to non-neutral conditions. It is argued that the primary effect of atmospheric stability, at least in stable conditions, is to modify the shear length scale Ls through changes in U(h) and U'(h), which in turn determines the streamwise spacing of the active, coherent motions.
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    ISSN: 1573-1472
    Keywords: Field measurements ; Thinned forests ; Turbulence ; Wind tunnel
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract Many forest management methods alterstand density uniformly. The effectsof such a change on the wind andturbulence regimes in the forest arecritical to a number of processes governingthe stability of the stand and itsmicroclimate. We measured wind speed andturbulence statistics with a Dantec tri-axialhot-film probe in model forests of variousdensities (31–333 trees m-2), created byremoving whole trees in a regular pattern in awind tunnel, and compared them with similarmeasurements made with propeller anemometers insimilarly thinned plots (156–625 trees ha-1)within a Sitka spruce stand in Scotland. The results agree well, in general, with measurements made inother such studies with diverse canopy types.The systematic variations with density and verticalleaf-area distribution (which differed betweenwind-tunnel and field trees) in our work can explainmuch of the variability shown in scaled profiles ofbasic turbulence statistics reported in theliterature. The wind tunnel and field results are shown to be in good agreement overalldespite the difference in vertical leaf-areadistribution. Within-canopy and isolated-treedrag coefficients in the wind tunnel showthat tree-scale shelter effects increase astree density increases. The measurements indicatethat turbulence in the canopy is dominated bylarge-scale structures with dimensions of the sameorder as the height of the canopy as found inother studies but suggest that inter-tree spacing also modulates the size of these structures. These structures are associated with the sweeps that dominatemomentum exchange in the canopy and it is thisfact that allows the tri-axial probe to operate sowell despite the relatively narrow range of anglesin which the wind vector is correctly measured. Theratio of streamwise periodicity of these structuresto vorticity thickness varies systematically withtree density in the range 2.7–5.1, which spans theexpected range of 3.5–5 found in a laboratorymixing-layer, suggesting that tree spacing imposes another relevant length scale. This test andothers show that the results are in agreement withthe idea that canopy turbulence resembles that of a mixing layer even though they disagree with, and challenge the linear relationship between, streamwise periodicity andshear length scale presented recently in theliterature. The measurements are also in goodoverall agreement with simple drag models presented recently by other researchers.
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    Boundary layer meteorology 96 (2000), S. 211-255 
    ISSN: 1573-1472
    Keywords: Canopy ; Characteristic eddy ; Empirical-orthogonal-function ; Shear flow ; Turbulence ; Wind tunnel
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract We have conducted an Empirical Orthogonal Functionanalysis (EOF) of a three dimensional, 2-pointvelocity covariance field, measured in a wind tunnel. The rate of convergence of the EOF sequence was usedas an objective test for the presence of distinctlarge turbulent structures. We found that in theroughness sublayer (2h 〉 z 〉 0) the sequence convergedmuch more rapidly than in the lower surface layer(6h 〉 z 〉 0), 75% of the total velocity variancebeing captured by the first three of 42 eigenmodes;h is the canopy height. The analysis was extended to three dimensions, whereover 50% of the variance and most of the spatialstructure of the covariance fields were captured by aneven smaller fraction of the total number ofeigenmodes. With some relatively weak additionalassumptions we were able to construct the velocityfield of a ‘characteristic eddy’ or large coherentstructure. This consisted of a pair ofcounter-rotating streamwise vortices centred above thecanopy. The sense of rotation of the vortex pair wasopposite to that found in the wall region of boundarylayers but matched that found in plane mixing layers. A strong gust or sweep motion generated between thevortices was responsible for most of the shear stresscarried by the large eddies. The region of significanttransport of streamwise momentum by the characteristiceddy is much smaller than the region of coherence of theeddy's velocity field.
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    Boundary layer meteorology 97 (2000), S. 331-357 
    ISSN: 1573-1472
    Keywords: Aircraft measurements ; Entrainment ; Stratocumulus ; Structure functions ; Turbulence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract During the Dynamics and Chemistry of the MarineStratocumulus (DYCOMS) experiment in July–August 1985, the NCAR Electra aircraft flew a series of flight legs just at the top of the marinestratocumulus cloud decks that cap the mixed layer off the coast of southernCalifornia. Because of the corrugated structure of the cloud-top, the aircraft, which was flown at a nearly constant level and adjusted only to maintain its altitude at the average cloud-top height, was alternately within and above the clouds – roughly half the time in each domain. These legs were used to examine the structure of the cloud-top by compositing the segments on either side of the cloud/clear-air interface, which was identified by the transitions of liquid water measured by the Forward Scattering Spectrometer Probe (either increasing or decreasing) through a threshold of 0.04 × 10-3 kg m-3.An equivalent vertical distance (EVD) from the cloud-top was obtained from the horizontal flight legs by estimating the average slope of the cloud-top from the cloud-top radiation temperature. The results show that a near discontinuity occurs in variables across cloud top over an EVD of 0.3 m, but that above this, the air has already been modified by boundary-layer air. Thus, cloud-top is not the limit of mixing of boundary-layer air. This mixing may extend to tens of metres or more. The bulk Richardson number in the vicinity of cloud-top increases from near zero within the cloud to about 1.2 at an EVD of 3–6 m above cloud. Fluctuations of the three velocity components within cloud are nearly equal; above cloud the vertical component structure function is about half the horizontal components. The scalar structure functions are about an order of magnitude higher above cloud than in cloud. The structure parameters of temperature and humidity measured just below cloud-top agree reasonably well with predicted values based on a previously-developed model for the clear convective boundary layer. Above cloud, the scalar structure parameters are much larger, but their interpretation is questionable, since this region does notcontain isotropic turbulence.
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