Marquardt inverse modeling of the residual gravity anomalies due to simple geometric structures: A case study of chromite deposit

Abstract In this paper, an inversion method based on the Marquardt’s algorithm is presented to invert the gravity anomaly of the simple geometric shapes. The inversion outputs are the depth and radius parameters. We investigate three different shapes, i.e. the sphere, infinite horizontal cylinder and semi-infinite vertical cylinder for modeling. The proposed method is used for analyzing the gravity anomalies from assumed models with different initial parameters in all cases as the synthetic data are without noise and also corrupted with noise to evaluate the ability of the procedure. We also employ this approach for modeling the gravity anomaly due to a chromite deposit mass, situated east of Sabzevar, Iran. The lowest error between the theoretical anomaly and computed anomaly from inverted parameters, determine the shape of the causative mass. The inversion using different initial models for the theoretical gravity and also for real gravity data yields approximately consistent solutions. According to the interpreted parameters, the best shape that can imagine for the gravity anomaly source is the vertical cylinder with a depth to top of 7.4 m and a radius of 11.7 m.

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Depth and shape solutions from residual gravity anomalies due to simple geometric structures using a statistical approach

Contributions to Geophysics and Geodesy ◽

10.1515/congeo-2017-0008 ◽

2017 ◽

Vol 47 (2) ◽

pp. 113-132 ◽

Cited By ~ 1

Author(s):

El-Sayed Abdelrahman ◽

Mohamed Gobashy

Keyword(s):

Gravity Data ◽

Random Noise ◽

Synthetic Data ◽

Vertical Cylinder ◽

Gravity Anomalies ◽

Horizontal Cylinder ◽

Random Errors ◽

Residual Gravity ◽

Characteristic Points ◽

The Usa

AbstractWe have developed a simple and fast quantitative method for depth and shape determination from residual gravity anomalies due to simple geometrical bodies (semi-infinite vertical cylinder, horizontal cylinder, and sphere). The method is based on defining the anomaly value at two characteristic points and their corresponding distances on the anomaly profile. Using all possible combinations of the two characteristic points and their corresponding distances, a statistical procedure is developed for automated determination of the best shape and depth parameters of the buried structure from gravity data. A least-squares procedure is also formulated to estimate the amplitude coefficient which is related to the radius and density contrast of the buried structure. The method is applied to synthetic data with and without random errors and tested on two field examples from the USA and Germany. In all cases examined, the estimated depths and shapes are found to be in good agreement with actual values. The present method has the capability of minimizing the effect of random noise in data points to enhance the interpretation of results.

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Shape and depth solutions from gravity data using correlation factors between successive least‐squares residuals

Geophysics ◽

10.1190/1.1443393 ◽

1993 ◽

Vol 58 (12) ◽

pp. 1785-1791 ◽

Cited By ~ 13

Author(s):

El‐Sayed M. Abdelrahman ◽

Hesham M. El‐Araby

Keyword(s):

Least Squares ◽

Shape Factor ◽

Gravity Data ◽

Bouguer Anomaly ◽

Vertical Cylinder ◽

Gravity Anomalies ◽

Horizontal Cylinder ◽

Amplitude Coefficient ◽

Regional Component ◽

Correlation Factors

The gravity anomaly expression produced by most geologic structures can be represented by a continuous function in both shape (shape factor) and depth variables with an amplitude coefficient related to the mass. Correlation factors between successive least‐squares residual gravity anomalies from a buried vertical cylinder, horizontal cylinder, and sphere are used to determine the shape and depth of the buried geologic structure. For each shape factor value, the depth is determined automatically from the correlation value. The computed depths are plotted against the shape factor representing a continuous correlation curve. The solution for the shape and depth of the buried structure is read at the common intersection of correlation curves. This method can be applied to a Bouguer anomaly profile consisting of a residual component caused by local structure and a regional component. This is a powerful technique for automatically separating the Bouguer data into residual and regional polynomial components. This method is tested on theoretical examples and a field example. In both cases, the results obtained are in good agreement with drilling results.

