Linear geophysical inversion via the discrete cosine pseudo‐inverse: application to potential fields

In this paper, we present a methodology to perform geophysical inversion of large‐scale linear systems via a covariance‐free orthogonal transformation: the discrete cosine transform. The methodology consists of compressing the matrix of the linear system as a digital image and using the interesting properties of orthogonal transformations to define an approximation of the Moore–Penrose pseudo‐inverse. This methodology is also highly scalable since the model reduction achieved by these techniques increases with the number of parameters of the linear system involved due to the high correlation needed for these parameters to accomplish very detailed forward predictions and allows for a very fast computation of the inverse problem solution. We show the application of this methodology to a simple synthetic two‐dimensional gravimetric problem for different dimensionalities and different levels of white Gaussian noise and to a synthetic linear system whose system matrix has been generated via geostatistical simulation to produce a random field with a given spatial correlation. The numerical results show that the discrete cosine transform pseudo‐inverse outperforms the classical least‐squares techniques, mainly in the presence of noise, since the solutions that are obtained are more stable and fit the observed data with the lowest root‐mean‐square error. Besides, we show that model reduction is a very effective way of parameter regularisation when the conditioning of the reduced discrete cosine transform matrix is taken into account. We finally show its application to the inversion of a real gravity profile in the Atacama Desert (north Chile) obtaining very successful results in this non‐linear inverse problem. The methodology presented here has a general character and can be applied to solve any linear and non‐linear inverse problems (through linearisation) arising in technology and, particularly, in geophysics, independently of the geophysical model discretisation and dimensionality. Nevertheless, the results shown in this paper are better in the case of ill‐conditioned inverse problems for which the matrix compression is more efficient. In that sense, a natural extension of this methodology would be its application to the set of normal equations.     

Modelling Gravitational Instabilities: Slab Break–off and Rayleigh–Taylor Diapirism

A non-standard new code to solve multiphase viscous thermo–mechanical problems applied to geophysics is presented. Two numerical methodologies employed in the code are described: A level set technique to track the position of the materials and an enrichment of the solution to allow the strain rate to be discontinuous across the interface. These techniques have low computational cost and can be used in standard desktop PCs. Examples of phase tracking with level set are presented in two and three dimensions to study slab detachment in subduction processes and Rayleigh–Taylor instabilities, respectively. The modelling of slab detachment processes includes realistic rheology with viscosity depending on temperature, pressure and strain rate; shear and adiabatic heating mechanisms; density including mineral phase changes and varying thermal conductivity. Detachment models show a first prolonged period of thermal diffusion until a fast necking of the subducting slab results in the break–off. The influence of several numerical and physical parameters on the detachment process is analyzed: The shear heating exerts a major influence accelerating the detachment process, reducing the onset time to one half and lubricating the sinking of the detached slab. The adiabatic heating term acts as a thermal stabilizer. If the mantle temperature follows an adiabatic gradient, neglecting this heating term must be included, otherwise all temperature contrasts are overestimated. As expected, the phase change at 410 km depth (olivine–spinel transition) facilitates the detachment process due to the increase in negative buoyancy. Finally, simple plume simulations are used to show how the presented numerical methodologies can be extended to three dimensions.     

Carbon Dioxide Discharged through the Las Cañadas Aquifer, Tenerife, Canary Islands

Carbon dioxide is one of the first gases to escape the magmatic environment due to its low solubility in basaltic magmas at low pressures. The exsolved CO₂ gas migrates towards the surface through rock fractures and high permeability paths. If an aquifer is located between the magmatic environment and the surface, a fraction of the CO₂ emitted is dissolved in the aquifer. In this paper, an estimation of the water mass balance and the CO₂ budget in Las Cañadas aquifer, Tenerife, Canary Islands, is presented. Magmatic CO₂ is transported by groundwater and discharged through man-made sub-horizontal drains or galleries that exist in this island, and by the flow of groundwater discharged laterally towards other aquifers or to the ocean. In addition, the pCO₂ at the gallery mouth (or entrance) and at the gallery bottom (internal and deepest discharge point where the gallery starts) are calculated and mapped. The total CO₂ advectively transported by groundwater is estimated to range from 143 to 211 t CO₂ d^?1. Considering that the diffuse soil emission of CO₂ for the same area is 437 t d^?1, the diffuse/dissolved CO₂ flux ratio varies between 2 and 3. The high dissolved inorganic carbon content of groundwater explains the ability of this low temperature hydrothermal water to dissolve and transfer magmatic CO₂ at volcanoes, even during quiescence periods.     

