Transformation of global spherical harmonic models of the gravity field to a local adjusted spherical cap harmonic model

The new global gravity models represented by global spherical harmonics like EGM2008 require a high degree and order in their coefficients to resolve the gravity field in local areas; therefore, there are interests to represent the regional or local field by less parameters and to develop a parameter transformation from the global model to a local kind of spherical harmonic model. The authors use local spherical cap harmonics for the regional gravity potential representation related to a local pole and a local spherical coordinate system. This allows to model regional gravity potential with less parameters and less memory requirements in computation and storage. From different kinds of representations of spherical cap harmonics, we have selected the so-called adjusted spherical cap harmonics (ASCH). This is the most appropriate for the presented mathematical model of deriving its coefficients from global gravity models. In that way, the global gravity models can fully be exploited and mapped to regional ASCH, in particular with respect to the computation of regional geoid models with improved solution.     

Optimization fitting of the eccentric dipole models to the observed geomagnetic field

We have obtained formulas to calculate the field components of an eccentric dipole (ED) with an arbitrarily directed moment from specified coordinates of the dipole center and from its moment vector. With these formulas, the model dipole position was fitted to the observed geomagnetic field, and the approximation accuracy was estimated according to the standard deviation from IGRF along the X, Y, and Z components; the fitting procedure was checked in several tests.We computed the ED parameters (position and moment orientation) and harmonic coefficients of the ED field for 50 years using surface IGRF components, discovered some changes, and compared them with those according to the Schmidt eccentric dipole model. It was found that the nondipolar contributions to the geomagnetic field increase as the dipole field decays with time, and the dipole center drifts northand eastward away from the Earth's gravity center.The main contribution of the dipole part in the Gaussian spherical harmonic expansion of the geomagnetic field turned out to come from the terms with n of 1 to 5 rather than from two first terms, and the contribution of higher harmonics increases with time. Therefore, the Schmidt ED approximation based on the first eight Gauss coefficients (n ≤ 2) must have only relative significance indicating changes of the parameters. On the other hand, world magnetic anomalies (WMA) show up already in two first Gauss terms rather than since the third term as it has been commonly assumed.     

Model of the MAGSAT magnetic anomaly field over Europe using spherical cap harmonic analysis

Based on the MAGSAT magnetic anomaly fields over Europe and the adjacent areas spherical cap models have been derived. The method of spherical cap harmonic analysis, due to Haines (1985), has been applied for the modelling of the MAGSAT magnetic anomalies. The data set used in the analysis included the 1 ° × 1 ° gridded values of the MAGSAT anomaly fields between latitudes ϑ = 6 ° to 60 °N and longitudes λ = 19 °W to 70 °E. The pole of the cap is at ϑ = 33 °N and λ = 26 °E and its half-angle is 40 ° . The maximum index (K_m) of the model is 18 and the total number of model coefficients is 361. A minimum wavelength corresponding to this index at the Earth's surface is ~ 1000 km. The RMS deviations between the calculated and observed values are ~ 4 nT for δX, ~ 3 nT for ΔY and 3,5 nT for ΔZ respectively. The spherical cap harmonic model was used for the construction of magnetic anomaly maps for all components and at different altitudes.     

Spherical harmonics in texture analysis

The objective of this contribution is to emphasize the fundamental role of spherical harmonics in constructive approximation on the sphere in general and in texture analysis in particular. The specific purpose is to present some methods of texture analysis and pole-to-orientation probability density inversion in a unifying approach, i.e. to show that the classic harmonic method, the pole density component fit method initially introduced as a distinct alternative, and the spherical wavelet method for high-resolution texture analysis share a common mathematical basis provided by spherical harmonics. Since pole probability density functions and orientation probability density functions are probability density functions defined on the sphere Ω^{3 3} or hypersphere Ω^{4 4}, respectively, they belong at least to the space of measurable and integrable functions ¹(Ω^d), d=3, 4, respectively.