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Exploring viable geologic interpretations of gravity models using distance-based global sensitivity analysis and kernel methods

Geophysics ◽

10.1190/geo2017-0742.1 ◽

2018 ◽

Vol 83 (5) ◽

pp. G79-G92 ◽

Cited By ~ 2

Author(s):

Geoff Phelps ◽

Celine Scheidt ◽

Jef Caers

Keyword(s):

Sensitivity Analysis ◽

Gravity Anomaly ◽

Gravity Data ◽

Gravity Anomalies ◽

Gravity Models ◽

Normal Faults ◽

Reverse Fault ◽

Density Values ◽

Generalized Sensitivity

ABSTRACT We have explored ways to integrate alternative geologic interpretations into the modeling of gravity data. These methods are applied to the Vaca Fault east of Fairfield, California, USA, where the structure across the fault is in question, and the Vaca Fault is used as a case study to demonstrate the method. The Vaca Fault is modeled using gravity data collected along a 10 km line perpendicular to the strike of the fault. Of particular interest is how the gravity data might inform on the dip of the Vaca Fault and thickness of the nonmarine section and whether spatial autocorrelation of density internal to the geologic units significantly influences the resulting gravity anomaly. We approach these questions by creating a suite of structural geologic models, which we then populate with geostatistically generated densities and from which the respective synthetic gravity anomalies are calculated. We perform distance-based generalized sensitivity analysis to identify which model inputs most leverage the calculated gravity anomaly. We then use multidimensional scaling to transform the gravity anomalies into a metric space and estimate the posterior probabilities of each structural geologic model using a Bayesian approach. We find that the gravity anomalies are particularly sensitive to zones of autocorrelated density values generated from geostatistical modeling. The structural geologic models most likely to produce gravity anomalies that match the observed data are the moderately dipping normal faults, 45° and 60°, although the probability that the fault dips more steeply, including in a strike slip or reverse fault orientation, is approximately 30%. The probability of a thicker nonmarine unit is 67%, more probable than a thinner nonmarine unit. This suggests that the Vaca Fault dips moderately to the east and truncates a thicker nonmarine unit, but that any further process modeling should include alternatives of the geologic structures.

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Basin Depth Mapping Beneath Wellington City Based on Residual Gravity Anomalies

10.26686/wgtn.15176061 ◽

2021 ◽

Author(s):

◽

Alistair Stronach

Keyword(s):

Gravity Anomaly ◽

Sedimentary Basin ◽

Gravity Data ◽

Three Dimensional ◽

Maximum Amplitude ◽

Central Business District ◽

Gravity Anomalies ◽

Depth Map ◽

Capital City ◽

Detailed Knowledge

New Zealand’s capital city of Wellington lies in an area of high seismic risk, which is further increased by the sedimentary basin beneath the Central Business District (CBD). Ground motion data and damage patterns from the 2013 Cook Strait and 2016 Kaikōura earthquakes indicate that two- and three-dimensional amplification effects due to the Wellington sedimentary basin may be significant. These effects are not currently accounted for in the New Zealand Building Code. In order for this to be done, three-dimensional simulations of earthquake shaking need to be undertaken, which requires detailed knowledge of basin geometry. This is currently lacking, primarily because of a dearth of deep boreholes in the CBD area, particularly in Thorndon and Pipitea where sediment depths are estimated to be greatest. A new basin depth map for the Wellington CBD has been created by conducting a gravity survey using a modern Scintrex CG-6 gravity meter. Across the study area, 519 new high precision gravity measurements were made and a residual anomaly map created, showing a maximum amplitude anomaly of -6.2 mGal with uncertainties better than ±0.1 mGal. Thirteen two-dimensional geological profiles were modelled to fit the anomalies, then combined with existing borehole constraints to construct the basin depth map. Results indicate on average greater depths than in existing models, particularly in Pipitea where depths are interpreted to be as great as 450 m, a difference of 250 m. Within 1 km of shore depths are interpreted to increase further, to 600 m. The recently discovered basin bounding Aotea Fault is resolved in the gravity data, where the basement is offset by up to 13 m, gravity anomaly gradients up to 8 mGal/km are observed, and possible multiple fault strands identified. A secondary strand of the Wellington Fault is also identified in the north of Pipitea, where gravity anomaly gradients up to 18 mGal/km are observed.