Ground deformation in a viscoelastic medium composed of a layer overlying a half-space: a comparison between point and extended sources

We obtain and compare analytical and numerical solutions for ground displacement caused by an overpressurized magma chamber placed in a linear viscoelastic medium composed of a layer over a half-space. Different parameters such as size, depth and shape of the chamber, crustal rheology and topography are considered and discussed. Numerical solutions for an axisymmetric extended source are computed using a finite element method (FEM). Analytical solutions for a point source are obtained using the dislocation theory and the propagator matrix technique. In both cases, the elastic solutions are used together with the correspondence principle of linear viscoelasticity to obtain the solution in the Laplace transform domain. Viscoelastic solutions in the time domain are derived inverting the Laplace transform using the Prony series method. The differences between the results allow us to constrain the applicability of the point source and the flat surface hypothesis, which are usually implicitly assumed when analytical solutions are derived. The effect of the topography is also considered. The results obtained show that neglecting the topographic effects may, in some cases, introduce an error greater than that implicit in the point-source hypothesis. Therefore, for an adequate modelling and interpretation of the time-dependent displacements, topography must be considered.     

The Velocity Field of Young Stars in the Solar Neighbourhood

A sample of O- and B-type stars with Hipparcos astrometric data, ages computed from Strömgren photometry and radial velocities, has been used to characterize the structure, age and kinematics of the Gould Belt system. The local spiral structure of our galaxy is determined from this sample, and also from a sample of Hipparcos Cepheid stars. The Gould Belt, with an orientation with respect to the galactic plane ofi _G = 16-22^° and Ω_G =275-295^°, extends up to a distance of 600 pc from the Sun. Roughly the 60-65% of the O and B stars younger than 60 Myr in the solar neighbourhood belong to this structure. Our results indicate that the kinematical behaviour of this system is complex, with an expansion motion in the solar neighbourhood (R<300 pc).In the frame of the Lin's theory, and analysing the O and B stars further than 600 pc and the Cepheids, we found a galactic spiral structure characterized by a 4-arm spiral pattern with the Sun located atψ_⊙ = 350-355 ± 30^° – near the Sagittarius-Carina arm– and outside the corotation circle. The angular rotation speed of the spiral pattern was found to be Ω_p = 31-32 ± 4 kms^-1 kpc^-1.     

An analysis of the convergence of Newton iterations for solving elliptic Kepler’s equation

In this note a study of the convergence properties of some starters \( E_0 = E_0(e,M)\) in the eccentricity–mean anomaly variables for solving the elliptic Kepler’s equation (KE) by Newton’s method is presented. By using a Wang Xinghua’s theorem (Xinghua in Math Comput 68(225):169–186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton’s method, we obtain for each starter \( E_0(e,M)\) a set of values \( (e,M) \in [0, 1) \times [0, \pi ]\) that lead to the q-convergence in the sense that Newton’s sequence \( (E_n)_{n \ge 0}\) generated from \( E_0 = E_0(e,M)\) is well defined, converges to the exact solution \(E^* = E^*(e,M)\) of KE and further \( \vert E_n - E^* \vert \le q^{2^n -1}\; \vert E_0 - E^* \vert \) holds for all \( n \ge 0\). This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014) by using Smale’s \(\alpha \)-test with \(q=1/2\). Also since in KE the convergence rate of Newton’s method tends to zero as \( e \rightarrow 0\), we show that the error estimates given in the Wang Xinghua’s theorem for KE can also be used to determine sets of q-convergence with \( q = e^k \; \widetilde{q} \) for all \( e \in [0,1)\) and a fixed \( \widetilde{q} \le 1\). Some remarks on the use of this theorem to derive a priori estimates of the error \( \vert E_n - E^* \vert \) after n Kepler’s iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained.     

