General line integrals for gravity anomalies of irregular 2D masses with horizontally and vertically dependent density contrast

Geophysics ◽  
2009 ◽  
Vol 74 (2) ◽  
pp. I1-I7 ◽  
Author(s):  
Xiaobing Zhou
Keyword(s):  
Gravity Anomaly ◽  
Gravity Anomalies ◽  
Density Contrast ◽  
Vertical Position ◽  
Horizontal Position ◽  
Surface Integral ◽  
Cross Sectional ◽  
Contrast Model ◽  
Contrast Function ◽  
Dependent Density

Line integrals (LIs) are an efficient tool in calculating the gravity anomaly caused by an irregular 2D mass body because the 2D surface integral is reduced to a 1D LI. Historically, LIs have been derived for 2D mass bodies of depth-dependent density contrast. I derive LIs for 2D mass bodies with density contrast dependent on (1) horizontal and (2) horizontal and vertical directions. Assuming the density contrast depends only on horizontal position, two types of representative LIs are derived: LIs with logarithmic kernel and density-integrated LIs for any integrable density-contrast function. A general density-contrast model that depends on horizontal and vertical directions is developed to include three components: a function of horizontal position, a function of vertical position, and a sum of crossterms of horizontal and vertical positions. Based on the general density-contrast model defined and proper selection of 2D vector gravity potentials, general LIs are derived to calculate the gravity anomaly. The newly developed LI method is then compared with two cases from the literature in calculating gravity anomaly, and agreement is obtained. However, the new LI method allows for more general 2D density-contrast variations and can be used to calculate the gravity anomaly of a 2D mass body. Such a mass body can have any cross-sectional profile that can be approximated by a polygonal cross section with any density-contrast function that can be approximated by a rich set of basis functions.

2D vector gravity potential and line integrals for the gravity anomaly caused by a 2D mass of depth-dependent density contrast

Geophysics ◽  
2008 ◽  
Vol 73 (6) ◽  
pp. I43-I50 ◽  
Author(s):  
Xiaobing Zhou
Keyword(s):  
Gravity Anomaly ◽  
Density Function ◽  
Case Studies ◽  
Two Dimensions ◽  
Density Contrast ◽  
Gravity Potential ◽  
Gravity Modeling ◽  
Desktop Computer ◽  
Contrast Function ◽  
Dependent Density

Using line integrals (LIs) used to calculate the gravity anomaly caused by a 2D mass of complicated geometry and spatially variable density contrast is a computationally efficient algorithm, that reduces the calculation from two dimensions to one dimension. This work has developed a mechanism for defining LIs systematically for different types of density functions. Two-dimensional vector gravity potential is defined as a vector, the net circulation of which, along the closed contour bounding a 2D mass, equals the gravity anomaly caused by the 2D mass. Two representative types of LIs are defined: an LI with an arctangent kernel for any depth-dependent density-contrast function, which has been studied historically; and an LI with a simple algebraic kernel for any integrable density-contrast function. The present work offers (1) a vectorial-based derivation of formulas that do not suffer from the arbitrary sign conventions found in some historical approaches; and (2) a simple algebraic kernel in line integrals as an alternative to the historical arctangent kernel, with the possibility of extension to more general cases. The concept of 2D vector gravity potential provides a useful tool for defining LIs systematically for any mass density function, helping us understand how dimensions can be reduced in a calculating gravity anomaly, especially when the density contrast varies with space. LIs have been tested in case studies. The maximum differences in calculated gravity anomalies by different LIs for the case studies were between [Formula: see text] and [Formula: see text]. Processing time required per station per segment of the 2D polygon of a 2D mass using LIs is [Formula: see text] on a Dell Optiplex GX 620 desktop computer, almost independent of the density function. The results indicate that the two types of LIs provide very fast, efficient, and reliable algorithms in gravity modeling or inversion for various types of density-contrast functions.

Stable inversion of gravity anomalies of sedimentary basins with nonsmooth basement reliefs and arbitrary density contrast variations

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 754-764 ◽  
Author(s):  
Valéria C. F. Barbosa ◽  
João B. C. Silva ◽  
Walter E. Medeiros
Keyword(s):  
Gravity Anomaly ◽  
Gravity Data ◽  
A Priori ◽  
Bouguer Anomaly ◽  
Gravity Anomalies ◽  
Sedimentary Basins ◽  
Upper And Lower Bounds ◽  
Density Contrast ◽  
Good Precision ◽  
Rift Basins

