Reply by the author to Th. Krey

Geophysics ◽  
1984 ◽  
Vol 49 (10) ◽  
pp. 1807-1807
Author(s):  
Peter Hubral
Keyword(s):  
Short Note ◽  
Normal Wave ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Geometrical Spreading ◽  
The Subject ◽  
Amplitude Reflection

I am grateful to Th. Krey for his derivation of formula (14) which is valid for the two‐dimensional case and which clearly shows what is actually obtained when subjecting primary reflections in neighbouring shot records to, what I called in the subject note, the process of normal wave stacking. Krey’s formula (14) confirms that the amplitude loss of primary shot record reflections, that is due to geometrical spreading, is removed by the normal wave stack as I stated in my short note. However, the formula also shows that the resulting event in the normal wave stack section does not, as I implied, provide a true amplitude reflection, which was correctly defined in my previous paper (Hubral, 1983) and in Krey’s short note (Krey, 1983). I am satisfied to see that Krey’s formula (14) however indicates that primary reflections in a normal wave stack section can very easily be transformed to the desired accurate true amplitude reflections with two simple corrections only.

On: “Simulating true amplitude reflections by stacking shot records,” by P. Hubral (GEOPHYSICS, v. 49, p. 303–306, 1984).

Geophysics ◽  
1984 ◽  
Vol 49 (10) ◽  
pp. 1806-1807
Author(s):  
Th. Krey
Keyword(s):  
Short Note ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Precise Result ◽  
Normal Waves ◽  
Analytical Derivation ◽  
The Subject ◽  
Line Survey ◽  
Zero Offset

Quite recently Peter Hubral published a short note in which he described a special, very perspicuous stacking method which, starting from the records of a line survey, produces true amplitude reflections for “normal waves,” as defined in his Introduction. In the following I want to supplement Hubral’s note by showing the analytical connection with Hubral’s earlier paper (Hubral, 1983) and the additional short note by Krey (Krey, 1983). My present investigation will be two‐dimensional (2-D) as is that in the subject paper; an extension to the three‐dimensional (3-D) case is conceptionally easy for the following analytical derivation as well as for Hubral’s note. Besides a basic confirmation of Hubral’s findings, I shall show that the result of Hubral’s method has still to be multiplied by [Formula: see text] in the 2-D case and by [Formula: see text] in the 3-D case in order to obtain the precise result. Here ω is the frequency. Moreover the angle of emergence α of the zero‐offset raypath has to be taken into account.

Spherical Geometries and Multigroups

1950 ◽  
Vol 2 ◽  
pp. 100-119 ◽  
Author(s):  
Walter Prenowitz
Keyword(s):  
Three Dimensional ◽  
Spherical Geometry ◽  
Great Circle ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Elliptic Geometry ◽  
The Subject ◽  
Case Based

1. Introduction. The notion spherical geometry is suggested by the familiar geometry of the Euclidean 2-sphere in which the role of path is played by “arc of great circle”. The first postulational treatment of the subject seems to be that of Halsted [10] for the two-dimensional case. Kline [11] under the name double elliptic geometry, gave a greatly simplified foundation for the three-dimensional case based on the primitive notions point and order. Halsted and Kline study not merely descriptive (that is positional, non-metrical) properties of figures but also introduce metrical notions by postulating or defining congruence. Kline includes a continuity postulate designed to yield real spherical geometry.

The formation of gas hydrates under shock wave impact on the bubble media (two-dimensional case)

2010 ◽  
Vol 7 ◽  
pp. 90-97
Author(s):  
M.N. Galimzianov ◽  
I.A. Chiglintsev ◽  
U.O. Agisheva ◽  
V.A. Buzina
Keyword(s):  
Shock Wave ◽  
Gas Hydrates ◽  
Hydrate Formation ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Wave Impact ◽  
Bubble Liquid ◽  
One Dimensional ◽  
Bubble Media ◽  
Main Factors

Formation of gas hydrates under shock wave impact on bubble media (two-dimensional case) The dynamics of plane one-dimensional shock waves applied to the available experimental data for the water–freon media is studied on the base of the theoretical model of the bubble liquid improved with taking into account possible hydrate formation. The scheme of accounting of the bubble crushing in a shock wave that is one of the main factors in the hydrate formation intensification with increasing shock wave amplitude is proposed.

Regions-based Two-dimensional Continua: The Euclidean Case

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Mathematical Induction ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Entire Space ◽  
Euclidean Case ◽  
Free Basis ◽  
One Dimensional ◽  
Archimedean Property ◽  
The One

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

Two-dimensional buoyant plumes in a uniform co-flow

2021 ◽  
Vol 932 ◽  
Author(s):  
Gary R. Hunt ◽  
Jamie P. Webb
Keyword(s):  
Theoretical Investigation ◽  
Analytical Solutions ◽  
Richardson Number ◽  
Relative Magnitude ◽  
Dynamical Behaviour ◽  
Finite Width ◽  
Two Dimensional ◽  
Buoyant Plumes ◽  
The Subject ◽  
Flow Case

The behaviour of turbulent, buoyant, planar plumes is fundamentally coupled to the environment within which they develop. The effect of a background stratification directly influences a plumes buoyancy and has been the subject of numerous studies. Conversely, the effect of an ambient co-flow, which directly influences the vertical momentum of a plume, has not previously been the subject of theoretical investigation. The governing conservation equations for the case of a uniform co-flow are derived and the local dynamical behaviour of the plume is shown to be characterised by the scaled source Richardson number and the relative magnitude of the co-flow and plume source velocities. For forced, pure and lazy plume release conditions the co-flow acts to narrow the plume and reduce both the dilution and the asymptotic Richardson number relative to the classic zero co-flow case. Analytical solutions are developed for pure plumes from line sources, and for highly forced and highly lazy releases from sources of finite width in a weak co-flow. Contrary to releases in quiescent surroundings, our solutions show that all classes of release can exhibit plume contraction and the associated necking. For entraining plumes, a dynamical invariance spatially only occurs for pure and forced releases and we derive the co-flow strengths that lead to this invariance.

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