AN INTERPRETATION THEORY FOR INDUCED POLARIZATION VERTICAL SOUNDINGS (TIME-DOMAIN)*

In this paper it is shown how one may obtain a generalized Ohm's law which relates the induced polarization electric field to the steady-state current density through the introduction of a fictitious resistivity defined as the product of the chargeability and the resistivity of a given medium. The potential generated by the induced polarization is calculated at any point in a layered earth by the same procedure as used for calculating the potential due to a point source of direct current. On the basis of the definition of the apparent chargeability m_a, the expressions of m_a for different stratigraphie situations are obtained, provided the IP measurements are carried out on surface with an appropriate AMNB array. These expressions may be used to plot master curves for IP vertical soundings. Finally some field experiments over sedimentary formations and the quantitative interpretation procedure are reported.     

A THEOREM FOR DIRECT CURRENT REGIMES AND SOME OF ITS CONSEQUENCES *

For any direct current regime, the theorem holds, where φ_p is the total measured or calculated potential at any point P, φ is the potential distribution known a priori, r is the distance between P and any volume element dV, the gradients are evaluated at the element, and the current sources and sinks have finite dimensions. Thus, each space element behaves as a dipole of moment (1/4π) ?φdV and contributes its share of signal or potential accordingly. By suitable summation or integration, the contribution from any assigned portion of space to the total measured signal can be determined. Except for the chargeability factor m, the formula also establishes Seigel's initial postulate for the time domain induced polarisation theory. The contribution depends on the potential gradient, not the current density, and the integration extends over the entire space. Although an insulating target carries no current, it contributes a signal that is in general larger than normal by virtue of its higher potential gradient, and thus helps in creating an overall positive anomaly or resistivity high. On the other hand, an infinitely conducting target—even though it supports a larger amount of current than normal—contributes nothing to the measured signal as the potential gradient is zero everywhere inside. Thus, by contributing less than normal, a conducting target promotes the creation of what is usually a resistivity low. In all cases, the contributions from the space elements add up exactly to the measured or total calculated value. Some other consequences of the theorem are also discussed, especially in relation to a simple two-layer earth. For instance, the contribution from the upper half-space (air) turns out to be equal to that from the lower (real ground), for all observation points on the ground surface and for any ground configuration.     

A NEW PARAMETER FOR THE INTERPRETATION OF INDUCED POLARIZATION FIELD PROSPECTING (time-domain) *

In a previous paper it has been shown that we can relate the transient IP electric field E_p , existing in a rock after a step wave of polarizing current, with the steady-state current density J_ss during the current step wave as follows: E_p =ρ' J_ss . This relation may be interpreted as a generalized Ohm's law, valid in linear cases, in which ρ’(fictitious resistivity) is defined as the product of the true resistivity ρ with the chargeability m. Supposing E _p=— grad U_p and applying the divergence condition div J_ss = o, one can, for a layered earth, obtain a general expression for the depolarization potential U_p as a solution of Laplace's equation ?²U_p= o. Since the mathematical procedure for the solution of this last equation is identical to that used in resistivity problems, we propose now the introduction of an apparent fictitious resistivity ρ'_a (defined as the product of the apparent resistivity ρ_a with the apparent chargeability m_a) as a new parameter for the interpretations of IP soundings carried out over layered structures with a common electrode array. The most general expression of ρ'_a as a function of the electrode distance turns out to be mathematically identical to the general expression of ρ'_a. Therefore it is possible to interpret a ρ'_a field curve using the same standard graphs for resistivity prospecting with the usual method of complete curve matching. In this manner a great deal of work is saved since there is no need to construct proper m_a graphs for the interpretation of IP soundings, as it has been done up to now. Finally some field examples are reported.     

A STUDY ON THE DIRECT INTERPRETATION OF RESISTIVITY SOUNDING DATA MEASURED BY WENNER ELECTRODE CONFIGURATION*

This paper describes certain procedures for deriving from the apparent resistivity data as measured by the Wenner electrode configuration two functions, known as the kernel and the associated kernel respectively, both of which are functions dependent on the layer resistivities and thicknesses. It is shown that the solution of the integral equation for the Wenner electrode configuration leads directly to the associated kernel, from which an integral expression expressing the kernel explicitly in terms of the apparent resistivity function can be derived. The kernel is related to the associated kernel by a simple functional equation where K₁(λ) is the kernel and B₁(λ) the associated kernel. Composite numerical quadrature formulas and also integration formulas based on partial approximation of the integrand by a parabolic arc within a small interval are developed for the calculation of the kernel and the associated kernel from apparent resistivity data. Both techniques of integration require knowledge of the values of the apparent resistivity function at points lying between the input data points. It is shown that such unknown values of the apparent resistivity function can satisfactorily be obtained by interpolation using the least-squares method. The least-squares method involves the approximation of the observed set of apparent resistivity data by orthogonal polynomials generated by Forsythe's method (Forsythe 1956). Values of the kernel and of the associated kernel obtained by numerical integration compare favourably with the corresponding theoretical values of these functions.     