Therefore, first a basic and simplified method to derive real symmetrized spherical harmonics with the mathematical property of providing a representation of rotations or orientations, respectively, is presented. Then, standard orientation or pole probability density functions, respectively, are introduced by summation processes of harmonic series expansions of ¹(Ω^d) functions, thus avoiding resorting to intuition and heuristics. Eventually, it is shown how a rearrangement of the harmonics leads quite canonically to spherical wavelets, which provide a method for high-resolution texture analysis. This unified point of view clarifies how these methods, e.g. standard functions, apply to texture analysis of EBSD orientation measurements.     

Analytical and numerical solutions for carbonated waterflooding

We develop a Riemann solver for transport problems including geochemistry related to oil recovery. The example considered here concerns one-dimensional incompressible flow in porous media and the transport for several chemical components, namely H₂O, H⁺, OH^?, CO₂, \(\text {CO}_{3}^{2-}\), \(\text {HCO}_{3}^{-}\), and decane; they are in chemical equilibrium in the aqueous and oleic phases, leading to mass transfer of CO₂ between the oleic and aqueous phases. In our ionic model, we employ equations with zero diffusion coefficients. We do so because it is well known that for upscaled equations, the convection terms dominate the diffusion terms. The Riemann solution for this model can therefore be applied for upscaled transport processes in enhanced oil recovery involving geochemical aspects. In our example, we formulate the conservation equations of hydrogen, oxygen, hydrogen, and decane, in which we substitute regression expressions that are obtained by geochemical software. This can be readily done because Gibbs phase rule together with charge balance shows that all compositions can be rewritten in terms of a single composition, which we choose to be the hydrogen ion concentration (p H). In our example, we use the initial and boundary conditions for the carbonated aqueous phase injection in an oil reservoir containing connate water with some carbon dioxide. We compare the Riemann solution with a numerical solution, which includes capillary and diffusion effects. The significant new contribution is the effective Riemann solver we developed to obtain solutions for oil recovery problems including geochemistry and a variable total Darcy velocity, a situation in which fractional flow theory does not readily apply. We thus obtain an accurate solution for a carbonated waterflood, which elucidates some mechanisms of low salinity carbonated waterflooding.     

Nonlocal response functions in processing global electromagnetic data

We discuss approaches to mapping lateral apparent conductivity variations at different periods from magnetic observatory data using multipoint transfer operators as nonlocal functions of the EM response. The multipoint operators provide correlation of three magnetic field components recorded at all observatories. The inversion procedure was applied to S_q observatory data for 1964–68 and records of 30 magnetic storms for 1957–2001. To obtain lateral conductivity patterns, data on diurnal S_q variations and global magnetic storms were processed with the spherical harmonic analysis. The same data were used to estimate the coefficients of first harmonics in the SH series of diurnal lateral variations of Earth's apparent conductivity.     

Correlations between global features of terrestrial fields

Studies of correlation coefficients between different sets of global geophysical data may lead to useful inferences concerning their relationship or independence. If one data set is allowed to rotate with respect to another, the statistical theory is complicated and extra care is required before one can conclude that there is any statistical significance to a maximized correlation coefficient. If, for some relative rotation, two spherical harmonic fields are significantly correlated, then their individual degree component harmonics of dominant power must also be significantly correlated. Rotations can be found that result in high correlations between the dominant low-degree spherical harmonics of the geomagnetic and tertestrial gravity field potentials, but rotations can also be found that result in equally high, yet meaningless, correlations if the lunar gravity field is substituted for the geomagnetic field. To explain such high correlations, the theoretical correlation distribution function between normally distributed component harmonics is derived and then verified for lowdegree harmonics by using a Monte Carlo technique which takes into account the three-dimensional rotation group. Some curious properties surface: (1)the correlation distribution function for all possible relative orientations is almost the same between identical and uncorrelated fields; and (2)a system for determining the correlation distribution function from randomly selected fields or from randomly rotated fields is almost ergodic.     