High resolution characterization of northwest Mediterranean coastal waters thermal regimes: To better understand responses of benthic communities to climate change

In the North West Mediterranean (NWM), mass mortality events (MME) of long-lived benthic species that have occurred over the last two decades have been related to regional warming trend. Gaining robust data sets on thermal regimes is critical to assess conditions to which species have adapted, detect extreme events and critically evaluate biological impacts. High resolution temperature (T) time series obtained during 1999–2006 from 5 to 40 m depth at four contrasted sites of the NWM were analyzed: Area Marina Protegida de les Illes Medes (NE Spain), Riou (Marseilles, France), Parc National de Port-Cros (France), and Réserve Naturelle de Scandola (Corsica, France). The seasonal pattern showed winter T around 11–13 °C, and summer T mainly around 22–24 °C near surface to 18–20 °C at depth. Stratification dynamics showed recurrent downwellings (>40 m) at Medes, frequent observation (1/3rd of the summer) of deep and cold upwelled waters at Riou, while Scandola exhibited stable summer stratification and highest suprathermoclinal T. Port-Cros showed an intermediate regime that oscillated between Riou and Scandola depending on the occurrence of northern winds. Data distribution study permitted to identify and to characterize 3 large scale positive anomalies concomitant with the mass mortality outbreaks of summers 1999, 2003 and 2006. The analysis of biological surveys on gorgonian populations showed significant impacts during the 3 years with temperature anomalies. Besides the degree of impact showed inter-annual differences which could be related to different T conditions concomitant to mortality events, from slight increase in T extreme of only 1–2 °C over short duration, to lengthened more classical summer conditions. Our results therefore support the hypothesis that shallow NWM populations of long-lived benthic species are living near their upper thermal thresholds. Given actual trends and projections in NWM, the repetition of new MMEs in the next decades is extremely likely. In such context, the acquisition of dedicated high resolution T series proves to be crucial for increasing our detection, understanding and forecasting abilities.     

Effect of mixed toxic diets (Microcystis and Chlorella) on the rotifers Brachionus calyciflorus and Brachionus havanaensis cultured alone and together

Brachionus havanaensis is a common planktonic rotifer sometimes found together with Brachionus calyciflorus in Microcystis-infected waterbodies in México. Here, we evaluate the impact of mixed diets (dry weight basis, 0%, 25%, 50%, 75% and 100% Microcystis or Chlorella) on the growth of and competition between B. calyciflorus and B. havanaensis. In general, a higher proportion of Microcystis aeruginosa in the diet resulted in decreased population growth rate of both rotifer species. Whether grown separately or together, B. havanaensis was always numerically more abundant than B. calyciflorus, regardless of the proportion of Microcystis in the diet. However, when both species were grown together, the impact of M. aeruginosa in the diet had a more adverse effect on B. calyciflorus than on B. havanaensis. When grown alone, both rotifer species showed better population growth on a diet of 75% Chlorella and 25% Microcystis than either 100% Chlorella or Microcystis.     

GPS Monitoring of the Tropical Storm Delta along the Canary Islands Track, November 28-29, 2005

In this article we analyze the variations of the Zenith Tropospheric Delay (ZTD) and its components, Zenith Hydrostatic Delay (ZHD) and Zenith Wet Delay (ZWD) recorded by the GPS reference stations in the area of the Canary Islands during the passing of the tropical storm Delta on November 28 and 29, 2005. During this event, we observed that all GPS stations experienced significant increases of the ZWD value of over 100 mm and a decrease in the ZHD values of about 30 mm. The increase of the normal ZWD values was detected several hours prior to the manifestation of the weather phenomena on the ground. We also noticed a connection between the maximum ZWD values observed and the temporal distribution of the rain. The observed variations of the tropospheric slant directional gradients correlate significantly with the variations in direction and intensity of the observed winds. The relation noted between the ZWD values and the tropospheric slant delay gradients with meteorological observables highlights the convenience of using existing or new GPS networks when studying weather phenomena such as severe cyclones.