We present a new, stable method for interpreting the basement relief of a sedimentary basin which delineates sharp discontinuities in the basement relief and incorporates any law known a priori for the spatial variation of the density contrast. The subsurface region containing the basin is discretized into a grid of juxtaposed elementary prisms whose density contrasts are the parameters to be estimated. Any vertical line must intersect the basement relief only once, and the mass deficiency must be concentrated near the earth’s surface, subject to the observed gravity anomaly being fitted within the experimental errors. In addition, upper and lower bounds on the density contrast of each prism are introduced a priori (one of the bounds being zero), and the method assigns to each elementary prism a density contrast which is close to either bound. The basement relief is therefore delineated by the contact between the prisms with null and nonnull estimated density contrasts, the latter occupying the upper part of the discretized region. The method is stabilized by introducing constraints favoring solutions having the attributes (shared by most sedimentary basins) of being an isolated compact source with lateral borders dipping either vertically or toward the basin center and having horizontal dimensions much greater than its largest vertical dimension. Arbitrary laws of spatial variations of the density contrast, if known a priori, may be incorporated into the problem by assigning suitable values to the nonnull bound of each prism. The proposed method differs from previous stable methods by using no smoothness constraint on the interface to be estimated. As a result, it may be applied not only to intracratonic sag basins where the basement relief is essentially smooth but also to rift basins whose basements present discontinuities caused by faults. The method’s utility in mapping such basements was demonstrated in tests using synthetic data produced by simulated rift basins. The method mapped with good precision a sequence of step faults which are close to each other and present small vertical slips, a feature particularly difficult to detect from gravity data only. The method was also able to map isolated discontinuities with large vertical throw. The method was applied to the gravity data from Reco⁁ncavo basin, Brazil. The results showed close agreement with known geological structures of the basin. It also demonstrated the method’s ability to map a sequence of alternating terraces and structural lows that could not be detected just by inspecting the gravity anomaly. To demostrate the method’s flexibility in incorporating any a priori knowledge about the density contrast variation, it was applied to the Bouguer anomaly over the San Jacinto Graben, California. Two different exponential laws for the decrease of density contrast with depth were used, leading to estimated maximum depths between 2.2 and 2.4 km.

Analytic solution of the gravity anomaly of irregular 2D masses with density contrast varying as a 2D polynomial function

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. I11-I19 ◽  
Author(s):  
Xiaobing Zhou
Keyword(s):  
Gravity Anomaly ◽  
Density Function ◽  
Coordinate Transformation ◽  
Numerical Stability ◽  
Analytic Solution ◽  
Polynomial Function ◽  
Transformation Method ◽  
Density Contrast ◽  
Vertical Position ◽  
Analytic Methods

The analytic solution of the gravity anomaly caused by a 2D irregular mass body with the density contrast varying as a polynomial function in the horizontal and vertical directions is extrapolated from a historical version in which the analytic solution for the gravity anomaly was given only at the origin of the coordinate system to any point for the density function in terms of variables relative to that origin. To calculate the gravity anomaly at stations that are not at origins, a coordinate transformation is performed, in which case the polynomial density contrast function must also be expressed in the transformed coordinates, or a transformed solution must be obtained. These analytic solutions can be obtained at any station using (1) a solution transformation method, in which the density function and boundary of a mass body are kept intact, or (2) a coordinate transformation method, in which polynomial coefficient and boundary of a mass body are transformed accordingly. The issue of singularity and instability of the analytic methods has been related to case studies. Caution should be exercised in modeling or interpreting the gravity survey data using the analytic methods for large target-distance-to-target-size ratios outside the range of numerical stability. Compared with other published methods, the analytic solution results agree very well with other numerical or seminumerical methods, indicating the solution is correct and can be applied for any gravity anomaly calculation caused by an irregular 2D mass body with the density-contrast approximated as a polynomial function of horizontal position and/or vertical position when the observation is within the range of numerical stability.

Gravity inversion of basement relief and estimation of density contrast variation with depth

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. J51-J58 ◽  
Author(s):  
João B. Silva ◽  
Denis C. Costa ◽  
Valéria C. Barbosa
Keyword(s):  
Gravity Anomaly ◽  
Surface Density ◽  
Prior Information ◽  
Bouguer Anomaly ◽  
Gravity Anomalies ◽  
Sedimentary Basins ◽  
Density Contrast ◽  
Gravity Inversion ◽  
Contrast Variation ◽  
Interpretation Model

We present a method to estimate the basement relief as well as the density contrast at the surface and the hyperbolic decaying factor of the density contrast with depth, assuming that the gravity anomaly and the depth to the basement at a few points are known. In both cases, the interpretation model is a set of vertical rectangular 2D prisms whose thicknesses are parameters to be estimated and that represent the depth to the interface separating sediments and basement. The solutions to both problems are stable because of the incorporation of additional prior information about the smoothness of the estimated relief and the depth to the basement at a few locations, presumably provided by boreholes. The method was tested with synthetic gravity anomalies produced by simulated sedimentary basins with smooth relief, providing not only well-resolved estimated relief, but also good estimates for the density contrasts at the surface and for the decaying factors of the density contrast with depth. The method was applied to the Bouguer anomaly from Recôncavo Basin, estimating the surface density contrast and the decaying factor of the density contrast with depth as [Formula: see text] and [Formula: see text], respectively.