The system Fe-Si-O: Oxygen buffer calibrations to 1,500K

The five solid-phase oxygen buffers of the system Fe-Si-O, iron-wuestite (IW), wuestite-magnetite (WM), magnetite-hematite (MH), quartz-iron-fayalite (QIF) and fayalite-magnetite-quartz (FMQ) have been recalibrated at 1 atm pressure and temperatures from 800°–1,300° C, using a thermogravimetric gas mixing furnace. The oxygen fugacity, \(f_{{\text{O}}_{\text{2}} }\) was measured with a CaO-doped ZrO₂ electrode. Measurements were made also for wuestite solid solutions in order to determine the redox behavior of wuestites with O/Fe ratios varying from 1.05 to 1.17. For FMQ, additional determinations were carried out at 1 kb over a temperature range of 600° to 800° C, using a modified Shaw membrane. Results agree reasonably well with published data and extrapolations. The reaction parameters K, ΔG _r ^o , ΔH _r ^o , and ΔS _r ^o were calculated from the following log \(f_{{\text{O}}_{\text{2}} }\) /T relations (T in K): $$\begin{gathered} {\text{IW }}\log f_{{\text{O}}_{\text{2}} } = - 26,834.7/T + 6.471\left( { \pm 0.058} \right) \hfill \\ {\text{ }}\left( {{\text{800}} - 1,260{\text{ C}}} \right), \hfill \\ {\text{WM }}\log f_{{\text{O}}_{\text{2}} } = - 36,951.3/T + 16.092\left( { \pm 0.045} \right) \hfill \\ {\text{ }}\left( {{\text{1,000}} - 1,300{\text{ C}}} \right), \hfill \\ {\text{MH }}\log f_{{\text{O}}_{\text{2}} } = - 23,847.6/T + 13.480\left( { \pm 0.055} \right) \hfill \\ {\text{ }}\left( {{\text{1,040}} - 1,270{\text{ C}}} \right), \hfill \\ {\text{QIF }}\log f_{{\text{O}}_{\text{2}} } = - 27,517.5/T + 6.396\left( { \pm 0.049} \right) \hfill \\ {\text{ }}\left( {{\text{960}} - 1,140{\text{ C}}} \right), \hfill \\ {\text{FMQ }}\log f_{{\text{O}}_{\text{2}} } = - 24,441.9/T + 8.290\left( { \pm 0.167} \right) \hfill \\ {\text{ }}\left( {{\text{600}} - 1,140{\text{ C}}} \right). \hfill \\ \end{gathered}$$ These experimentally determined reaction parameters were combined with published 298 K data to determine the parameters G_f, H_f, and S_f for the phases wuestite, magnetite, hematite, and fayalite from 298 K to the temperatures of the experiments. The T? \(f_{{\text{O}}_{\text{2}} }\) data for wuestite solid solutions were used to obtain activities, excess free energies and Margules mixing parameters. The new data provide a more reliable, consistent and complete reference set for the interpretation of redox reactions at elevated temperatures in experiments and field settings encompassing the crust, mantle and core as well as extraterrestrial environments.     