MATHEMATICAL MODELS OF CONDUCTING ORE BODIES FOR DIRECT CURRENT ELECTRICAL PROSPECTING *

The authors generalize a method expounded in a previous paper (1971, Geoph. Prosp. 18, 786-799) to the case of a local conductivity σ(M) of the infinite medium satisfying the relation where the R_i's are the distances from the point M to n fixed points S_i (i= 1,. n), k is a positive real constant and C_i, C_ii are constants ensuring the condition α > O. The sub-surface conductivity distributions (half-spaces) complying with (1) provide a wide variety of conducting structures, which can fit quite successfully the rather complicated distributions of conductivity occurring in natural ore bodies. An exact algebraic calculation of the apparent resistivity for these grounds, valid for any dc electrical prospecting devices (Wenner, Schlumberger, dipole, etc.) leads to a set of simultaneous linear equations, with a matrix which is invariant with respect to the position of the quadrupole being used. This greatly simplifies the numerical computation. We also present some examples of cross sections for the real and apparent resistivity obtained by this method.     

MODEL STUDIES ON SOME ASPECTS OF RESISTIVITY AND MEMBRANE POLARIZATION BEHAVIOUR OVER A LAYERED EARTH*

Electrolytic model tank experiments to study resistivity and time domain induced polarization (IP) response over layered earth models were initiated primarily to facilitate the understanding of field results. Alternate layers of clay and sand (or clay-coated sand) with, in some cases, a surficial layer of water were assembled in the tank and resistivity and IP measurements made for a range of electrode spacings using the Wenner configuration. Graphite and silver-silver chloride electrodes were used as current and potential electrodes respectively. Clay-coated (3% by weight) sand was found to generate stronger polarization than either clay or sand alone. Apparent chargeability m_a was observed to be positive for a nonpolarizable surface layer. For a polarizable surface layer, the sign of IP was controlled by the polarizability, the thickness of the second layer, and the spacing of the electrode spreads. The apparent chargeability m_a can theoretically change sign from positive to negative and vice versa with a gradual increase in electrode spacing, and such negative IP effects were obtained in a few observations. A simultaneous decrease in IP and an increase in resistivity, which is a qualitative diagnostic feature for the occurrence of clean freshwater sand aquifers, could also be generated in the model tank experiment. Combined resistivity and IP soundings were carried out near Fredericton Junction and Tracy, New Brunswick, Canada. Field curves are presented along with the model curves for qualitative comparison and understanding of IP behaviour over a layered earth. Twenty-five out of twenty-seven soundings show only positive apparent chargeabilities, whereas two show chargeability sign changes (positive/negative/positive). The model study gives reason to believe that surface soils and Quaternary gravel boulder deposits near Fredericton Junction are relatively non-polarizable. As an auxiliary experiment, sand and clay were taken in different proportions by weight and mixed thoroughly with water in a cement mixer. The mixtures were then compressed with a suitable die and plunger under 3.6 Pa pressure to prepare cylindrical samples of height 18 cm and diameter 15.5 cm. IP measurements were done on the flat faces using the Wenner configuration with a= 2 cm. Chargeability was found to be negative for 100 and 90% clay mixtures. It reached a positive maximum for an 80% clay-20% sand mixture and then decreased gradually with increasing sand and decreasing clay content.     

A COMPARISON OF IP TRANSIENT SHAPES OVER DISSEMINATED AND MASSIVE SULFIDE SHEETS IN THE LOWER PILLOW LAVAS OF CYPRUS*

A number of time-domain IP traverses were carried out across two parallel mineralized sheets in the Lower Pillow Lavas, near Mitsero, Cyprus with Huntec Mark III equipment using the pole-dipole array. In one sheet the mineralization was disseminated (2%S), and in the other it was massive (30%S). The transients were recorded at separation n= 2 at a number of points to give the complete shape of the curves. The normalized time integrals were anomalous over the two sheets, but were not significantly different; the highest values being observed over the disseminated sheet. Both sheets were also associated with high electromagnetic components of the decay curve. The chargeability and resistivity values obtained over the disseminated body were considerably higher. The metal factor was also of value in discriminating between massive ore, disseminated mineralization, and barren rock. The values of P₂ and P₃ for the two bodies were also compared (P₂ and P₃ are defined by where M₁ to M₄ are the amplitudes of the decay curve at 55, 130, 280 and 580 ms respectively). For the massive ore, P was inversely related to M, but for the disseminated ore P was independent of M. Four simple parameters from the decay curves show that indices of curve shape offer the best prospect of grade discrimination.     

TRAITEMENT AUTOMATIQUE DES SONDAGES ELECTRIQUES AUTOMATIC PROCESSING OF ELECTRICAL SOUNDINGS*

I. Introduction In this section the problem is stated, its physical and mathematical difficulties are indicated, and the way the authors try to overcome them are briefly outlined. Made up of a few measurements of limited accuracy, an electrical sounding does not define a unique solution for the variation of the earth resistivities, even in the case of an isotropic horizontal layering. Interpretation (i.e. the determination of the true resistivities and thicknesses of the ground-layers) requires, therefore, additional information drawn from various more or less reliable geological or other geophysical sources. The introduction of such information into an automatic processing is rather difficult; hence the authors developped a two-stage procedure:

• a) the field measurements are automatically processed, without loss of information, into more easily usable data;
• b) some additional information is then introduced, permitting the determination of several geologically conceivable solutions.

