In order to obtain the gravitational field of a general finite body inside its Brillouin sphere, we developed a new method to compute the field accurately. First, the body is assumed to consist of some layers in a certain spherical polar coordinate system and the volume mass density of each layer is expanded as a Maclaurin series of the radial coordinate. Second, the line integral with respect to the radial coordinate is analytically evaluated in a closed form. Third, the resulting surface integrals are numerically integrated by the split quadrature method using the double exponential rule. Finally, the associated gravitational acceleration vector is obtained by numerically differentiating the numerically integrated potential. Numerical experiments confirmed that the new method is capable of computing the gravitational field independently of the location of the evaluation point, namely whether inside, on the surface of, or outside the body. It can also provide sufficiently precise field values, say of 14–15 digits for the potential and of 9–10 digits for the acceleration. Furthermore, its computational efficiency is better than that of the polyhedron approximation. This is because the computational error of the new method decreases much faster than that of the polyhedron models when the number of required transcendental function calls increases. As an application, we obtained the gravitational field of 433 Eros from its shape model expressed as the 24 × 24 spherical harmonic expansion by assuming homogeneity of the object.

Precise and Fast Computation of the Gravitational Field of a General Finite Body and Its Application to the Gravitational Study of Asteroid Eros

We developed an accurate method to compute the gravitational field of a tesseroid. The method numerically integrates a surface integral representation of the gravitational potential of the tesseroid by conditionally splitting its line integration intervals and by using the double exponential quadrature rule. Then, it evaluates the gravitational acceleration vector and the gravity gradient tensor by numerically differentiating the numerically integrated potential. The numerical differentiation is conducted by appropriately switching the central and the single-sided second-order difference formulas with a suitable choice of the test argument displacement. If necessary, the new method is extended to the case of a general tesseroid with the variable density profile, the variable surface height functions, and/or the variable intervals in longitude or in latitude. The new method is capable of computing the gravitational field of the tesseroid independently on the location of the evaluation point, namely whether outside, near the surface of, on the surface of, or inside the tesseroid. The achievable precision is 14–15 digits for the potential, 9–11 digits for the acceleration vector, and 6–8 digits for the gradient tensor in the double precision environment. The correct digits are roughly doubled if employing the quadruple precision computation. The new method provides a reliable procedure to compute the topographic gravitational field, especially that near, on, and below the surface. Also, it could potentially serve as a sure reference to complement and elaborate the existing approaches using the Gauss–Legendre quadrature or other standard methods of numerical integration.

Accurate computation of gravitational field of a tesseroid

We have developed a generalized solution for computing the gravity anomalies of 3D irregular-mass bodies with complicated density-contrast variation. The 3D irregular-shaped bodies can be approximated flexibly by a collection of finite-juxtaposed right-rectangular prisms. The complicated density-contrast variation of each prism can be well-represented by a depth-dependent polynomial function. A novel analytic solution of gravity anomalies due to a right-rectangular prism with an arbitrary order of polynomial density-contrast function of depth is then derived. The solution is singularity free in the upper half-space over the prism, and its singularity in the lower half-space containing the prism is resolved by assigning their limit values to the singular terms. The numerical stability of the solution is also evaluated through numerical tests. Hence, the solution can be used to compute the gravity anomalies of 3D irregular bodies with variable density contrasts without singularities when computation...

A versatile solution for the gravity anomaly of 3D prism-meshed bodies with depth-dependent density contrast

Rigorous modelling of the spherical gravitational potential spectra from the volumetric density and geometry of an attracting body is discussed. Firstly, we derive mathematical formulas for the spatial analysis of spherical harmonic coefficients. Secondly, we present a numerically efficient algorithm for rigorous forward modelling. We consider the finite-amplitude topographic modelling methods as special cases, with additional postulates on the volumetric density and geometry. Thirdly, we implement our algorithm in the form of computer programs and test their correctness with respect to the finite-amplitude topography routines. For this purpose, synthetic and realistic numerical experiments, applied to the gravitational field and geometry of the Moon, are performed. We also investigate the optimal choice of input parameters for the finite-amplitude modelling methods. Fourth, we exploit the rigorous forward modelling for the determination of the spherical gravitational potential spectra inferred by lunar crustal models with uniform, laterally variable, radially variable, and spatially (3D) variable bulk density. Also, we analyse these four different crustal models in terms of their spectral characteristics and band-limited radial gravitation. We demonstrate applicability of the rigorous forward modelling using currently available computational resources up to degree and order 2519 of the spherical harmonic expansion, which corresponds to a resolution of ~ 2.2 km on the surface of the Moon. Computer codes, a user manual and scripts developed for the purposes of this study are publicly available to potential users.

/pdf/forward-modelling-of-global-gravity-fields-with-3d-density-1bgznrm9xj.pdf

Forward modelling of global gravity fields with 3D density structures and an application to the high-resolution (~ 2 km) gravity fields of the Moon

Selection protocols such as SELEX, where molecules are selected over multiple rounds for their ability to bind to a target molecule of interest, are popular methods for obtaining binders for diagnostic and therapeutic purposes. With the increasing amount of such high-throughput experimental data available, machine learning techniques have become increasingly popular for molecular datasets analysis. Here, we show that Restricted Boltzmann Machines (RBMs), a two-layer neural network architecture, can successfully be trained on sequence ensembles from SELEX experiments for thrombin aptamers, and used to estimate the fitness of the sequences obtained through the experimental protocol. As a direct consequence, we show that trained RBMs can be exploited to classify as well as generate novel molecules. To confirm our findings, we experimentally verify the generated sequences from RBM.

