Equivalence of the generalized Lie-Hori method and the method of averaging

In this investigation, a comparison is made of two methods for developing perturbation theories for non-canonical dynamical systems. The methods compared are the generalized Lie-Hori method and the method of averaging. In the comparison presented here, the equivalence of the methods up to the second order in the small parameter is shown. However, the approach used can be extended to demonstrate the equivalence for higher orders. To illustrate the equivalence Duffing's equation, the van der Pol equation and the oscillator with quadratic damping problem are solved using each method.     

Regularization and the computation of optimal trajectories

A completely regular form for the differential equations governing the three-dimensional motion of a continuously thrusting space vehicle is obtained by using the Kustaanheimo-Stiefel regularization. The differential equations for the thrusting rocket are transformed using the K-S transformation and an optimal trajectory problem is posed in the transformed space. The canonical equations for the optimal motion in the transformed space are regularized by a suitable change of the independent variable. The transformed equations are regular in the sense that the differential equations do not possess terms with zero divisors when the motion encounters a gravitational force center. The resulting equations possess symmetry in form and the coefficients of the dependent variables are slowly varying quantities for a low-thrust space vehicle.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Isotopic evidence of Holocene climatic change in the San Juan Mountains, Colorado

The δD of cellulose from ¹⁴C-dated wood, collected in the San Juan Mountains of southwestern Colorado, decreased by about 45‰ from 9600 to 3100 yr B.P. and an additional 25‰ to the present. The wood samples are from trees that grew above present-day tree line and reflect a time of warmer average summer temperatures. These changes in δD are interpreted to indicate a major change during the Holocene in the sources of moisture, in the seasonality of precipitation, or in both.     

Vegetation change detection from NOAA polar orbiting satellites

A Method is proposed for detecting vegetation change using global area coverage data from the NOAA-7 Advanced Very High Resolution Radiometer. Other methods of vegetation change detection, such as baseline images, are discussed, but not demonstrated. Vegetation Change Detection Images are produced by combining three separate weekly composites of Channel 2 minus Channel 1 AVHRR data. This compositing method takes the highest value of the differenced data for each week period and each pixel location. Three such weekly composites red imaged by individual red, green and blue CRT color guns respectively and then combined to form a color composite. The Vegetation Change Detection Image has 20 km resolution. It can be used to show an entire continent on a single image when the data point to CRT pixel ratio is 1:1. The vector vegetation change (magnitude and direction) is depicted by the color pattern resulting from the relative amount of red, green and blue placed in the pixel due to the individual weekly color composites. If there is no greenness change at a pixel location the amount of red, green and blue will be the same and a gray scale color results. The VCDI could provide a timely first alert for identifying regional and global agricultural changes.Doubts have been expressed about the reliability of this satellite data because of the unknown (and variable) effects of atmospheric attenuation. A basic qualitative rather than quantitative analysis of the data has been suggested as the correct approach.     

Optimal solutions of unobservable orbit determination problems

The method of data augmentation, in the form ofa priori covariance information on the reference solution, as a means to overcome the effects of ill-conditioning in orbit determination problems has been investigated. Specifically, for the case when ill-conditioning results from parameter non-observability and an appropriatea priori covariance is unknown, methods by which thea priori covariance is optimally chosen are presented. In problems where an inaccuratea priori covariance is provided, the optimal weighting of this data set is obtained. The feasibility of these ridge-type solution methods is demonstrated by their application to a non-observable gravity field recovery simulation. In the simulation, both ridge-type and conventional solutions are compared. Substantial improvement in the accuracy of the conventional solution is realized by the use of these ridge-type solution methods. The solution techniques presented in this study are applicable to observable, but ill-conditioned problems as well as the unobservable problems directly addressed. For the case of observable problems, the ridge-type solutions provide an improvement in the accuracy of the ordinary least squares solutions.     

GIS-based multicriteria evaluation for earthquake response: a case study of expert opinion in Vancouver,Canada

Natural Hazards - GIS-based multicriteria evaluation (MCE) provides a framework for analysing complex decision problems by quantifying variables of interest to score potential locations according...     

Relative size of fluvial and glaciated valleys in central Idaho

Quantitative comparisons of the morphometry of glaciated and fluvial valleys in central Idaho were used to investigate the differences in valley relief and width in otherwise similar geologic and geomorphic settings. The local relief, width, and cross-sectional area of valleys were measured using GIS software to extract information from USGS digital elevation models. Hillslope gradients were also measured using GIS software. Power-law relationships for local valley relief, width, and cross-sectional area as a function of drainage area were developed. Local valley relief in glaciated valleys relates to drainage area with a power-law exponent similar to fluvial valleys, but glaciated valleys are deeper for a given drainage area. Local valley width in glaciated valleys is greater than in fluvial valleys, but the exponent of the power-law relationship to drainage area is similar in both valley types. Local valley cross-sectional area in glaciated valleys increases with drainage area with a power-law exponent similar to fluvial valleys, however, glacial valleys have roughly 80% greater cross-sectional area. Steep valley walls in glaciated basins increase the potential for bedrock landsliding relative to fluvial basins. Both the Olympic Mountains of Washington and valleys in central Idaho show relationships in which glaciated valleys are up to 30% deeper than fluvial valleys despite differences in lithology, tectonic setting, and climate.     

Modification of δD values in eastern Nevada granitoid rocks spatially related to thrust faults

Stable isotope data have been determined for 13 Mesozoic and Tertiary plutons in eastern Nevada and nearby Utah. In the southern Snake Range of eastern Nevada, where relations are best exposed and have been most intensively studied, D, ¹⁸O, and apparent K-Ar ages depend on proximity to the Snake Range decollement. Where stresses resulting from late movement on the decollement have caused cataclasis of Oligocene (37 Ma) granitoid rock, ¹⁸O, D, and K-Ar age values as low as –2.5, –155, and 18 Ma, respectively, have been determined. Where there has been no cataclasis, ¹⁸O values of Jurassic, Cretaceous, and Oligocene granitoid rocks are apparently unaffected, but both D values and K-Ar ages have been modified for distances of tens of meters below the decollement.Results similar to those in the southern Snake Range have been observed in other eastern Nevada granitoid rocks spatially related to regional thrust faults, as in the Kern Mountains, the Toana Range, and the northern Egan Range. In each of these areas cataclasis or deformation of granitoid rocks has resulted in lowered ¹⁸O, D, and K-Ar age values. Where there has been no cataclasis or deformation, ¹⁸O values are unaffected, but both D and K-Ar age values have been lowered by stresses resulting from postcrystallization movement along overlying thrust faults.Many of the plutons discussed have not been deeply eroded, and spatially related thrust faults crop out. Where thrust faults are not in evidence and the granitoid rocks give D values lower than about –130 along with spuriously low K-Ar age results, modification of the D and K-Ar age values may have been caused by stresses related to late movement along an overlying (now eroded) thrust fault.