The final interpretation remains with the geophysicist who has to adjust the results of the processing to all the specific conditions of his actual problem. II. Principles of the procedure In this section the fundamental idea of the procedure is given as well as an outline of its successive stages. Since the early thirties, geophysicists have been working on direct methods of interpreting E.S. related to a tabular ground (sequence of parallel, homogeneous, isotropic layers of thicknesses h_i and resistivities ρ_i). They generally started by calculating the Stefanesco (or a similar) kernel function, from the integral equation of the apparent resistivity: where r is the distance between the current source and the observation point, S₀ the Stefanesco function, ρ(z) the resistivity as a function of the depth z, J₁ the Bessel function of order 1 and λ the integration variable. Thicknesses and resistivities had then to be deduced from S₀ step by step. Unfortunately, it is difficult to perform automatically this type of procedure due to the rapid accumulation of the errors which originate in the experimental data that may lead to physically impossible results (e.g. negative thicknesses or resistivities) (II. 1). The authors start from a different integral representation of the apparent resistivity: where K₁ is the modified Bessel function of order I. Using dimensionless variables t = r/2h₀ and y(t)=ζ (r)/ρ₁ and subdividing the earth into layers of equal thicknesses h₀ (highest common factor of the thicknesses h_i), ø becomes an even periodic function (period 2π) and the integral takes the form: The advantage of this representation is due to the fact that its kernel ø (function of the resistivities of the layers), if positive or null, always yields a sequence of positive resistivities for all values of θ and thus a solution which is surely convenient physically, if not geologically (II.3). Besides, it can be proved that ø(θ) is the Fourier transform of the sequence of the electric images of the current source in the successive interfaces (II.4). Thus, the main steps of the procedure are: a) determination of a non-negative periodic, even function ø(θ) which satisfies in the best way the integral equation of apparent resistivity for the points where measurements were made; b) a Fourier transform gives the electric images from which, c) the resistivities are obtained. This sequence of resistivities is called the “comprehensive solution”; it includes all the information contained in the original E.S. diagram, even if its too great detail has no practical significance. Simplification of the comprehensive solution leads to geologically conceivable distributions (h, ρ) called “particular solutions”. The smoothing is carried out through the Dar-Zarrouk curve (Maillet 1947) which shows the variations of parameters (transverse resistance R_i= h_i.ρ_i–as function of the longitudinal conductance C_i=h_i/ρ_i) well suited to reflect the laws of electrical prospecting (principles of equivalence and suppression). Comprehensive and particular solutions help the geophysicist in making the final interpretation (II.5). III. Computing methods In this section the mathematical operations involved in processing the data are outlined. The function ø(θ) is given by an integral equation; but taking into account the small number and the limited accuracy of the measurements, the determination of ø(θ) is performed by minimising the mean square of the weighted relative differences between the measured and the calculated apparent resistivities: minimum with inequalities as constraints: where t_l are the values of t for the sequence of measured resistivities and p_l are the weights chosen according to their estimated accuracy. When the integral in the above expression is conveniently replaced by a finite sum, the problem of minimization becomes one known as quadratic programming. Moreover, the geophysicist may, if it is considered to be necessary, impose that the automatic solution keep close to a given distribution (h, ρ) (resulting for instance from a preliminary interpretation). If φ(θ) is the ø-function corresponding to the fixed distribution, the quantity to minimize takes the form: where: The images are then calculated by Fourier transformation (III.2) and the resistivities are derived from the images through an algorithm almost identical to a procedure used in seismic prospecting (determination of the transmission coefficients) (III.3). As for the presentation of the results, resorting to the Dar-Zarrouk curve permits: a) to get a diagram somewhat similar to the E.S. curve (bilogarithmic scales coordinates: cumulative R and C) that is an already “smoothed” diagram where deeper layers show up less than superficial ones and b) to simplify the comprehensive solution. In fact, in arithmetic scales (R versus C) the Dar-Zarrouk curve consists of a many-sided polygonal contour which múst be replaced by an “equivalent” contour having a smaller number of sides. Though manually possible, this operation is automatically performed and additional constraints (e.g. geological information concerning thicknesses and resistivities) can be introduced at this stage. At present, the constraint used is the number of layers (III.4). Each solution (comprehensive and particular) is checked against the original data by calculating the E.S. diagrams corresponding to the distributions (thickness, resistivity) proposed. If the discrepancies are too large, the process is resumed (III.5). IV. Examples Several examples illustrate the procedure (IV). The first ones concern calculated E.S. diagrams, i.e. curves devoid of experimental errors and corresponding to a known distribution of resistivities and thicknesses (IV. 1). Example I shows how an E.S. curve is sampled. Several distributions (thickness, resistivity) were found: one is similar to, others differ from, the original one, although all E.S. diagrams are alike and characteristic parameters (transverse resistance of resistive layers and longitudinal conductance of conductive layers) are well determined. Additional informations must be introduced by the interpreter to remove the indeterminacy (IV.1.1). Examples 2 and 3 illustrate the principles of equivalence and suppression and give an idea of the sensitivity of the process, which seems accurate enough to make a correct distinction between calculated E.S. whose difference is less than what might be considered as significant in field curves (IV. 1.2 and IV. 1.3). The following example (number 4) concerns a multy-layer case which cannot be correctly approximated by a much smaller number of layers. It indicates that the result of the processing reflects correctly the trend of the changes in resistivity with depth but that, without additional information, several equally satisfactory solutions can be obtained (IV. 1.4). A second series of examples illustrates how the process behaves in presence of different kinds of errors on the original data (IV.2). A few anomalous points inserted into a series of accurate values of resistivities cause no problem, since the automatic processing practically replaces the wrong values (example 5) by what they should be had the E.S. diagram not been wilfully disturbed (IV.2.1). However, the procedure becomes less able to make a correct distinction, as the number of erroneous points increases. Weights must then be introduced, in order to determine the tolerance acceptable at each point as a function of its supposed accuracy. Example 6 shows how the weighting system used works (IV.2.2). The foregoing examples concern E.S. which include anomalous points that might have been caused by erroneous measurements. Geological effects (dipping layers for instance) while continuing to give smooth curves might introduce anomalous curvatures in an E.S. Example 7 indicates that in such a case the automatic processing gives distributions (thicknesses, resistivities) whose E.S. diagrams differ from the original curve only where curvatures exceed the limit corresponding to a horizontal stratification (IV.2.3). Numerous field diagrams have been processed (IV. 3). A first case (example 8) illustrates the various stages of the operation, chiefly the sampling of the E.S. (choice of the left cross, the weights and the resistivity of the substratum) and the selection of a solution, adapted from the automatic results (IV.3.1). The following examples (Nrs 9 and 10) show that electrical prospecting for deep seated layers can be usefully guided by the automatic processing of the E.S., even when difficult field conditions give original curves of low accuracy. A bore-hole proved the automatic solution proposed for E.S. no 10, slightly modified by the interpreter, to be correct.     

