EXPANSION OF PAPER AS AFFECTING AREA

Abstract

For the sake of the junior reader we may repeat an old and simple investigation. Let us suppose that the paper on which a map is printed undergoes a regular expansion p in one direction, say the X direction, and another regular expansion q in the Y direction, perpendicular to the former; it is required to know the effect of these expansions on the area of any parcel on the map. Note that, so far as the mathematics are affected, X and Y are not necessarily parallel to the margins of the sheet; we shall take them here as axes of any rectangular coordinate system. The symbols p and q are regarded as ratios, so that 100p and 100p represent the percentage expansions; if the paper contracts instead of expanding, no more is necessary than to change the sign.     

THE TRANSVERSE MERCATOR PROJECTION

Abstract

I. Introduction.—Map projection is a branch of applied mathematics which owes much to J. H. Lambert (v. this Review, i, 2, 91). In his “Beyträge zum Gebrauche der Mathematik und deren Anwendung” (Berlin, 1772) he arrived at a form of projection whereof the Transverse Mercator is a special case, and pointed out that this special case is adapted to a country of great extent in latitude but of small longitudinal width. Germain (“Traité des Projections”, Paris, 1865) described it as the Projection cylindrique orthomorphe de Lambert, but he also introduced the name Projection de Mercator transverse or renversée; he shows that Lambert's treatment of the projection was remarkably simple.     

BOWDITCH TRAVERSE ADJUSTMENT AND A MODIFICATION: NOTES

Abstract

In the second part of the paper on this subject in the last issue (30, 483) the references to the relative angular and linear closures are rather misleading. Mr Clendinning points out that the probable angular error at a station must be considered; the mean error is clearly different.     

SOME PROBLEMS OF RESECTION AND COUNTERSECTION-I

Abstract

“A Well-Defined mountain, though miles inland and never visited by the surveyors, will often prove the very keystone of a chart which cannot be regularly and theoretically triangulated” (“Hydrographic Surveying”, by Rear-Adm. Sir Wm J. L. Wharton, K.C.B., and Rear-Adm. Mostyn Field, F.R.S. 3rd Ed. 1909, p. 128). To many the reasons prohibiting the occupation of inland stations may be unknown; it may suffice to state that, in the past, British hydrographers have mapped many coastal waters where penetration of the land was at least inadvisable. Since the charts so made were in general sold to the world, seamen of all nations have benefited from the surveys.     

Population growth and the development of a central place system

This paper describes the spatial and functional evolution of a central place system as market conditions change with population growth. Utilizing a partial equilibrium optimization model, we examine the spatial response of two economic sectors to increases in market populations resulting from natural increase and migration. Response in both sectors is conditioned by threshold demand, with factor prices also affecting one of the sectors. As the central place system evolves it exhibits spatial and functional characteristics that are initially consistent with a Löschian landscape, then a Christallerian landscape at higher populations, while at even larger populations Krugman’s landscape emerges.     

Unsupervised Change Detection From Multichannel SAR Images

Multichannel synthetic aperture radar (SAR) data present a good potential for environmental monitoring and disaster management, owing both to their insensitivity to atmospheric and sun-illumination conditions, and to the improved discrimination capability they may provide as compared to single-channel SAR. However, this requires accurate and possibly automatic techniques to generate change maps from multichannel SAR images acquired from the same geographic area at different times. In this letter, an automatic unsupervised contextual change-detection method is proposed for two-date multichannel SAR images, by integrating a SAR-specific extension of the Fisher transform with a variant of the expectation-maximization algorithm and with Markov random fields. The method is validated by experiments on SIR-C/XSAR data     

On the Existence of Coverage and Integration Time Regimes in Bistatic SAR Configurations

Bistatic synthetic aperture radar (SAR) is an extension of traditional monostatic SAR, which increases the flexibility in designing SAR missions. We describe a scheme for the computation of integration time and azimuth coverage of bistatic SARs based on space-time diagrams. A classification of bistatic SAR configurations is introduced in terms of size and velocity on the ground of antenna footprints. Bistatic SAR regimes are also identified.     

High-Resolution Forecast Models of Water Vapor Over Mountains: Comparison With MERIS and Meteosat Data

Propagation delay due to variable tropospheric water vapor (WV) is one of the most intractable problems for radar interferometry, particularly over mountains. The WV field can be simulated by an atmospheric model, and the difference between the two fields is used to correct the radar interferogram. Here, we report our use of the U.K. Met Office Unified Model in a nested mode to produce high-resolution forecast fields for the 3-km-high Mount Etna volcano. The simulated precipitable-water field is validated against that retrieved from the Medium-Resolution Imaging Spectrometer (MERIS) radiometer on the Envisat satellite, which has a resolution of 300 m. Two case studies, one from winter (November 24, 2004) and one from summer (June 25, 2005), show that the mismatch between the model and the MERIS fields ( rms = 1.1 and 1.6 mm, respectively) is small. One of the main potential sources of error in the models is the timing of the WV field simulation. We show that long-wavelength upper tropospheric troughs of low WV could be identified in both the model output and Meteosat WV imagery for the November 24, 2004 case and used to choose the best time of model output.     

Best prediction in linear models with mixed integer/real unknowns: theory and application

In this contribution, we extend the existing theory of minimum mean squared error prediction (best prediction). This extention is motivated by the desire to be able to deal with models in which the parameter vectors have real-valued and/or integer-valued entries. New classes of predictors are introduced, based on the principle of equivariance. Equivariant prediction is developed for the real-parameter case, the integer-parameter case, and for the mixed integer/real case. The best predictors within these classes are identified, and they are shown to have a better performance than best linear (unbiased) prediction. This holds true for the mean squared error performance, as well as for the error variance performance. We show that, in the context of linear model prediction, best predictors and best estimators come in pairs. We take advantage of this property by also identifying the corresponding best estimators. All of the best equivariant estimators are shown to have a better precision than the best linear unbiased estimator. Although no restrictions are placed on the probability distributions of the random vectors, the Gaussian case is derived separately. The best predictors are also compared with least-squares predictors, in particular with the integer-based least-squares predictor introduced in Teunissen (J Geodesy, in press, 2006).     

Modeling long-period noise in kinematic GPS applications

We develop and test an algorithm for modeling and removing elevation error in kinematic GPS trajectories in the context of a kinematic GPS survey of the salar de Uyuni, Bolivia. Noise in the kinematic trajectory ranges over 15 cm and is highly autocorrelated, resulting in significant contamination of the topographic signal. We solve for a noise model using crossover differences at trajectory intersections as constraints in a least-squares inversion. Validation of the model using multiple realizations of synthetic/simulated noise shows an average decrease in root-mean-square-error (RMSE) by a factor of four. Applying the model to data from the salar de Uyuni survey, we find that crossover differences drop by a factor of eight (from an RMSE of 5.6 to 0.7 cm), and previously obscured topographic features are revealed in a plan view of the corrected trajectory. We believe that this algorithm can be successfully adapted to other survey methods that employ kinematic GPS for positioning.