Gravity anomalies of two‐dimensional bodies of irregular cross‐section with density contrast varying with depth

Geophysics ◽  
1979 ◽  
Vol 44 (9) ◽  
pp. 1525-1530 ◽  
Author(s):  
I. V. Radhakrishna Murthy ◽  
D. Bhaskara Rao
Keyword(s):  
Gravity Anomaly ◽  
Cross Section ◽  
Cross Sections ◽  
Integral Method ◽  
Gravity Anomalies ◽  
Density Contrast ◽  
Two Dimensional ◽  
Gravity Effect ◽  
Line Integral ◽  
Exponential Increase

The line‐integral method of Hubbert (1948) is extended to obtain the gravity anomalies of two‐dimensional bodies of arbitrary cross‐sections with density contrast varying linearly with depth. The cross‐section is replaced by an N‐sided polygon. The coordinates of two vertices of any given side are used to determine the associated contribution to the gravity anomaly. The gravity contribution of each side is then summed to yield the total gravity effect. The case where density contrast varies exponentially with depth is also considered. This technique is used to obtain the structure of the San Jacinto Graben, California, where sediments filling the graben have an exponential increase in density with depth.

scholarly journals Implementation of Two-point Quadrature Gauss-Legendre Method on 2D Gravity Anomaly Modeling in Basins with Density Distribution Varied Polynomially as a Function of Depth

2018 ◽  
Vol 16 (2) ◽  
pp. 11
Author(s):  
Wahyu Srigutomo ◽  
Sesri Santurima ◽  
Cahyo Aji Hapsoro ◽  
Hairil Anwar ◽  
I Gede Putu Fadjar Soerya Djaja
Keyword(s):  
Density Distribution ◽  
Gravity Anomaly ◽  
Surface Topography ◽  
2D Gravity ◽  
Gravity Anomalies ◽  
Density Contrast ◽  
Quadrature Method ◽  
Gravity Method ◽  
Basin Geometry ◽  
Legendre Quadrature

Study of basin geometry basin is important in geosciences and geophysical exploration. Gravity method can be used to address this issue by measuring gravity anomalies on the surface caused by density contrast between the bedrock and the sediment that fills the basin, geometry of the basin and surface topography. Numerically, gravity anomaly modeling can be conducted using two-point rule Gauss-Legendre Quadrature method, for a case where density contrast varies with depth exponentially. Within the scope of this study, gravity anomalies on the surface are significantly affected by the geometry of the curvature of the bedrock as well as the topographic elevation of the surface and the selected density contrast, and are not significantly affected by the undulation of the bedrock curvature.  

Peripheral representation of antennal orientation by the scapal hair plate of the cockroach Periplaneta americana

2001 ◽  
Vol 204 (24) ◽  
pp. 4301-4309 ◽  
Author(s):  
J. Okada ◽  
Y. Toh
Keyword(s):  
Single Unit ◽  
Periplaneta Americana ◽  
Vertical Component ◽  
Horizontal Component ◽  
Vertical Position ◽  
Horizontal Position ◽  
Joint Movement ◽  
Basal Segment ◽  
Dynamic Phase ◽  
Hair Sensilla

SUMMARY Arthropods have hair plates that are clusters of mechanosensitive hairs, usually positioned close to joints, which function as proprioceptors for joint movement. We investigated how angular movements of the antenna of the cockroach (Periplaneta americana) are coded by antennal hair plates. A particular hair plate on the basal segment of the antenna, the scapal hair plate, can be divided into three subgroups: dorsal, lateral and medial. The dorsal group is adapted to encode the vertical component of antennal direction, while the lateral and medial groups are specialized for encoding the horizontal component. Of the three subgroups of hair sensilla, those of the lateral scapal hair plate may provide the most reliable information about the horizontal position of the antenna, irrespective of its vertical position. Extracellular recordings from representative sensilla of each scapal hair plate subgroup revealed the form of the single-unit impulses in response to hair deflection. The mechanoreceptors were characterized as typically phasic-tonic. The tonic discharge was sustained indefinitely (>20 min) as long as the hair was kept deflected. The spike frequency in the transient (dynamic) phase was both velocity- and displacement-dependent, while that in the sustained (steady) phase was displacement-dependent.

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