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Introduction

Klaus Helbig is now (2017) 90 years old. He currently works as an independent researcher. Klaus does research in Mathematical Physics and Geophysics. His most recent publication is '2. Early Developments in Diffraction Theory'.

## Publications

Publications (106)

One hundred and fifty five years ago, Kelvin published the first part of a fundamental analysis of the elastic tensor, in which he proposed a coordinate‐free representation through its eigensystem. His thoughts were apparently far ahead of his time, since it took 125 years before the paper elicited a positive reaction (it is now accessible through...

Traditionally, input acquired in the field consisted of the original paper records; output submitted to the client consisted of structural sections and depth-contour maps of selected interfaces. Before the introduction of magnetic recording, it was common practice to do the conversion in the field office. Tools for this conversion ranged from slide...

Correct interpretation and processing of seismic data must integrate a correct description of the mechanical behavior of rocks, taking into account facts such as the presence of anisotropy and porosity with or without a saturating fluid. This work discusses elasticity of porous media of arbitrary anisotropy type, with emphasis on the study of defor...

Elastic waves can, in principle, be classified according to their propagation velocity, e.g., through the values of the slowness or the group velocity surfaces along a given direction. We show that there are media with the same velocity distribution but drastically different polarization behaviour. Such media are kinematically identical but dynamic...

Anisotropy is the dependence of the velocity on direction; dispersion is the dependence of the velocity on frequency (or wavelength λ). These apparently disjoint phenomena can be dealt with together if one uses as the common independent variable the wave vector k = ki, a vector in the direction of the wave and normal with length proportional to the...

The diclinic class of anisotropic media was suggested some year ago by Francis Muir on formal grounds. Figure 1 shows the pattern of the 6×6 representation of the elastic tensors of orthotropic, monoclinic, diclinic, and triclinic media in their natural coordinate system: Note that these patterns hold for both stiffness-and compliance tensors. With...

In 1821, Fresnel obtained the wave surface of an optically biaxial crystal, assuming that light waves are vibrations of the ether in which longitudinal vibrations (P waves) do not propagate. An anisotropic elastic medium mathematically analogous to Fresnel's crystal exists. The medium has four elastic constants: a P-wave modulus, associated with a...

Lamellations — layering on a scale small compared to the wavelengths — result in anisotropy (transverse isotropy) of elastic wave propagation. Since many sedimentary sequences can be regarded as lamellated, one might expect that exploration seismics would be beset with problems resulting from straightforward processing under the assumption of isotr...

The thermostat fitted to the Worden “Master” Gravity Meter results in much improved drift characteristics, and gravity differences under difficult conditions can now be measured with a precision of 0.2 mgal. This paper describes its use in West Pakistan to establish a base network of 8 air links connected to Woollard's Karachi airport station where...

The arrival-time curve of a reflection from a horizontal interface, beneath a homogeneous isotropic layer, is a hyperbola in the x - t-domain.
If the subsurface is one-dimensionally inhomogeneous (horizontally layered), or if some or all of the layers are transversely isotropic with vertical axis of symmetry, the statement is no longer strictly tru...

In seismic underground surveys carried out by Seismos GmbH in siderite mines of Siegerland (Germany) anomalies in velocities have been found which could be explained only by the assumption of slates being aeolotropic. In this paper some of the peculiarities connected with the propagation of elastic waves in aeolotropic media–especially those consis...

Wavefront charts in anisotropic gradient media are a useful tool in ray geometric constructions, particular in shear-wave exploration. They can be constructed by: (i) a family of wavefronts that contains a vertical plane as member - it is convenient to choose constant time increments; (ii) tracing one ray that makes everywhere the angle with the no...

An analytical expression for the time-distance curve of seismic waves travelling in a medium consisting of intrinsically anisotropic layers with arbitrarily dipping plane interfaces can be given in terms of the “co-ordinates” of the interfaces (length h of the perpendicular from the shotpoint to the interface, strike ν and dip α of the interface) i...

The acceptance of articles in Geophysical Prospecting is strictly based on the originality and relevance of the contents. In order to be published a paper has to satisfy additional requirements on its external form and on its style and language: it should be written preferably in English though articles in French and German are permitted. A paper s...

On page 18ND, right column, the phrase “in the Valhall oil field in Norway” was incorrect. We apologize for any inconvenience caused by this error.

