On the prediction error variance of three common spatial interpolation schemes

Three forms of linear interpolation are routinely implemented in geographical information science, by interpolating between measurements made at the endpoints of a line, the vertices of a triangle, and the vertices of a rectangle (bilinear interpolation). Assuming the linear form of interpolation to be correct, we study the propagation of error when measurement error variances and covariances are known for the samples at the vertices of these geometric objects. We derive prediction error variances associated with interpolated values at generic points in the above objects, as well as expected (average) prediction error variances over random locations in these objects. We also place all the three variants of linear interpolation mentioned above within a geostatistical framework, and illustrate that they can be seen as particular cases of Universal Kriging (UK). We demonstrate that different definitions of measurement error in UK lead to different UK variants that, for particular expected profiles or surfaces (drift models), yield weights and predictions identical with the interpolation methods considered above, but produce fundamentally different (yet equally plausible from a pure data standpoint) prediction error variances.     

Nonparametric catchment clustering using the data depth function

ABSTRACT

The clustering of catchments is important for prediction in ungauged basins, model parameterization and watershed development and management. The aim of this study is to explore a new measure of similarity among catchments, using a data depth function and comparing it with catchment clustering indices based on flow and physical characteristics. A cluster analysis was performed for each similarity measure using the affinity propagation clustering algorithm. We evaluated the similarity measure based on depth–depth plots (DD-plots) as a basis for transferring parameter sets of a hydrological model between catchments. A case study was developed with 21 catchments in a diverse New Zealand region. Results show that clustering based on the depth–depth measure is dissimilar to clustering on catchment characteristics, flow, or flow indices. A hydrological model was calibrated for the 21 catchments and the transferability of model parameters among similar catchments was tested within and between clusters defined by each clustering method. The mean model performance for parameters transferred within a group always outperformed those from outside the group. The DD-plot based method was found to produce the best in-group performance and second-highest difference between in-group and out-group performance.

EDITOR D. Koutsoyiannis; ASSOCIATE EDITOR A. Viglione     

The dynamic response of a fluid‐filled borehole under a normal point load on the free surface

The dynamic response of a semi‐infinite fluid‐filled borehole embedded in an elastic half‐space under a concentrated normal surface load is analysed in the long‐wavelength limit. The solution of the problem is obtained with integral transforms in the form of a double integral with respect to the slowness and frequency. The partial P‐ and SV‐wave responses are further transformed to path integrals along Cagniard paths in the complex slowness plane. Unlike the traditional Cagniard‐de Hoop technique based on the Laplace transform of time dependence, this paper is based on the Fourier transform. The tube‐wave response is presented as a causal integral over a slowness range. The resultant representation in the time‐domain is suitable for the numerical evaluation of the complete response in the fluid‐filled borehole, especially at large distances. Asymptotic analysis of seismic phases arising in the borehole is performed on the basis of the obtained solution. The complete asymptotic wavefield consists in P‐ and SV‐waves, the Rayleigh wave and the low‐frequency Stoneley (tube) wave. Pressure synthetics obtained by the use of the asymptotic formulas are shown to be in good agreement with straightforward calculations.     