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Towards an updated, enhanced regional gravity field solution for Antarctica

10.5194/egusphere-egu21-9873 ◽

2021 ◽

Author(s):

Mirko Scheinert ◽

Philipp Zingerle ◽

Theresa Schaller ◽

Roland Pail ◽

Martin Willberg

Keyword(s):

Gravity Anomaly ◽

Gravity Field ◽

Gravity Data ◽

Gravity Anomalies ◽

Data Set ◽

Gravity Field Model ◽

Terrestrial Gravity ◽

Field Solution ◽

Gravity Disturbances ◽

Regional Gravity

In the frame of the IAG Subcommission 2.4f &#8220;Gravity and Geoid in Antarctica&#8221; (AntGG) a first Antarctic-wide grid of ground-based gravity anomalies was released in 2016 (Scheinert et al. 2016). That data set was provided with a grid space of 10 km and covered about 73% of the Antarctic continent. Since then a considerably amount of new data has been made available, mainly collected by means of airborne gravimetry. Regions which were formerly void of any terrestrial gravity observations and have now been surveyed include especially the polar data gap originating from GOCE satellite gravimetry. Thus, it is timely to come up with an updated and enhanced regional gravity field solution for Antarctica. For this, we aim to improve further aspects in comparison to the AntGG 2016 solution: The grid spacing will be enhanced to 5 km. Instead of providing gravity anomalies only for parts of Antarctica, now the entire continent should be covered. In addition to the gravity anomaly also a regional geoid solution should be provided along with further desirable functionals (e.g. gravity anomaly vs. disturbance, different height levels).We will discuss the expanded AntGG data base which now includes terrestrial gravity data from Antarctic surveys conducted over the past 40 years. The methodology applied in the analysis is based on the remove-compute-restore technique. Here we utilize the newly developed combined spherical-harmonic gravity field model SATOP1 (Zingerle et al. 2019) which is based on the global satellite-only model GOCO05s and the high-resolution topographic model EARTH2014. We will demonstrate the feasibility to adequately reduce the original gravity data and, thus, to also cross-validate and evaluate the accuracy of the data especially where different data set overlap. For the compute step the recently developed partition-enhanced least-squares collocation (PE-LSC) has been used (Zingerle et al. 2021, in review; cf. the contribution of Zingerle et al. in the same session). This method allows to treat all data available in Antarctica in one single computation step in an efficient and fast way. Thus, it becomes feasible to iterate the computations within short time once any input data or parameters are changed, and to easily predict the desirable functionals also in regions void of terrestrial measurements as well as at any height level (e.g. gravity anomalies at the surface or gravity disturbances at constant height).We will discuss the results and give an outlook on the data products which shall be finally provided to present the new regional gravity field solution for Antarctica. Furthermore, implications for further applications will be discussed e.g. with respect to geophysical modelling of the Earth&#8217;s interior (cf. the contribution of Schaller et al. in session G4.3).