The 11th International Workshop on Seismic Anisotropy (11IWSA), hosted by Memorial University of Newfoundland and chaired by Michael Slawinski, was held July 25–30, 2004, at St. John's, Newfoundland, Canada. Attendees presented more than 40 papers covering recent developments of the theory of seismic wave propagation in anisotropic media and its va...

The idea that the propagation of elastic waves can be anisotropic, i.e., that the velocity may depend on the direc-tion, is about 175 years old. The first steps are connected with the top scientists of that time, people such as Cauchy, Fresnel, Green, and Kelvin. For most of the 19th century, anisotropic wave propagation was studied mainly by math-...

Shortly after his appointment to the first geophysical professorship (1895 at the Jagiellonian University of Cracow), Rudzki had published two papers in which he made a strong case for anisotropy of crustal rocks [Beitr. Geophys. 2 (1898); Bull. Acad. Sci. Crac. (1899)]. He had solved the Christoffel equation for transversely isotropic media in ter...

We obtain the energy velocities and quality factors of anisotropic reservoir rocks as a function of pore pressure, partial saturation and frequency. The model is based on Biot's poroelastic theory for anisotropic media. The directional dependence of attenuation is obtained by generalizing the eigenstiffnesses of the undrained medium to relaxation f...

Using a lightweight portable vibrator, we have evaluated the accuracy of the ‘weighted sum’ method for calculating ground force. Experiments in which the vibrator was suspended elastically have shown that, contrary to expectations based on standard theory, the amplitude of the weighted sum ground force was significantly above zero at high frequenci...

The torsion balance is an ingenious instrument. The deviation of the vertical has been used since antiquity to study the geometry of the earthy's gravity, but only after the introduction of the torsion balance early in the 20th century did the accuracy become high enough to detect the effect of geologic features of interest in the exploration for o...

Many experimental studies on the elastic properties of rocks have unambiguously established a number of robust results. These are the most important: • rocks are generally anisotropic, e.g., their elastic response depends on the direction of the observation (e.g. Babuska and Cara, 1991, or Bourbié et al., 1987) , • rocks exhibit a nonlinear respons...

Elastic waves can, in principle, be classified according to their propagation velocity, e.g., using the values of the slowness or the group velocity surfaces along a given direction. The investigation is restricted to media with a distinct outer (P) sheet of the wave surface. We show that there are media with the same velocity distribution but dras...

This contribution describes a consistent nomenclature for anisotropic nonlinear media. Without requiring any basic change, its application would make existing theories and results more transparent. .

In Part 1 of this paper (Helbig, 1998 - Rev. Bras. Geof. 16 (2-3):103-114) it was shown that a medium consisting of a periodic sequence of layers is, in the long-wavelength approximation, equivalent to a homogeneous compound medium with elastic parameters that are generalized averages of the constituents' stiffnesses. Though the matrix-algorithm de...

The Eighth International Workshop on Seismic Anisotropy (8IWSA) was held on April 20-24, 1998, in Boussens, France, at the CISFAB Training Center of Elf Aquitaine (for more information, see http://8IWSA.seg.org ). This conference, held every two years since 1984, brings together many of the world's experts on seismic anisotropy for a week of intens...

The propagation of elastic waves is generally treated under four assumptions: - that the medium is isotropic, - that the medium is homogeneous, - that there is a one-to-one relationship between stress and strain, - that stresses are linearly related to strains (equivalently, that strains are linearly related to stresses). Real media generally viola...

Planes of symmetry are often identified by the existence of pure cross-plane polarization. However, this type of polarization can occur without the plane being a plane of symmetry. Planes that support cross-plane polarization are called "decoupling planes", since the system of three coupled linear equations in the direction cosines of the polarizat...

An important reason for elastic anisotropy of geological media is layering on a scale that is small to the wavelengths. If the parameters of the layering are known, the "effective parameters" of the compound medium can be derived from the parameters of the constituents. If the layering is periodic with the period small compared to the wavelengths,...

The internal structure of a material (as, e.g., expressed by the stiffnesses) dictates the dependence of the quality factor Q on direction. Therefore, the attenuation should have at least the symmetry of the crystallographic form of the material. To investigate this property, we compare three different constitutive equations for modeling anisotropi...

The amplitudes of seismic waves have always been a foremost concern of the seismologist to which considerable ingenuity was devoted. In the 1920s the problem was to magnify the ground motion sufficiently for detection. This was done at first by simple levers that moved mechanical pens. But at the start of exploration seismology, this had already be...