Inversion of 2D spectral induced polarization imaging data

Laboratory measurements of various materials suggest that more information can be obtained by measuring the in‐phase and out‐of‐phase potentials at a number of frequencies. One common model used to describe the variation of the electrical properties with frequency is the Cole‐Cole model. Apart from the DC resistivity (ρ) and chargeability (m) parameters used in conventional induced‐polarization (IP) surveys, the Cole‐Cole model has two additional parameters, i.e. the time (τ) and relaxation (c) constants. Much research has been conducted on the use of the additional Cole‐Cole parameters to distinguish between different IP sources. Here, we propose a modified inversion method to recover the Cole‐Cole parameters from a 2D spectral IP (SIP) survey. In this method, an approximate inversion method is initially used to construct a non‐homogeneous starting model for the resistivity and chargeability values. The 2D model consists of a number of rectangular cells with constant resistivity (ρ), chargeability (m), time (τ) and relaxation (c) constant values in each cell. A regularized least‐squares optimization method is then used to recover the time and relaxation constant parameters as well as to refine the chargeability values in the 2D model. We present results from tests carried out with the proposed method for a synthetic data set as well as from a laboratory tank experiment.     

Regional study of the anomalous change in apparent resistivity before the Tangshan earthquake (M=7.8, 1976) in China

Electrical resistivity measurements have been conducted as a possible means for obtaining precursory earthquake information. Before five great earthquakes (M>7,h<25 km) in China, the apparent resistivity _a showed systematic variations within a region 200 km from the epicenters. In particular, 9 stations in the Tangshan-Tianjin-Beijing region prior to the Tangshan earthquake (M=7.8,h=11 km, 27 July 1976) showed a consistent decrease of apparent resistivity around the epicenter, with a maximum resistivity change of 6% and a period of variation of 2–3 years. Simultaneous water table observations in this region showed a declining water table, and ground surface observations indicated a slight (5 mm) uplift in the epicenter region relative to its surroundings.In order to develop an explanation for the observed change of apparent resistivity associated with these great earthquakes, we have used Archie's Law to explore the effects of changes in rock porosity, water content and electrolyte resistivity on measured resistivity.Tentative conclusions of this study are as follows: (1) the apparent resistivity change is opposite to the effect expected from the simultaneous water table trend; (2) the dilatancy needed to give such resistivity variations (assuming Archie's Law holds) is much larger than that needed to explain the observed uplift (by 2–3 orders of magnitude); (3) salinity change in the pore electrolyte is a possible explanation for the variation in resistivity: an increase in the salinity would cause a proportional decrease in resistivity; the data needed to test this hypothesis, however, are lacking; and (4) the effect of changing geometry of rock pores or cracks due to pressure solution may provide an explanation for the decrease in apparent resistivity; it is different in nature from the effect of a volume change in response to stress although the geometry change is also closely related to the stress change.     