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Warping least-squares inversion for 4D velocity change: Gulf of Mexico case study

Interpretation ◽

10.1190/int-2018-0145.1 ◽

2019 ◽

Vol 7 (2) ◽

pp. SB23-SB31

Author(s):

Chang Li ◽

Mark Meadows ◽

Todd Dygert

Keyword(s):

Gulf Of Mexico ◽

Least Squares ◽

Synthetic Data ◽

Inversion Method ◽

Velocity Change ◽

Data Set ◽

Phase Constraints ◽

Free Case ◽

Velocity Changes

We have developed a new trace-based, warping least-squares inversion method to quantify 4D velocity changes. There are two steps to solve for these velocity changes: (1) dynamic warping with phase constraints to align the baseline and monitor traces and (2) least-squares inversion for 4D velocity changes incorporating the time shifts and 4D amplitude differences (computed after trace alignment by warping). We have demonstrated this new inversion workflow using simple synthetic layered models. For the noise-free case, phase-constrained warping is superior to standard, amplitude-based warping by improving trace alignment, resulting in more accurate inverted velocity changes (less than 1% error). For synthetic data with 6% rms noise, inverted velocity changes are reasonably accurate (less than 10% error). Additional inversion tests with migrated finite-difference data shot over a realistic anticline model result in less than 10% error. The inverted velocity changes on a 4D field data set from the Gulf of Mexico are more interpretable and consistent with the dynamic reservoir model than those estimated from the conventional time-strain method.

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A Fast Interpretation Method for Inverse Modeling of Residual Gravity Anomalies Caused by Simple Geometry

Journal of Geological Research ◽

10.1155/2012/327037 ◽

2012 ◽

Vol 2012 ◽

pp. 1-10 ◽

Cited By ~ 10

Author(s):

Khalid S. Essa

Keyword(s):

Gravity Anomaly ◽

Shape Factor ◽

Vertical Cylinder ◽

Gravity Anomalies ◽

Depth Estimation ◽

Model Parameters ◽

Fast Method ◽

Random Errors ◽

Residual Gravity ◽

Amplitude Coefficient

An inversion technique using a fast method is developed to estimate, successively, the depth, the shape factor, and the amplitude coefficient of a buried structure using residual gravity anomalies. By defining the anomaly value at the origin and the anomaly value at different points on the profile, the problem of depth estimation is transformed into a problem of solving a nonlinear equation of the form . Knowing the depth, the shape factor can be estimated and finally the amplitude coefficient can be estimated. This technique is applicable for a class of geometrically simple anomalous bodies, including the semiinfinite vertical cylinder, the infinitely long horizontal cylinder, and the sphere. The efficiency of this technique is demonstrated with gravity anomaly due to a theoretical model, in each case with and without random errors. Finally, the applicability is illustrated using the residual gravity anomaly of Mobrun ore body, situated near Noranda, QC, Canada. The interpreted depth and the other model parameters are in good agreement with the known actual values.

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GRAVITY ANOMALY ASSESSMENT USING GGMS AND AIRBORNE GRAVITY DATA TOWARDS BATHYMETRY ESTIMATION

ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences ◽

10.5194/isprs-archives-xlii-4-w1-287-2016 ◽

2016 ◽

Vol XLII-4/W1 ◽

pp. 287-297

Author(s):

A. Tugi ◽

A. H. M. Din ◽

K. M. Omar ◽

A. S. Mardi ◽

Z. A. M. Som ◽

...

Keyword(s):

Gravity Anomaly ◽

Gravity Field ◽

Ocean Circulation ◽

Gravity Data ◽

Gravity Anomalies ◽

Airborne Gravity ◽

Satellite Gravity ◽

Earth’S Gravity Field ◽

Explorer Mission ◽

Satellite Gravity Missions

The Earth’s potential information is important for exploration of the Earth’s gravity field. The techniques of measuring the Earth’s gravity using the terrestrial and ship borne technique are time consuming and have limitation on the vast area. With the space-based measuring technique, these limitations can be overcome. The satellite gravity missions such as Challenging Mini-satellite Payload (CHAMP), Gravity Recovery and Climate Experiment (GRACE), and Gravity-Field and Steady-State Ocean Circulation Explorer Mission (GOCE) has introduced a better way in providing the information on the Earth’s gravity field. From these satellite gravity missions, the Global Geopotential Models (GGMs) has been produced from the spherical harmonics coefficient data type. The information of the gravity anomaly can be used to predict the bathymetry because the gravity anomaly and bathymetry have relationships between each other. There are many GGMs that have been published and each of the models gives a different value of the Earth’s gravity field information. Therefore, this study is conducted to assess the most reliable GGM for the Malaysian Seas. This study covered the area of the marine area on the South China Sea at Sabah extent. Seven GGMs have been selected from the three satellite gravity missions. The gravity anomalies derived from the GGMs are compared with the airborne gravity anomaly, in order to figure out the correlation (R2) and the root mean square error (RMSE) of the data. From these assessments, the most suitable GGMs for the study area is GOCE model, GO_CONS_GCF_2_TIMR4 with the R2 and RMSE value of 0.7899 and 9.886 mGal, respectively. This selected model will be used in the estimating the bathymetry for Malaysian Seas in future.