Equilibrium thermodynamics of Al/Si ordering in anorthite

Variations in the equilibrium degree of Al/Si order in anorthite have been investigated experimentally over the temperature range 800-1535° C. Spontaneous strain measurements give the temperature dependence of the macroscopic order parameter, Q, defined with respect to the \(C\bar 1 \rightleftharpoons I\bar 1\) phase transition, while high temperature solution calorimetric data allow the relationship between Q and excess enthalpy, H, to be determined. The thermodynamic behaviour can be described by a Landau expansion in one order parameter if the transition is first order in character, with an equilibrium transition temperature, T _tr, of ～2595 K and a jump in Q from 0 to ～0.65 at T_tr. The coefficients in this Landau expansion have been allowed to vary with composition, using Q=1 at 0 K for pure anorthite as a reference point for the order parameter. Published data for H and Q at different compositions allow the calibration of the additional parameters such that the free energy due to the \(C\bar 1 \rightleftharpoons I\bar 1\) transition in anorthite-rich plagioclase feldspars may be expressed (in cal. mole^-1) as: \(\begin{gathered}G = \tfrac{1}{2} \cdot 9(T - 2283 + 2525X_{Ab} )Q^2 \\ {\text{ + }}\tfrac{1}{4}( - 26642 + 121100X_{Ab} )Q^4 \\ {\text{ + }}\tfrac{1}{6}(47395 - 98663X_{Ab} )Q^6 \\ \end{gathered}\) where X _Ab is the mole fraction of albite component. The nature of the transition changes from first order in pure anorthite through tricritical at ～An₇₈ to second order, with increasing albite content. The magnitude of the free energy of \()\) ordering reduces markedly as X _Ab increases. At ～700° C incommensurate ordering in crystals with compositions ～An₅₀–An₇₀ needs to have an associated free energy reduction of only a few hundred calories to provide a more stable structure. These results, together with a simple mixing model for the disordered ( \()\) ) solid solution, an assumed tricritical model for the incommensurate ordering and published data for ordering in albite have been used to calculate a set of possible free energy relations for the plagioclase system. The incommensurate structure should appear on the equilibrium phase diagram, but its apparent stability with respect to the assemblage albite plus anorthite at low temperatures depends on the values assigned to the mixing parameters of the $$$$ solid solution.     

Accessing the capability of TRMM 3B42 V7 to simulate streamflow during extreme rain events: Case study for a Himalayan River Basin

The paper examines the quality of Tropical Rainfall Monitoring Mission (TRMM) 3B42 V7 precipitation product to simulate the streamflow using Soil Water Assessment Tool (SWAT) model for various rainfall intensities over the Himalayan region. The SWAT model has been set up for Gandak River Basin with 41 sub-basins and 420 HRUs. Five stream gauge locations are used to simulate the streamflow for a time span of 10 years (2000–2010). Daily streamflow for the simulation period is collected from Central Water Commission (CWC), India and Department of Hydrology and Meteorology (DHM), Nepal. The simulation results are found good in terms of Nash–Sutcliffe efficiency \((\hbox {NSE}) {>}0.65\), coefficient of determination \((R^{2}) {>}0.67\) and Percentage Bias \(\hbox {(PBIAS)}{<}15\%\), at each stream gauge sites. Thereafter, we have calculated the PBIAS and RMSE-observations standard deviation ratio (RSR) statistics between TRMM simulated and observed streamflow for various rainfall intensity classes, viz., light (\({<}7.5 \, \hbox {mm}/\hbox {d}\)), moderate (7.5 to 35.4 mm/d), heavy (35.5 to 124.4 mm/d) and extremely heavy (\({>}124.4 \, \hbox {mm}/\hbox {d}\)). The PBIAS and RSR show that TRMM simulated streamflow is suitable for moderate to heavy rainfall intensities. However, it does not perform well for light- and extremely-heavy rainfall intensities. The finding of the present work is useful for the problems related to water resources management, irrigation planning and hazard analysis over the Himalayan regions.     

The estimation of the pK1HA of acids in seawater using the Pitzer equations

The equations of Pitzer have been used to calculate the stoichiometric ionization constants, pK¹_HA, for acids in NaCl media at 25°C. The calculated results for the ionization of HAc, H₂O, B(OH)₃, H₂CO^?₃, H₃PO₄, H₂PO^?₄, HPO^2?₄, H₃AsO₄, H₂AsO^?₄ and HAsO^2?₄ are in good agreement with the measured values, providing higher order interaction terms (θ and ψ) are used. The pK¹_HA measurements of these acids in NaCl media containing Mg²⁺ and Ca²⁺ were used to determine Pitzer specific interaction parameters at I = 0.7. With these Pitzer coefficients, it was possible to make reliable estimates for the activity coefficients of anions in seawater (S = 35) that form strong interactions with Mg²⁺ and Ca²⁺. The calculated activity coefficients yield reliable estimates for the pK¹_HA of acids in seawater.     