Uncertainty due to DEM error in landslide susceptibility mapping

Terrain attributes such as slope gradient and slope shape, computed from a gridded digital elevation model (DEM), are important input data for landslide susceptibility mapping. Errors in DEM can cause uncertainty in terrain attributes and thus influence landslide susceptibility mapping. Monte Carlo simulations have been used in this article to compare uncertainties due to DEM error in two representative landslide susceptibility mapping approaches: a recently developed expert knowledge and fuzzy logic-based approach to landslide susceptibility mapping (efLandslides), and a logistic regression approach that is representative of multivariate statistical approaches to landslide susceptibility mapping. The study area is located in the middle and upper reaches of the Yangtze River, China, and includes two adjacent areas with similar environmental conditions – one for efLandslides model development (approximately 250 km²) and the other for model extrapolation (approximately 4600 km²). Sequential Gaussian simulation was used to simulate DEM error fields at 25-m resolution with different magnitudes and spatial autocorrelation levels. Nine sets of simulations were generated. Each set included 100 realizations derived from a DEM error field specified by possible combinations of three standard deviation values (1, 7.5, and 15 m) for error magnitude and three range values (0, 60, and 120 m) for spatial autocorrelation. The overall uncertainties of both efLandslides and the logistic regression approach attributable to each model-simulated DEM error were evaluated based on a map of standard deviations of landslide susceptibility realizations. The uncertainty assessment showed that the overall uncertainty in efLandslides was less sensitive to DEM error than that in the logistic regression approach and that the overall uncertainties in both efLandslides and the logistic regression approach for the model-extrapolation area were generally lower than in the model-development area used in this study. Boxplots were produced by associating an independent validation set of 205 observed landslides in the model-extrapolation area with the resulting landslide susceptibility realizations. These boxplots showed that for all simulations, efLandslides produced more reasonable results than logistic regression.     

Rotation,activity, and lithium abundance in cool binary stars

We have used two robotic telescopes to obtain time‐series high‐resolution optical echelle spectroscopy and VI and/or by photometry for a sample of 60 active stars, mostly binaries. Orbital solutions are presented for 26 double‐lined systems and for 19 single‐lined systems, seven of them for the first time but all of them with unprecedented phase coverage and accuracy. Eighteen systems turned out to be single stars. The total of 6609 R = 55000 échelle spectra are also used to systematically determine effective temperatures, gravities, metallicities, rotational velocities, lithium abundances and absolute Hα‐core fluxes as a function of time. The photometry is used to infer unspotted brightness, V – I and/or b – y colors, spot‐induced brightness amplitudes and precise rotation periods. An extra 22 radial‐velocity standard stars were monitored throughout the science observations and yield a new barycentric zero point for our STELLA/SES robotic system. Our data are complemented by literature data and are used to determine rotation‐temperature‐activity relations for active binary components. We also relate lithium abundance to rotation and surface temperature. We find that 74% of all known rapidly‐rotating active binary stars are synchronized and in circular orbits but 26% (61 systems) are rotating asynchronously of which half have P_rot > P_orb and e > 0. Because rotational synchronization is predicted to occur before orbital circularization active binaries should undergo an extra spin‐down besides tidal dissipation. We suspect this to be due to a magnetically channeled wind with its subsequent braking torque. We find a steep increase of rotation period with decreasing effective temperature for active stars, P_rot α T^–7_eff, for both single and binaries, main sequence and evolved. For inactive, single giants with P_rot > 100 d, the relation is much weaker, P_rot α T^‐1.12_eff. Our data also indicate a period‐activity relation for Hα of the form R_Hα α P_0.24^rot for binaries and R_Hα α P_‐0.14^rot for singles. Its power‐law difference is possibly significant. Lithium abundances in our (field‐star) sample generally increase with effective temperature and are paralleled with an increase of the dispersion. The dispersion for binaries can be 1–2 orders of magnitude larger than for singles, peaking at an absolute spread of 3 orders of magnitude near T_eff ≈ 5000 K. On average, binaries of comparable effective temperature appear to exhibit 0.25 dex less surface lithium than singles, as expected if the depletion mechanism is rotation dependent. We also find a trend of increased Li abundance with rotational period of form log n (Li) α –0.6 log P_rot but again with a dispersion of as large as 3‐4 orders of magnitude (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Explicit versus implicit front advancing schemes for the simulation of hydraulic fracture growth

The proper computation of the time evolution of the fracture front is the main challenge of three-dimensional (3D) hydraulic fracture growth simulation. We discuss explicit and implicit variants of a hydraulic fracture propagation scheme based on a level set representation of the fracture. Such a scheme couples a finite discretization of the governing equations and the near-tip hydraulic fracture asymptotes. We benchmark the accuracy, robustness, and stability of these different front advancing schemes on a number of test cases. Our results indicate a large computational gain of the explicit scheme at the expense of a slightly less accurate solution (few percent less accuracy over few time steps) when crossing heterogeneities. The predictor corrector scheme combines at least an approximately 25% computational gain while retaining the stability and accuracy of the fully implicit version of the scheme in all cases.     