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Regional Recovery of Gravity Anomaly from the Inversion of Diagonal Components of GOCE Gravitational Tensor: A Case Study in Ethiopia

Artificial Satellites ◽

10.2478/arsa-2018-0006 ◽

2018 ◽

Vol 53 (2) ◽

pp. 55-74 ◽

Cited By ~ 1

Author(s):

Mehdi Eshagh ◽

Andenet A. Gedamu ◽

Tulu B. Bedada

Keyword(s):

Gravity Anomaly ◽

Gravity Field ◽

Ocean Circulation ◽

Opposite Sign ◽

Joint Inversion ◽

Gravity Anomalies ◽

The Earth ◽

Tikhonov Regularisation ◽

The Difference

Abstract The tensor of gravitation is traceless as the gravitational field of the Earth is harmonic outside the Earth’s surface. Therefore, summation of the 2nd-order horizontal derivatives on its diagonal components should be equal to the radial one but with the opposite sign. The gravity field can be recovered locally from either of them, or even their combination. Here, we use the in-orbit diagonal components of the gravitational tensor measured by the gravity field and steady state ocean circulation explorer (GOCE) mission for recovering gravity anomaly with a resolution of 1°×1° at sea level in Ethiopia. In order to solve the system of equations, derived after discretisation of integral equations, the Tikhonov regularisation is applied and the bias of this regularisation is estimated and removed from the estimated gravity anomalies. The errors of the anomalies are estimated and their significance of recovery from these diagonal components is investigated. Statistically, the difference between the recovered anomalies from each scenario is not significant comparing to their errors. However, their joint inversion of the diagonal components improved the solution by about 1 mGal. Furthermore, the inversion processes are better stabilised when using errors of the input data compared with its exclusion, but at the penalty of degradation in accuracy of the estimates.

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Bathymetry Inversion Using Vertical Deflections: a Case Study in South China Sea

10.21203/rs.3.rs-185040/v1 ◽

2021 ◽

Author(s):

Xiaoyun Wan ◽

Bo Liu ◽

Xiaohong Sui ◽

Richar Fiifi Annan ◽

Yijun Min

Keyword(s):

South China Sea ◽

Gravity Anomaly ◽

South China ◽

Additional Data ◽

Gravity Anomalies ◽

Local Area ◽

China Sea ◽

The North ◽

East Component

Abstract As an alternative method, an algorithm for bathymetry inversion using vertical deflections is proposed. Firstly, the formulas for the bathymetry inversion from north and east components of vertical deflections are derived and the data processing is introduced. Then a local area in the South China Sea is selected as an example to experiment the method. The bathymetry inversion based on gravity anomaly was also conducted for a comparison. The results show that the bathymetry derived from the north component of the vertical deflections have almost the same accuracy as that derived from gravity anomalies and the results derived from the east component have the poorest accuracy. The experiment’s results also show that accuracy of the derived bathymetry can be improved if the fitting parameters are adjusted according to the water depths. In summary, among the gravity field products used in this study, although the gravity anomaly yielded the best performance in the bathymetry inversion, the vertical defections can still be used as supplements, especially in areas where accurate vertical deflections exist. This is because deriving gravity anomaly from altimetry observations needs additional data and calculation efforts.

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