The thermodynamic properties of reciprocal solid solutions

A multisite solid solution of the type (A, B) (X, Y) has the four possible components AX, AY, BX, BY. Taking the standard state to be the pure phase at the pressure and temperature of interest, the mixing of these components is shown not to be ideal unless the condition: $$\Delta G^0 = (\mu _{AX}^0 + \mu _{BY}^0 - \mu _{AY}^0 - \mu _{BX}^0 = 0$$ applies. Even for the case in which mixing on each of the individual sublattices is ideal, ΔG ⁰ contributes terms of the following form to the activity coefficients of the constituent components: $$RT\ln \gamma _{AX} = - X_{B_1 } X_{Y_2 } \Delta G^0$$ (X _Ji refers to the atomic fraction of J on sublattice i). The above equation, which assumes complete disorder on (A, B) sites and on (X, Y) sites is extended to the general n-component case. Methods of combining the “cross-site” or reciprocal terms with non-ideal terms for each of the individual sites are also described. The reciprocal terms appear to be significant in many geologically important solid solutions, and clinopyroxene, garnet and spinel solid solutions are all used as examples. Finally, it is shown that the assumption of complete disorder only applies under the condition: $$\Delta G^0 \ll zn_1 RT$$ where z is the number of nearest-neighbour (X, Y) sites around A and n ₁ is the number of (A, B) sites in the formula unit. If ΔG ⁰ is relatively large, then substantial short range oder must occur and the activity coefficient is given by (ignoring individual site terms): $$\gamma _{AX} = \left( {\frac{{1 - X'_{Y_2 } }}{{1 - X_{Y_2 } }}} \right)^{zn_1 }$$ where X′_Y2 is the equilibrium atomic fraction of Y atoms surrounding A atoms in the structure. The ordered model may be developed for multicomponent solutions and individual site interactions added, but numerical methods are needed to solve the simultaneous equations involved.     

The relationship between the adiabatic bulk modulus and enthalpy for mantle-related minerals

It is found that the adiabatic bulk modulus, K_S, is linear with enthalpy over a wide temperature range: up to at least 1825 K, the present limit of the measurement of the bulk modulus. This correlation is shown to hold for Al₂O₃, MgO, and Mg₂SiO₄. Since the enthalpy is listed in thermodynamic tables up to 3000 K, one can reasonably safely extrapolate K_S up to lower mantle temperatures using this correlation. This correlation was anticipated in a theoretical 1966 paper, where the definition of the anharmonic parameter δ _S was made in terms of properties which vary with temperature, \(\delta _s = - \left( {\frac{1}{{\alpha K_S }}} \right)\left( {\frac{{\delta K}}{{\delta T}}} \right)_{P'}\) where α is the volume coefficient of thermal expansion. The correlation was first confirmed for polycrystalline oxides in an experimental 1966 paper. Since the isotropic shear modulus, G, is linear with T, it is possible to estimate the sound velocities in the temperature regime just below the melting point.     

Heat capacity of hydrous trachybasalt from Mt Etna: comparison with CaAl<Subscript>2</Subscript>Si<Subscript>2</Subscript>O<Subscript>8</Subscript> (An)–CaMgSi<Subscript>2</Subscript>O<Subscript>6</Subscript> (Di) as basaltic proxy compositions