/pdf/generative-and-interpretable-machine-learning-for-aptamer-1dsn84zq.pdf

Generative and interpretable machine learning for aptamer design and analysis of in vitro sequence selection

Restricted Boltzmann machines (RBM) are bilayer neural networks used for the unsupervised learning of model distributions from data. The bipartite architecture of RBM naturally defines an elegant sampling procedure, called alternating Gibbs sampling (AGS), where the configurations of the latent-variable layer are sampled conditional to the data-variable layer and vice versa. We study here the performance of AGS on several analytically tractable models borrowed from statistical mechanics. We show that standard AGS is not more efficient than classical Metropolis-Hastings (MH) sampling of the effective energy landscape defined on the data layer. However, RBM can identify meaningful representations of training data in their latent space. Furthermore, using these representations and combining Gibbs sampling with the MH algorithm in the latent space can enhance the sampling performance of the RBM when the hidden units encode weakly dependent features of the data. We illustrate our findings on three datasets: Bars and Stripes and MNIST, well known in machine learning, and the so-called lattice proteins dataset, introduced in theoretical biology to study the sequence-to-structure mapping in proteins.

Barriers and dynamical paths in alternating Gibbs sampling of restricted Boltzmann machines.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Deep Tempering with Nested Restricted Boltzmann Machines Clément Roussel, Jorge Fernandez-De-Cossio-Diaz, Simona Cocco, Rémi Monasson

Deep Tempering with Nested Restricted Boltzmann Machines

The purpose of this paper is to present the design, development and testing of an innovative instrument called GraviMob, which allows performing dynamic measurements of underwater gravity anomalies. After recalling the interest in underwater gravimetry, we describe the system, the core of which consists of triads of accelerometers rigidly attached to an Autonomous Underwater Vehicle (AUV). The article also presents the mathematical methods for estimating the east, north and vertical components of the local gravity vector. An unscented Kalman filter, integrating AUV position and orientation data, performs estimation of gravity in a frame adapted to its interpretation. To assess its performance, GraviMob was tested in the Mediterranean Sea during the year 2016. A comparison of the surface gravimetric signal previously acquired by the French Navy indicates that the maximum discrepancy between the vertical gravity component and its reference is below 4 mGal. Components of the vertical deflection calculated from GraviMob’s measurements were compared with those calculated from recent gravity field models. While a remarkable agreement was found on the north component, there remains a discrepancy (7 arcsec) on the east component which can be largely reduced by refining the estimation of the orientation of GraviMob’s sensitive axes in the AUV.

/pdf/development-of-a-lightweight-inertial-gravimeter-for-use-on-1fpnslb0.pdf

Development of a Lightweight Inertial Gravimeter for Use on Board an Autonomous Underwater Vehicle: Measurement Principle, System Design and Sea Trial Mission

The increasing availability of geophysical models of the Earth’s lithosphere and mantle has
generated renewed interest in computation of theoretical gravity effects at global and regional
scales. At the same time, the increasing availability of gravity gradient anomalies derived
from satellite measurements, such as those provided by GOCE satellite, requires mathematical
methods that directly model the gravity gradient anomalies in the same reference frame as
GOCE gravity gradients. Our main purpose is to interpret these anomalies in terms of source
and density distribution. Numerical integrationmethods for calculating gravity gradient values
are generally based on a mass discretization obtained by decomposing the Earth’s layers into
a finite number of elementary solid bodies. In order to take into account the curvature of
the Earth, spherical prisms or ‘tesseroids’ have been established unequivocally as accurate
computation tools for determining the gravitational effects of large-scale structures. The
question which then arises from, is whether gravity calculation methods using spherical
prisms remain valid when factoring in the ellipticity of the Earth. In the paper, we outline
a comprehensive method to numerically compute the complete gravity field with the help of
the Gauss–Legendre quadrature involving ellipsoidal shaped prisms. The assessment of this
new method is conducted by comparison between the gravity gradient values of simple sources
obtained by means of numerical and analytical calculations, respectively. A comparison of the
gravity gradients obtained from PREM and LITHO1.0 models using spherical- and ellipsoidalprism-
based methods is also presented. Numerical results indicate that the error on gravity
gradients, caused by the use of the spherical prism instead of its ellipsoidal counterpart to
describe an ellipsoidally shaped Earth, is useful for a joint analysis with those deduced from
GOCE satellite measurements. Provided that a suitable scaling of prism densities has been
performed, the spherical approximation error at GOCE height hardly reaches 1mE for the
entire Earth’s lithosphere. The error attains 6mE at a peak for a complete modeling of the
Earth, from the crust down to the internal core.

https://hal.archives-ouvertes.fr/hal-01947048/document

Complete gravity field of an ellipsoidal prism by Gauss–Legendre quadrature

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Papers

Barriers and dynamical paths in alternating Gibbs sampling of restricted Boltzmann machines.

Deep Tempering with Nested Restricted Boltzmann Machines

Development of a Lightweight Inertial Gravimeter for Use on Board an Autonomous Underwater Vehicle: Measurement Principle, System Design and Sea Trial Mission

Complete gravity field of an ellipsoidal prism by Gauss–Legendre quadrature