Sharpening of Observational Data in Two Dimensions

Burr, Edmund John.

doi:10.1071/ph550030

Cited by 19 publications

(3 citation statements)

References 2 publications

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“…But it is not possible to take measurements at points. Some finite aperture must be used, and post-observational correction compensates for the instrumental broadening [7,27]. In the geographical case the census scans the country with a variable aperture instrument!…”

Section: Geographical Enhancementmentioning

confidence: 99%

Geographical Filters and their Inverses*

Tobler

1969

Geographical Analysis

110

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Science proceeds by detecting structures embedded in observational data. It is no simple matter to separate these general structures from the specific details. Many analytical techniques have been designed to do just this; to separate the important from the unimportant. Among the best known of these are statistical methods of accounting for the variability in the observations. The popularity of techniques which yield ordered results (stepwise regression, principal components analysis, or clustering methods, for example) thus becomes clear.Geographical versions of data filtering occur frequently in cartography. Maps are generalized representations since it is clearly impossible to display reality in all its complexity at a reduced scale. Not too surprisingly, map makers are unable to provide explicit abstract statements about the process of map generalization, but some empirical evidence suggests constancy of information content per map centimeter squared, regardless of map scale [5'7]. As Perkal[58] and Tobler [5Z] point out, this is an area for interesting research. Clustering techniques have obvious applicability to dot map generalization [SI] ; network simplification and algorithms do not appear nearly as well developed [Z, 16,40,59]. Even the venerable techniques of regionalization, as forms of data simplification, can probably be improved [&I. Trend AnalysisMost recently popularized have been filtering methods for the analysis of geographical trends [lo]. These are simple extensions of

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Section: Geographical Enhancementmentioning

confidence: 99%

Geographical Filters and their Inverses*

Tobler

1969

Geographical Analysis

110

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“…At regional to teleseismic distances the seismic wavefield is sufficiently coherent that array processing beam forming can be applied to dense samplings of the wavefield. While array processing is common and well understood in seismology [e.g., Burr , 1955; Burg , 1964; Green et al , 1965, Frosch and Green , 1966; Whiteway , 1966; Gangi and Disher , 1968; Capon , 1969], it is worthwhile to present a short review here to point out some differences in its application to HRGPS time series “seismograms.” Array processing analyzes multiple samplings of a wavefield in both time, using seismograms, and space, using simultaneous sampling from seismograms at different locations. To illustrate the basics of array processing, consider the D'Alembert traveling wave solution to the wave equation, f ( • ± vt ).…”

Section: Discussionmentioning

confidence: 99%

Love wave dispersion in central North America determined using absolute displacement seismograms from high‐rate GPS

Davis

Smalley

2009

J. Geophys. Res.

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[1] We use seismic array processing of high-rate GPS (HRGPS) displacement time series from the Great, 2004, M w 9+, Sumatra-Andaman earthquake recorded at over 90 nonuniformly distributed HRGPS stations in central North America to determine the fundamental Love wave phase velocity dispersion curve there. These measurements were performed using frequency domain beam forming, which we show reduces the effects of GPS multipath and common mode noise on the measurement of surface wave phase velocity and azimuth. Our HRGPS based results for surface wave phase velocity agree well with those obtained from 28 broadband seismometers between periods of 20 to 300 s. By separating waves based on their apparent velocity, beam forming supports the simple model for relative displacement HRGPS time series as being the difference between the reference and kinematic station's absolute displacements. Beam forming also demonstrates that for differential GPS processing the infinite apparent velocity beam of an array is composed of the sum of common mode noise and the reference station's absolute displacements, multipath and noise. We show the infinite apparent velocity beam, which we call the generalized spatial filter, is similar to the spatial filter commonly used to remove common mode noise from HRGPS seismograms and can be used to produce absolute displacement time series for the kinematic stations with reduced common mode noise. Beam forming also suggests a filtering method, complimentary to sidereal filtering, to produce GPS multipath reduced HRGPS time series.Citation: Davis, J. P., and R. Smalley Jr. (2009), Love wave dispersion in central North America determined using absolute displacement seismograms from high-rate GPS,

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“…29 The condition is that V and H have inverses, equivalently that the transfer function does not contain zeros in the frequency domain. 3,30 . This, then, is an example of a transformation applied to the substantive, rather than the locative, data.…”

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A Transformational View of Cartography

Tobler¹

1979

The American Cartographer

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Cartographic transformations are applied to locative geographic data and to substantive geographic data. Conversion between locative aliases are between points, lines, and areas. Substantive transformations occur in map interpolation, filtering, and generalization, and in map reading. The theoretical importance of the inverses is in the study of error propagation effects. Leonard Bernstein, in a recent television lecture, made an exciting, and largely successful, attempt to describe musical concepts in terms of Noam Chomsky's ideas concerning transformational grammars, as originally devised for linguistics. 2, 4 A similar, though less ambitious, attempt is made here to look at a range of cartographic activities from a transformational point of view. The treatment is not particularly Chompskian, although some work on picture languages is available. 7, 11, 21, 23 The idea of transformations is hardly new to cartography. The well-known example is the procedure by which we associate locations on the two-dimensional surface of the earth with locations on the two-dimensional surface of a piece of paper. 16 The study of the subject of map projections once constituted the bulk of what was known as "mathematical cartography." An extension 'was suggested in 1958 by Julian Perkal: "Cartographic transformations, employed to represent the surface of the earth on maps, are of two simple types: map projections and generalization. Map projections are obtained by an objective mathematical operation. The subject of this paper is the second cartographic transformation-map generalization." Perkal then proceeded to derive mathematical rules for a particular method of generalizing lines on maps. As can be seen in his paper the method embodies a notion of spatial resolution. 20 Since that time considerable further work has been done on line generalization, and it is now known that the problem admits of more solutions than were recognized in Perkal's early paper. 9, 25, 28, 41 At the time that Perkal wrote, computer cartography had not reached the large scale implementation which it now enjoys. Automatic plotters were not commonplace and he did not have available the computer tapes, now in widespread use, of world coastlines and hydrography, of political boundaries, of city street patterns, or of world topography. These fundamental geographical data can now be processed automatically, either to solve geographical problems directly, or to provide geographic illustration in the form of pictures. Thus, a broader view is required of mathematical cartography and has elsewhere been labeled analytical cartography. 35 This is the changing paradigmatic world of cartography. Perkal recognized only two classes of cartographic transformation: map projections and map generalization. But the entire process of making, and using, a map can be viewed as a sequence of transformations. Original observations are manipulated and digested in various ways to obtain the data going into a map. In the design phase these are converted to a graphic representation,...

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Sharpening of Observational Data in Two Dimensions

Abstract: Distributions in two dimensions as measured are always blurred or smoothed by limitations in the observing technique. Recovery of the true distribution involves the solution of an integral equation of the form where the functions g, h are known from observation

Cited by 19 publications

References 2 publications

Geographical Filters and their Inverses*

Geographical Filters and their Inverses*

Love wave dispersion in central North America determined using absolute displacement seismograms from high‐rate GPS

A Transformational View of Cartography

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