The specific heat capacity (C _p) of six variably hydrated (~3.5 wt% H₂O) iron-bearing Etna trachybasaltic glasses and liquids has been measured using differential scanning calorimetry from room temperature across the glass transition region. These data are compared to heat capacity measurements on thirteen melt compositions in the iron-free anorthite (An)–diopside (Di) system over a similar range of H₂O contents. These data extend considerably the published C _p measurements for hydrous melts and glasses. The results for the Etna trachybasalts show nonlinear variations in, both, the heat capacity of the glass at the onset of the glass transition (i.e., C _p ^g ) and the fully relaxed liquid (i.e., C _p ^l ) with increasing H₂O content. Similarly, the “configurational heat capacity” (i.e., C _p ^c  = C _p ^l  ? C _p ^g ) varies nonlinearly with H₂O content. The An–Di hydrous compositions investigated show similar trends, with C _p values varying as a function of melt composition and H₂O content. The results show that values in hydrous C _p ^g , C _p ^l and C _p ^c in the depolymerized glasses and liquids are substantially different from those observed for more polymerized hydrous albitic, leucogranitic, trachytic and phonolitic multicomponent compositions previously investigated. Polymerized melts have lower C _p ^l and C _p ^c and higher C _p ^g with respect to more depolymerized compositions. The covariation between C _p values and the degree of polymerization in glasses and melts is well described in terms of SM_hydrous and NBO/T _hydrous. Values of C _p ^c increase sharply with increasing depolymerization up to SM_hydrous ~ 30–35 mol% (NBO/T _hydrous ~ 0.5) and then stabilize to an almost constant value. The partial molar heat capacity of H₂O for both glasses (\( C_{{{\text{p}}\;{\text{H}}_{2} {\text{O}}}}^{\text{g}} \)) and liquids (\( C_{{{\text{p}}\;{\text{H}}_{2} {\text{O}}}}^{\text{l}} \)) appears to be independent of composition and, assuming ideal mixing, we obtain a value for \( C_{{{\text{p}}\;{\text{H}}_{2} {\text{O}}}}^{\text{l}} \) of 79 J mol^?1 K^?1. However, we note that a range of values for \( C_{{{\text{p}}\;{\text{H}}_{2} {\text{O}}}}^{\text{l}} \) (i.e., ~78–87 J mol^?1 K^?1) proposed by previous workers will reproduce the extended data to within experimental uncertainty. Our analysis suggests that more data are required in order to ascribe a compositional dependence (i.e., nonideal mixing) to \( C_{{{\text{p}}\;{\text{H}}_{2} {\text{O}}}}^{\text{l}} \).     

Atmospheric pressure changes due to volcanic eruptions and possible water level fluctuations

The data of Reed (1983) are analysed to produce the following empirical equations for the amplitude p ₀ (overall fluctuation) in Pascals of the air pressure wave associated with a volcanic eruption of volume V km³ or a nuclear explosion of strength M Mt: Here s is the distance from the source in km. $$\begin{gathered} \log _{10} p_0 = 4.44 + \log _{10} V - 0.84\log _{10} s \hfill \\ {\text{ }} = 3.44 + \log _{10} M - 0.84\log _{10} s. \hfill \\ \end{gathered} $$ Garrett's (1970) theory is examined on the generation of water level fluctuations by an air pressure wave crossing a water depth discontinuity such as a continental shelf. The total amplitude of the ocean wave is determined to be where c ₂ ¹ = gh ₁, c ₂ ² = gh ₂, g is acceleration of gravity, h ₁ and h ₂ are the water depths on the ocean and shore side of the depth discontinuity, c is the speed of propagation of the air pressure wave, and ? is the water density. $$B = \left[ {\frac{{c_2^2 }}{{c^2 - c_2^2 }} + \frac{{c^2 (c_1 - c_2 )}}{{(c - c_1 )(c^2 - c_2^2 )}}} \right]\frac{{p_0 }}{{g\varrho }}$$ It is calculated that a 10 km³ eruption at Mount St. Augustine would cause a 460 Pa air pressure wave and a discernible water level fluctuation at Vancouver Island of several cm amplitude.     

The notion of “distinguishability” between bulk elastic parameters on the basis of the Gibbs deformation energy