Physical properties of the Earth's core

Calculations of the compression and temperature gradient of the core are facilitated by the use of the thermodynamic Grüneisen ratio, =3Ks/C _P . A pressure-dependent factor in is found to have the same numerical value for the core as for laboratory iron, justifying the use of a constant value for (1.6) in core calculations. The density of the outer core is satisfied by the assumption that it contains about 15% of light elements, particularly sulphur, whereas the inner core is probably ironnickel with very little lighter component. The presence of sulphur in the outer core reduces its liquidus at least 600° below pure iron, so that the adiabatic gradient does not intersect the liquidus, as Higgins and Kennedy have shown would occur in a pure iron core. The inner core is probably close to its melting point, 4700 K, and the adiabatic temperature gradient of the outer is calculated with this as a fixed point, giving 3380 K at the core-mantle boundary. The estimated electrical resistivity of the outer core, 3×10^–6 m, corresponds to a thermal conductivity of 28 W·m^–1·deg^–1, which, with the adiabatic core gradient gives a minimum of 3.9×10¹² W of heat conduction to the mantle. The only plausible source of this much heat is the radioactive decay of potassium in the core. As pointed out by Goles, Lewis, and Hall and Murthy, the presence of potassium becomes geochemically probable once sulphur is admitted as a core constituent. Thus it appears that the recognition of sulphur in the core resolves the two major difficulties which we have faced in attempting to understand the core.List of Symbols a equilibrium atomic spacing at zero pressure, also a constant - A surface area of core - b a constant - c a constant - C _V ,C _P specific heat at constant volume, constant pressure - D dimension of core (or core eddy) - E(r) atomic interaction energy - E energy due to atomic displacement from equilibrium - lattice energy of material - f ₁,f ₂ structure-dependent constants - F(P) pressure dependent factor in Grüneisen's ratio - g gravitational acceleration; also a constant (Equation (13)) - H latent heat of solidification - I integral (Equation (23)) - k Boltzmann's constant - K incompressibility (bulk modulus) - K _T ,K _S isothermal, adiabatic incompressibilities - N number of atoms in a volume of material - P pressure - dQ/dt core to mantle heat flux - r atomic spacing - r _e equilibrium value ofr under pressure - R _m magnetic Reynolds number - T temperature - T _c critical temperature - T _R reduced temperature (Equation (39)) - U specific internal energy of a material - v velocity of internal core motion - V volume - 3 volume expansion coefficient - compressibility - thermodynamic Grüneisen ratio (Equation(2)) - magnetic diffusivity - thermal conductivity - _e electronic contribution to - ₀ permeability of free space - density - _e electrical resistivity - R reduced conductivity,eM/e     

Copper dispersion in ephemeral stream sediments

Sediment transport related parameters in ephemeral streams may be used to model and delineate: (1) average dispersion patterns of copper-laden sediments; (2) differences in dispersion of copper in bedload and suspended sediments; and (3) variability in the copper-sediment dispersion patterns. A model that effectively describes dispersion of copper in ephemeral stream sediments in a simple mixing model: where C_r is the resultant concentration beneath the confluence of the main channel with a tributary, C_t is the concentration of metal in sediments of the tributary, C_m is the metal concentration in main channel sediments, and X_m and X_t are the basin areas or sediment yields of the main channel and tributary channel at their confluence. Variability in metal concentrations about values predicted by this model may be due to the different responses of bedload and suspended load to changes in stream hydraulics, the dynamics of bedload transport, the spatial and temporal variability rainfall within the drainage basin, and chemical mobility of the copper.     

Water uptake and equilibrium sizes of aerosol particles at high relative humidities: Their dependence on the composition of the water-soluble material

Equilibrium water uptake and the sizes of atmospheric aerosol particles have for the first time been determined for high relative humidities, i.e., for humidities above 95 percent, as a function of the particles chemical composition. For that purpose a new treatment of the osmotic coefficient has been developed and experimentally confirmed. It is shown that the equilibrium water uptake and the equilibrium sizes of atmospheric aerosol particles at large relative humidities are significantly dependent on their chemical composition.List of symbols A proportionality factor - a _w activity of water in a solution - c _{p v} specific heat of water vapour at constant pressure - c _w specific heat of liquid water - f relative humidity - l _w specific heat of evaporation of water - M _i molar mass of solute speciesi - M _s mean molar mass of all the solute species in a solution - M _w molar mass of water - m ₀ mass of an aerosol particle in dry state - m _i mass of solute speciesi - m _s mass of solute - m _w mass of water taken up by an aerosol particle in equilibrium state - m total molality=number of mols of solute species in 1000 g of water - m _i molality of solute speciesi - m _k total molality of a pure electrolytek - O(m ²) remaining terms being of the second and of higher powers ofm - p ⁺ standard pressure - p total pressure of the gas phase - p pressure within a droplet - p ₁,p ₂,p ₃ coefficients in the expansion of M - p _1i, p_2i, p_3i specific parameters of ioni - p _s saturation vapour pressure - p _w water vapour pressure - R _w individual gas constant of water - r radius of a droplet - r ₀ equivalent volume radius of an aerosol particle in dry state - T temperature - T ₀ standard temperature - T ₁ temperature of the pure water drop in the osmometer - v _w specific volume of pure water - z _i valence of ioni - _i relativenumber concentration of ioni in a solution - correction term due to the adsorption of ions at liquid-solid interfaces - activity coefficient of solute speciesi in a solution, related to molalities - I bridge current - T temperature difference between solution and pure water drop in the osmometer - exponential mass increase coefficient - _w specific chemical potential of water vapour - _w specific chemical potential of water - ⁰ _w specific chemical potential of pure water vapour - ⁰ _w specific chemical potential of pure water - ₀ density of an aerosol particle in dry state - _w density of pure water - surface tension of a droplet - ₀ surface tension of pure water, i.e., at infinite dilution of the solute - osmotic coefficient - _k osmotic coefficient of a solution of a pure electrolytek - _k osmotic coefficient of a solution of a mixed solute - _M fugacity coefficient of water vapour - ^s _{i=1 i} z ² _i This work is part of a Ph.D. thesis carried out at the Meteorological Institute of the Johannes Gutenberg-Universität, Mainz.     