The book covers all aspects of acquisition, processing and interpretation of shallow reflection seismic data for geotechnical and environmental purposes. Chapter 1 provides a thorough exposition of (i) the mathematical foundations of digital seismics, (ii) the physics of wave propagation in solids, (iii) the mathematical tools for the analysis of w...

Due to its relatively high sensitivity to the distribution of elastic parameters, the seismic technique is quite suitable for geotechnical subsurface investigations. However, practical implementation of the P-wave exploration technique for soft-soil and shallow-groundwater conditions, as often encountered in the Netherlands, may often prove problem...

Seismic traces obtained with a vibratory source must be compressed during pre-processing before they can be compared with impulsive seismic traces. The two techniques available for compression are cross correlation and deconvolution with an approximation of the expected far-field signal. In both methods, the estimation of the far-field signal is ba...

The high-velocity branch in refraction seismic observations is generally considered as a head wave traveling along the plane interface between a homogeneous overborden and a homogeneous half space (the substratum). Head waves satisfy Snell's law, and the entire ray path consists of straight segments. Travel times of refracted arrivals can thus be d...

Across the many disciplines of the oil field, a nearly universal phenomenon is anisotropy —the variation of a property with the direction in which it is measured. Where anisotropy arises, convenient assumptions fall. Seismic reflectors appear at the wrong depth. Seismic lines don't tie. Waterflood programs fail. Induction logs are misinterpreted an...

The average (or mean) velocity va (of seismic waves, of trains, or downhill skiers) is “total distance divided by total time.” The rms‐velocity vrms (useful only for seismic waves) is the “root of the mean of the square of the velocity”; it involves squares of velocities, individual traveltime increments, total traveltime and the square root of the...

A longitudinal direction is one in which three pure modes can propagate. It is known that every medium has at least three longitudinal directions. Every axis of symmetry is such a ''bound'' longitudinal direction, but in most media there are additional ''free'' longitudinal directions which do not coincide with symmetry directions. In particular, t...

Real media are likely to display some degree of elastic anisotropy. The internal preferential orientation can be due to a variety of causes, among them fine layering, orientated cracks caused by non-lithostatic stress and the direction of cooling or flow. Isotropy is a special case, thus rules derived from wave propagation in isotropic conditions a...

The elastic stiffness tensor relates stress to strain (both tensors of rank two). It is of rank foor and thus has 81 components. The symmetry of both stress and strain and the existence of an elastic potential (the independence of the elastic energy from the strain-history) result in at most 21 of these being independent. Material symmetry reduces...

Anisotropy in the Earth's crust seems to be well modeled by orthorhombic symmetry (three mutually orthogonal planes of symmetry. In many places rocks can be regarded as a transversely isotropie (t.i.) background with a vertical symmetry axis (due mainly to fine horizontal layering ) in which is embedded a system of vertical parallel fractures (equi...

The propagation of elastic waves in the 1-3 plane of an orthorhombic medium (3-axis vertically downwards) is controlled by six elastic stiffnesses: c11, c13, c33, c44, c55, and c66. For propagation in the 2-3 plane c1 and c13 are replaced by c22 and c23. The remaining elastic stiffness c12 influences propagation in the horizontal (symmetry) plane....

Vertical seismic profiling using shear waves showed seismic anisotropy in the shallow 'Pot-Clay' sequence in the northern parts of The Netherlands. Shear-wave splitting, a key identifier for anisotropy, was observed at various depth using a multi-offset/multi-azimuth data acquisition technique. For the first time in such investigations, a three-com...

The orientation of the displacement vector U of a plane wave in a homogeneous anisotropicelastic medium is the polarization of that plane wave. For transversely isotropic media, U of the fastest plane wave propagating in a given direction need not be in or near the direction of propagation, i.e., the direction of the slowness vector s. Moreover, wi...

It is generally recognised that the broad-band nature of time-domain electromagnetic systems offers definite
advantages in comparison with frequency-domain systems. A balanced view is given by Parasnis (1979), who
also indicates the problems involved in the analysis. The high price of available time-domain field instruments has not contributed towa...

The behavior of transversely isotropic elastic media is analyzed from both the kinematic (slowness surface) and dynamic (particle displacement) point of view. The relations for the slowness surfaces and wave front surfaces are derived in polar coordinates. Examination of the eigenvectors of the displacement equations of motion gives the relation fo...