The present work aims in discussing a principle that distinguishes between elastic parameters sets, $ \{ \Upphi \} \equiv \{ K_{0} , \, K^{\prime}, \, V_{0} ,\ldots\} The present work aims in discussing a principle that distinguishes between elastic parameters sets, { \Upphi } o { K₀ ,  K^￠,  V₀ ,?} \{ \Upphi \} \equiv \{ K_{0} , \, K^{\prime}, \, V_{0} ,\ldots\} , on the basis of an energetic criterion: once a reference set, { \Upphi_R } \{ \Upphi_{R} \} , is given, another one can be fixed, { \Upphi _min } \left\{ {\Upphi_{ \min } } \right\} , so that they are as close as possible to each other, but yield non-equivalent deformation energy curves \Updelta G({ \Upphi } )_\textdeform \Updelta G(\{ \Upphi \} )_{\text{deform}} , i.e. they give \Updelta G({ \Upphi_R } )_\textdeform \Updelta G(\{ \Upphi_{R} \} )_{\text{deform}} and \Updelta G({ \Upphi _min } )_\textdeform \Updelta G(\{ \Upphi_{ \min } \} )_{\text{deform}} such that | \Updelta G({ \Upphi _min } )_\textdeform - \Updelta G({ \Upphi_R } )_\textdeform | 3 1×s[\Updelta G_\textdeform ]. \left| {\Updelta G(\{ \Upphi_{ \min } \} )_{\text{deform}} - \Updelta G(\{ \Upphi_{R} \} )_{\text{deform}} } \right| \ge 1\times \sigma [\Updelta G_{\text{deform}} ]. ΔG _deform, calculated using the equation of state (EoS), and its uncertainty σ[ΔG _deform], obtained by a propagation of the errors affecting { \Upphi } \{ \Upphi \} are crucial to fix which mineral assemblage forms at P–T conditions and allow one to assess the reliability of such a prediction. We explore some properties related to the principle introduced, using the average values of the elastic parameters found in literature and related uncertainties for di-octahedral mica, olivine, garnet and clinopyroxene. Two elementary applications are briefly discussed: the effect of refining V ₀ in fitting EoSs to P–V experimental data, in the case of garnet and omphacite, and the phengite 3T–2M ₁ relative stability, controlled by pressure.     

Spherical harmonic analysis of earth’s conductive heat flow

A reappraisal of the international heat flow database has been carried out and the corrected data set was employed in spherical harmonic analysis of the conductive component of global heat flow. Procedures used prior to harmonic analysis include analysis of the heat flow data and determination of representative mean values for a set of discretized area elements of the surface of the earth. Estimated heat flow values were assigned to area elements for which experimental data are not available. However, no corrections were made to account for the hypothetical effects of regional-scale convection heat transfer in areas of oceanic crust. New sets of coefficients for 12° spherical harmonic expansion were calculated on the basis of the revised and homogenized data set. Maps derived on the basis of these coefficients reveal several new features in the global heat flow distribution. The magnitudes of heat flow anomalies of the ocean ridge segments are found to have mean values of less than 150 mW/m². Also, the mean global heat flow values for the raw and binned data are found to fall in the range of 56–67 mW/m², down by nearly 25% compared to the previous estimate of 1993, but similar to earlier assessments based on raw data alone. To improve the spatial resolution of the heat flow anomalies, the spherical harmonic expansions have been extended to higher degrees. Maps derived using coefficients for 36° harmonic expansion have allowed identification of new features in regional heat flow fields of several oceanic and continental segments. For example, lateral extensions of heat flow anomalies of active spreading centers have been outlined with better resolution than was possible in earlier studies. Also, the characteristics of heat flow variations in oceanic crust away from ridge systems are found to be typical of conductive cooling of the lithosphere, there being little need to invoke the hypothesis of unconfined hydrothermal circulation on regional scales. Calculations of global conductive heat loss, compatible with the observational data set, are found to fall in the range of 29–34 TW, nearly 25% less than the 1993 estimate, which rely on one-dimensional conductive cooling models.     

External seismic stability of vertical geosynthetic-reinforced soil walls using pseudo-static method