PERFORMANCE OF 2000 AND 6000 PSI AIR GUNS: THEORY AND EXPERIMENT*

Air guns have been used in various applications for a number of years. They were first used in coal-mining operations and were operated at up to 16000 psi charge pressures. Later, single air guns, operated at 2000 psi, found application as an oceanographic survey tool. Air gun arrays were first used in offshore seismic exploration in the mid-1960's. These early arrays were several hundred cubic inches in total volume and were operated at 2000 psi; they were either tuned arrays or several large guns of the same size with wave-shape kits. Today's arrays have total volumes greater than 5000 cu in. and are typically operated at 2000 psi. Recently, higher-pressure, lower-volume arrays operated at 4000–5000 psi have been introduced; guns used in these arrays are descendants of the coal-mining gun. On first thought one would equate increased gun pressure linearly with the amplitude of the initial pulse. This is approximately true for the signature radiated by a “free-bubble” (no confining vessel) and recorded broadband. The exact relation depends on the depth at which the gun is operated; from solution of the free-bubble oscillation equation, the relation is If P_c,1= 6014.7 psia, P_c,2= 2014.7 psia and P_{O, 1}=P_{O, 2}= 25.8 psia (corresponding to absolute pressure at 25 ft water depth), then Experiments were conducted offshore California in deep water to determine the performance of several models of air guns at pressures ranging from 2000 to 6000 psi and gun volumes ranging from 5 to 300 cu in. At a given gun pressure, the initial acoustic pulse P_a correlated with gun volume V_c according to the classical relation For 1 ms sampled data the ratio varied between 4.5 and 5.5 dB depending on gun model. Pulse width of the 2000 psi signatures indicated they are compatible with 2 ms sample-rate recording while pulse width of the 6000 psi signatures was greater, indicating they are less compatible with 2 ms sample-rate recording. Conclusions reached were that 2000 psi air guns are more efficient than higher pressure guns and are more compatible with 2 ms sample-rate requirements.     

Equilibration of saltation

A two‐dimensional numerical model of the saltation process was developed on a parallel computer in order to investigate the temporal behaviour of transport rate as well as its downwind distribution. Results show that the effects of unsteady flow on the transportation of particulates (sediment) have to be considered in two spatial dimensions (x, y). Transport rate Q(x, t) appears in the transport equation for mass M(x, t): where A = ΔxW denotes unit area composed of unit streamwise length Δx and width W. S(x, t) (units kg m⁻² s⁻¹) stands for the balance over the splash process. A transport equation for transport rate itself is suggested with U _c (x, t) a mean particle velocity at location x as the characteristic velocity of the grain cloud. For a steadily blowing wind over a 50 m long sediment bed it was found that downwind changes in Q cease after roughly 10–40 m, depending on the strength of the wind. The onset of stationarity (∂/∂t=0) was found to be a function of the friction velocity and location. The local equilibrium between transport rate and wind was obtained at different times for different downstream locations. Two time scales were found. One fast response (in the order of 1) to incipient wind and a longer time for equilibrium to be reached throughout the simulation length. Transport rate also has different equilibrium values at different locations. A series of numerical experiments was conducted to determine a propagation speed of the grain cloud. It was found that this velocity relates linearly to friction velocity. Copyright © 2000 John Wiley & Sons, Ltd.     

FAST HANKEL TRANSFORMS*

Inspired by the linear filter method introduced by D. P. Ghosh in 1970 we have developed a general theory for numerical evaluation of integrals of the Hankel type: Replacing the usual sine interpolating function by sinsh (x) =a· sin (ρx)/sinh (aρx), where the smoothness parameter a is chosen to be “small”, we obtain explicit series expansions for the sinsh-response or filter function H*. If the input function f(λ exp (iω)) is known to be analytic in the region o < λ < ∞, |ω|≤ω₀ of the complex plane, we can show that the absolute error on the output function is less than (K(ω₀)/r) · exp (?ρω₀/Δ), Δ being the logarthmic sampling distance. Due to the explicit expansions of H* the tails of the infinite summation ((m?n)Δ) can be handled analytically. Since the only restriction on the order is ν > ? 1, the Fourier transform is a special case of the theory, ν=± 1/2 giving the sine- and cosine transform, respectively. In theoretical model calculations the present method is considerably more efficient than the Fast Fourier Transform (FFT).     