Transient analysis of full-space unbounded domains by reciprocal absorbing boundaries

In this paper, we explore a combination of the reciprocal absorbing boundary condition with perfectly matched discrete layers for the time-domain analysis of wave motion in the full plane. Coupling of these absorbing boundaries leads to solutions for out-of-plane as well as in-plane excitations of the full plane. Together with numerical transient simulations, harmonic analysis is also carried out for the purpose of verifying the results by comparing with available exact frequency-domain solutions. The proposed combination is investigated through several numerical examples, which demonstrate its accuracy and efficiency.     

岩石材料内部特征尺度对裂纹扩展机制的影响

为考察岩石材料细观结构特征对于裂纹扩展机制的影响,首先借助偏光显微镜和SEM观测了3类岩石材料中材料内部组织结构,分析了各自非均质程度和晶粒大小关系。之后,对3类岩石的带I型初始裂纹的3点弯曲试样进行了落重冲击试验,同时采用高速摄影获得了不同裂纹速度下的裂纹扩展形态。将这些结果对照分析,研究了3类岩石中裂纹扩展过程与各自材料组织结构的非均质程度和晶粒大小的关系,得出初步结论：组织结构内部特征尺度越大,动态裂纹所能达到的终极速度就相对越小。反之亦然。     

水-岩化学作用等效裂纹扩展细观力学模型

从化学腐蚀下岩石细观结构的变化出发,研究水化学溶液对岩石中裂纹的腐蚀作用。运用地球化学矿物-水反应的溶解动力学,考查水-岩反应化学溶液中不同时间段离子浓度的变化,从理论上探讨化学腐蚀下等效裂纹扩展的定量化分析方法。通过质量守恒定律计算出裂纹在迹长及隙宽方向的变化,将水化腐蚀等效成有效裂纹长度,建立了水-岩化学作用下等效裂纹扩展的计算公式并计算应力强度因子。通过对砂岩的研究结果表明,水-岩化学作用对岩石裂纹的扩展有显著影响,并且具有时间效应。同时,水化学溶液的离子浓度及pH值大小对裂纹的扩展存在影响,其中pH值的变化对岩石的断裂力学效应更加显著。当溶液的酸性越强或碱性越强,腐蚀越大,裂纹扩展越快;pH值为中性时,腐蚀作用得到较大缓解,裂纹发展缓慢。为定量研究裂隙岩体在水岩化学作用下的细观机理提供了一种新的思路。     

Born序列频散方程和Born-Kirchhoff传播算子

传统的Kirchhoff传播算子结构简洁,适用于描述横向均匀介质中波的传播.Ray-Kirchhoff传播算子较为精确地描述了波在非均匀介质中传播的运动学特征,其理论上的先天不足依赖于介质的复杂性.本文通过Born序列逼近波在非均匀介质中传播的大角度波分量,提出一种Born-Kirchhoff传播算子,将传统Kirchhoff传播算子的适用范围扩展至非均匀介质,同时描述波的运动学和动力学特征,其精度取决于Born序列逼近的阶数.利用Born序列频散方程,可以精确分析各阶Born-Kirchhoff传播算子对波长、传播角和非均质性的尺度依赖特征,其中,一阶Born-Kirchhoff传播算子的精度高于传统的相屏传播算子.波数域的Born-Kirchhoff传播算子对于高波数波是奇异的,导致波数域数值计算发散,但其空间域版本是非奇异的,无条件数值稳定,可通过Kirchhoff求和数值实施.本文给出各阶Born-Kirchhoff传播算子及其频散方程,可用于不同程度非均匀介质中的波传播模拟,复杂构造地震成像和速度估计.本文利用零阶和一阶Born-Kirchhoff传播算子计算简单二维模型的合成地震图,并与边界元法进行了比较.