Analysis of external stability of vertical geosynthetic-reinforced soil (GRS) walls is very important in the seismic prone zone. The scope of this paper is to obtain required minimum reinforcement length, L _min, for external seismic stability of vertical GRS walls by pseudo-static limit equilibrium method. Then, L _min can be calculated to resist sliding, eccentricity, and bearing capacity failure modes. The parameters considered include both horizontal and vertical seismic coefficients (k _h and k _v ), surcharge load (q), wall height (H) and the properties of retained backfill, GRS, and foundation soil. Results show that L _min against sliding failure mode, L _min,S , increases more quickly than that against the other two failure modes with the increase in k _h , q, or unit weight of retained backfill, γ _b , while L _min,S decreases more quickly than that against the other two failure modes with increase in friction angle of retained backfill, ? _b , or unit weight of GRS, γ _r . For the different failure modes, the effect of k _v on L _min is not identical with the change of k _h , and in addition, L _min/H will tend to remain unchanged with the increase in H. In general, L _min against bearing capacity failure mode, L _min,BC, is larger than L _min against the other two failure modes. However, L _min,BC will be less than L _min against eccentricity failure mode, L _min,E , for k _h exceeding 0.35, or friction angle of foundation soil, ? _f , exceeding 37°, and L _min,BC will also be less than L _min,S for friction angle of GRS, ? _r , being no more than 26°.     

An empirical Redlich-Kwong-type equation of state for water to 1,000 °C and 200 Kbar

An empirically derived Redlich-Kwong type of equation of state (ERK) is proposed for H₂O, expressing a, the term related to the attraction between the molecules, as a pressure-independent function of temperature, and b, the covolume, as a temperature-independent function of pressure. The coefficients of a(T) and b(P) were derived by least squares non-linear regression, using P-V-T data given by Burnham et al. (1969b) and Rice and Walsh (1957) in conjunction with more precise recent data obtained by Tanishita et al. (1976), Hilbert (1979) and Schmidt (1979): $$a(T) = 1.616 x 10^8 - 4.989 x 10^4 T - 7.358 x 10^9 T^{ - 1} $$ and $$ = \frac{{1 + 3.4505x 10^{--- 4} P + 3.8980x 10^{--- 9} P^2 - 2.7756x 10^{--- 15} P^3 }}{{6.3944x 10^{--- 2} + 2.3776x 10^{--- 5} + 4.5717x 10^{--- 10} P^2 }}$$ , where T is expressed in Kelvin and P in bars. The ERK works very well at upper mantle conditions, at least up to 200 kbar and 1,000 °C. At subcritical conditions and those somewhat above the critical point, it still reproduces the molar Gibbs energy, \(\tilde G_{{\text{H}}_{\text{2}} {\text{O}}} \) , with a maximum deviation of 400 joules. Thus, for the purpose of calculation of geologically interesting heterogeneous equilibria, it predicts the thermodynamic properties of H₂O well enough. The values of molar volume, \(\tilde V_{{\text{H}}_{\text{2}} {\text{O}}} \) , and \(\tilde G_{{\text{H}}_{\text{2}} {\text{O}}} \) are tabulated in the appendix over a considerable P-T range. A FORTRAN program generating these functions as well as a FORTRAN subroutine for calculating the fugacity values, \(f_{{\text{H}}_{\text{2}} {\text{O}}} \) for incorporation into existing programs, are available upon request.     

High-pressure elastic behavior of Ca4La6(SiO4)6(OH)2 a synthetic rare-earth silicate apatite: a powder X-ray diffraction study up to 9.33 GPa

The compression behavior of a synthetic Ca₄La₆(SiO₄)₆(OH)₂ has been investigated to about 9.33 GPa at 300 K using in situ angle-dispersive X-ray diffraction and a diamond anvil cell. No phase transition has been observed within the pressure range investigated. The values of zero-pressure volume V ₀, K ₀, and $K_{0}^{'}$ refined with a third-order Birch–Murnaghan equation of state are V ₀ = 579.2 ± 0.1 Å³, K ₀ = 89 ± 2 GPa, and $K_{0}^{'} = 10.9 \pm 0.8$ . If $K_{0}^{'}$ is fixed at 4, K ₀ is obtained as 110 ± 2 GPa. Analysis of axial compressible modulus shows that the a-axis (K _a0 = 79 ± 2 GPa) is more compressible than the c-axis (K _c0 = 121 ± 7 GPa). A comparison between the high-pressure elastic response of Ca₄La₆(SiO₄)₆(OH)₂ and the iso-structural calcium apatites is made. The possible reasons of the different elastic behavior between Ca₄La₆(SiO₄)₆(OH)₂ and calcium apatites are discussed.