Improved method to determine the molar volume and compositions of the NaCl-H<Subscript>2</Subscript>O-CO<Subscript>2</Subscript> system inclusion

On the basis of Parry’s method (1986), an improved method was established to determine the molar volume (Vm) and compositions (X) of the NaCl-H2O-CO2 (NHC) system inclusion. To use this method, the determination of Vm-X only requires three microthermometric data of a NHC inclusion: partial homog-enization temperature (Th ,CO2), salinity (S) and total homogenization temperature (Th). Theoretically, four associated equations are needed containing four unknown parameters: X CO2, XNaCl, Vm and F (volume fraction of CO2 phase in total inclusion when occurring partial homogenization). When they are released, the Vm-X are determined. The former three equations, only correlated with Th ,CO2, S and F, have simplified expressions:XCO2=f1(Th,CO2,S,F),XNaCl=f2(Th,CO2,S,F),Vm=f3(Th,CO2,S,F). The last one is the thermodynamic relationship of X CO2, XNaCl, Vm and Th:f4(XCO2,XNaCl,Vm,Th)=0.Since the above four associated equations are complicated, it is necessary to adopt iterative technique to release them. The technique can be described by:(i) Freely input a F value (0≤F≤1),with Th ,CO2 and S, into the former three equations. As a result,X CO 2,XNaCl and the molar volume value recorded as Vm1 are derived. (ii) Input the X CO2 and XNaCl gotten in the step above into the last equation, and another molar volume value recorded as Vm2 is determined. (iii) If Vm1 is unequal to Vm2, the calculation will be restarted from “(i)”. The iteration is completed until Vm1 is equal to Vm2, which means that the four associated equations are released. Compared to Parry’s (1986) solution method, the improved method is more convenient to use, as well as more accurate to determine X CO 2. It is available for a NHC inlusion whose partial homogenization temperature is higher than clatherate melting temperature and there are no solid salt crystals in the inclusion at parital homogenization.     

Planar charged-particle trajectories in multipole magnetic fields

This paper provides a complete generalization of the classic result that the radius of curvature () of a charged-particle trajectory confined to the equatorial plane of a magnetic dipole is directly proportional to the cube of the particles equatorial distance () from the dipole (i.e. ³). Comparable results are derived for the radii of curvature of all possible planar chargedparticle trajectories in an individual static magnetic multipole of arbitrary order m and degree n. Such trajectories arise wherever there exists a plane (or planes) such that the multipole magnetic field is locally perpendicular to this plane (or planes), everywhere apart from possibly at a set of magnetic neutral lines. Therefore planar trajectories exist in the equatorial plane of an axisymmetric (m = 0), or zonal, magnetic multipole, provided n is odd: the radius of curvature varies directly as ⁿ⁼². This result reduces to the classic one in the case of a zonal magnetic dipole (n = 1). Planar trajectories exist in 2m meridional planes in the case of the general tesseral (0 < m < n) magnetic multipole. These meridional planes are defined by the 2m roots of the equation cos[m()–_n^m)] = 0, where _n^m = (1/m) arctan (h_n^m/g_n^m); g_n^m and h_n^m denote the spherical harmonic coefficients. Equatorial planar trajectories also exist if (n – m) is odd. The polar axis ( = O,) of a tesseral magnetic multipole is a magnetic neutral line if m > I. A further 2_m(n – m) neutral lines exist at the intersections of the 2_m meridional planes with the (n – m) cones defined by the (n – m) roots of the equation P_n^m(cos ) = 0 in the range 0 < 9 < , where P_n^m(cos ) denotes the associated Legendre function. If (n – m) is odd, one of these cones coincides with the equator and the magnetic field is then perpendicular to the equator everywhere apart from the 2m equatorial neutral lines. The radius of curvature of an equatorial trajectory is directly proportional to ⁿ⁼² and inversely proportional to cos[m(–)]. Since this last expression vanishes at the 2m equatorial neutral ines, the radius of curvature becomes infinitely large as the particle approaches any one of these neutral lines. The radius of curvature of a meridional trajectory is directly proportional to rⁿ⁺², where r denotes radial distance from the multiple, and inversely proportional to P_n^m(cos )/sin . Hence the radius of curvature becomes infinitely large if the particle approaches the polar magnetic neutral ine (m > 1) or any one of the 2m(n – m) neutral ines located at the intersections of the 2m meridional planes with the (n – m) cones. Illustrative particle trajectories, derived by stepwise numerical integration of the exact equations of particle motion, are pressented for low-degree (n 3) magnetic multipoles. These computed particle trajectories clearly demonstrate the non-adiabatic scattering of charged particles at magnetic neutral lines. Brief comments are made on the different regions of phase space defined by regular and irregular trajectories.Also Visiting Reader in Physics, University of Sussex, Falmer, Brighton, BN1 9QH, UK     