Elastic waves propagating in a periodically layered medium exhibit transverse isotropy, provided the wavelength is long compared to the spatial period of the layer sequence. High-resolution techniques, the increasing use of shear waves, and attention to stratigraphic detail require a quantitative evaluation of what is sufficiently long, as well as...

Without further specification, the word ‘deep’ in connection with a seismic survey has no precise meaning: applied loosely, it means just ‘deeper than most surveys’. A seismic section or a seismogram without time scale contains no clue to the depth of the reflectors or the total time. As a matter of fact, the conceptual difficulties to expand the s...

Although the wavefronts of P and SV waves can never be ellipsoids if the anisotropy is the result of lamellation, pieces of the wavefront can be represented with sufficient accuracy by an ellipsoid. This representation allows a simple determination of the ratio 'zero-offset limit of stacking velocity/vertical velocity'. Constraints on the parameter...

The survey we describe here was to serve a double purpose. On the one hand to contribute to a study of the depositional history of shallow sediments in south-west Netherlands. As far as this objective is concerned, the survey was a direct continuation of one started by M. A. Herber and J. Runia in 1977 (see Herber, Runia and Helbig 1981). On the ot...

Shear waves can today be generated and observed, though not with the flexibility and the technical standard of compressional waves, and they can be identified in seismograms by various means. Their potential lies not so much in their lower velocity (corresponding—for the same frequency—to shorter wavelength and higher resolution) but in the fact th...

Waves propagating through a sequence of layers that are thin compared with the wavelength show effects of anisotropy: velocity and displacement direction depend on the angle between the plane of layering and the wave normal, and shear waves split up into two distinct types of different velocity. The layered medium can thus be replaced by a transver...

Assume that in a refraction survey arrival times of refracted waves have been observed along at least two lines of different azimuth, and on at least one of the lines, the “main” line, observations in reversed directions have been taken. The set of observed arrival times is complete (i.e. there is a one-to-one relationship between interfaces in the...

With the term raw data set we describe the data obtained in a seismic survey. It is implicitly assumed that they have been pre-processed to minimize the effects of near surface layers, variations in source strength, and variations in source-to-ground and receiver-to-ground coupling. In short, we assume the data to be essentially homogeneous if the...

The seismogram generated by plane waves falling vertically on a stack of plane parallel layers can be expressed in terms of the sequence of the reflection coefficients corresponding to the interfaces between individual layers. The reflection coefficients can be expressed through the wave impedances, which are parameters characterizing the layers an...

Berryman shows elegantly that the “inequality ( c 11 - c 44 ) ⋅ ( c 33 - c 44 ) ⩾ ( c 13 - c 44 ) 2 is true for any horizontally stratified, homogeneous material whose constituent layers are isotropic…” However, the final clause of this sentence “…, i.e., any homogeneous, transversely isotropic material,” is, if taken at face value, misleading. It...

High-resolution seismic profiling requires special instrumentation and special acquisition techniques, with a high-frequency source being the key factor. In a feasibility study on the Roggenplaat, a tidal flat in the mouth of the Oosterschelde (SW Netherlands), the CDP-method was applied using standard recording equipment together with a specially...

Levin treats the subject concisely and exhaustively. Nevertheless, I feel a few comments to be indicated. My first point is rather general: of the three surfaces mentioned in the Appendix, the phase velocity surface (or normal surface) is easiest to calculate, since it is nothing but the graphical representation of the plane‐wave solutions for each...

Since its foundation in 1968, the European Association of Earth Science Editors (EDITERRA) has regarded the preparation of a Handbook as a natural means by which the discussion of problems which editors encounter might be codified for the benefit of all editors. At early meetings the general pattern of the proposed Handbook was decided, and a numbe...

Leipzig, Phil. F., Diss. v. 25. April 1968 (Nicht f. d. Aust.).

Auch im Buchh. als: Bayer. Akad. d. Wiss. Abhandlungen. Math.-naturwiss. Kl. N. F. H. 122 München, Naturwiss. F., Hab.Schr. v. 30. Juli 1964 (Nicht f. d. Aust.).

The first part of this paper describes the investigation of thermomagnetic effects in three basalt samples. It was found that when a sample was heated in air the ferrimagnetic mineral component changed from a homogeneous member of the Ulvöspinell-Magnetite-series into two components. One of these had the same O/Fe-ratio, the other one the same Fe/T...