Structure and evolution of the terrestrial planets

Geo-scientific planetary research of the last 25 years has revealed the global structure and evolution of the terrestrial planets Moon, Mercury, Venus and Mars. The evolution of the terrestrial bodies involves a differentiation into heavy metallic cores, Fe-and Mg-rich silicate mantles and light Ca, Al-rich silicate crusts early in the history of the solar system. Magnetic measurements yield a weak dipole field for Mercury, a very weak field (and local anomalies) for the Moon and no measurable field for Venus and mars. Seismic studies of the Moon show a crust-mantle boundary at an average depth of 60 km for the front side, P- and S-wave velocities around 8 respectively 4.5 km s^–1 in the mantle and a considerable S-wave attenuation below a depth of 1000 km. Satellite gravity permits the study of lateral density variations in the lithosphere. Additional contributions come from photogeology, orbital particle, x-and -ray measurements, radar and petrology.The cratered surfaces of the smaller bodies Moon and Mercury have been mainly shaped by meteorite impacts followed by a period of volcanic flows into the impact basins until about 3×10⁹ yr before present. Mars in addition shows a more developed surface. Its northern half is dominated by subsidence and younger volcanic flows. It even shows a graben system (rift) in the equatorial region. Large channels and relics of permafrost attest the role of water for the erosional history. Venus, the most developed body except Earth, shows many indications of volcanism, grabens (rifts) and at least at northern latitudes collisional belts, i.e. mountain ranges, suggesting a limited plate tectonic process with a possible shallow subduction.List of Symbols and Abbreviations a=R _e mean equatorial radius (km) - A(r, t) heat production by radioactive elements (W m^–3) - A, B equatorial moments of inertia - b polar radius (km) - complex amplitude of bathymetry in the wave number (K) domain (m) - C polar moment of inertia - C _Fe moment of inertia of metallic core - C _Si moment of inertia of silicate mantle - C _p heat capacity at constant pressure (JK^–1 mole^–) - C _nm,J _nm,S _nm harmonic coefficients of degreen and orderm - C/(MR _e ² ) factor of moment of inertia - d distance (km) - d nondimensional radius of disc load of elastic bending model - D diameter of crater (km) - D flexural rigidity (dyn cm) - E Young modulus (dyn cm^–2) - E maximum strain energy - E energy loss during time interval t - f frequency (Hz) - f flattening - F magnetic field strength (Oe) (1 Oe=79.58A m^–1) - g acceleration or gravity (cms^–2) or (mGal) (1mGal=10^–3cms^–2) - mean acceleration - g _e equatorial surface gravity - complex amplitude of gravity anomaly in the wave number (K) domain - g free air gravity anomaly (FAA) - g Bouguer gravity anomaly - g _t gravity attraction of the topography - G gravitational constant,G=6.67×10^–11 m³kg^–1s^–2 - GM planetocentric gravitational constant - h relation of centrifugal acceleration (² R _e ) to surface acceleration (g _e ) at the equator - J magnetic flux density (magnetic field) (T) (1T=10⁹ nT=10⁹ =10⁴G (Gauss)) - J ₂ oblateness - J _nm seeC _nm - k ₍₀₎ (zero) pressure bulk modulus (Pa) (Pascal, 1 Pa=1 Nm^–2) - K wave number (km^–1) - K ^* thermal conductivity (Jm^–1s^–1K^–1) - L thickness of elastic lithosphere (km) - M mas of planet (kg) - M _Fe mass of metallic core - M _Si mass of silicate mantle - M(r) fractional mass of planet with fractional radiusr - m magnetic dipole moment (Am²) (1Am²=10³Gcm³) - m _b body wave magnitude - N crater frequency (km^–2) - N(D) cumulative number of cumulative frequency of craters with diameters D - P pressure (Pa) (1Pa=1Nm^–2=10^–5 bar) - P _z vertical (lithostatic) stress, see also _z (Pa) - P _n ^m (cos) Legendre polynomial - q surface load (dyn cm^–2) - Q seismic quality factor, 2E/E - Q _s ,Q _p seismic quality factor derived from seismic S-and P-waves - R=R ₀ mean radius of the planet (km) (2a+b)/3 - R _e =a mean equatorial radius of the planet - r distance from the center of the planet (fractional radius) - r _Fe radius of metallic core - S _nm seeC _nm - t time and age in a (years), d (days), h (hours), min (minutes), s (seconds) - T mean crustal thickness from Airy isostatic gravity models (km) - T temperature (°C or K) (0°C=273.15K) - T _m solidus temperature - T sideral period of rotation in d (days), h (hours), min (minutes), s (seconds), =2/T - U external potential field of gravity of a planet - V volume of planet - V _p ,V _s compressional (P), shear (S) wave velocity, respectively (kms^–1) - w deflection of lithosphere from elastic bending models (km) - z, Z depth (km) - z (K) admittance function (mGal m^–1) - thermal expansion (°C^–1) - viscosity (poise) (1 poise=1gcm^–1s^–1) - co-latitude (90°-) - longitude - Poisson ratio - density (g cm^–3) - mean density - ₀ zero pressure density - _m , _Si average density of silicate mantle (fluid interior) - average density of metallic core - _t , _top density of the topography - density difference between crustal and mantle material - electrical conductivity (^–1 m^–1) - _r , radial and azimuthal surface stress of axisymmetric load (Pa) - _z vertical (lithostatic) stress (seeP _z ) - _II second invariant of stress deviation tensor - latitude - angular velocity of a planet (=2/T) - ages in years (a), generally 0 years is present - B.P. before present - FAA Free Air Gravity Anomaly (see g - HFT High Frequency Teleseismic event - LTP Lunar Transient Phenomenon - LOS Line-Of-Sight - NRM Natural Remanent Magnetization Contribution No. 309, Institut für Geophysik der Universität, Kiel, F.R.G.