Open access peer-reviewed chapter - ONLINE FIRST

New Semi-Inversion Method of Bouguer Gravity Anomalies Separation

Written By

Abdel Fattah

Submitted: June 12th, 2021Reviewed: November 11th, 2021Published: May 10th, 2022

DOI: 10.5772/intechopen.101593

IntechOpen
Gravitational FieldEdited by Khalid S. Essa

From the Edited Volume

Gravitational Field [Working Title]

Prof. Khalid S. Essa

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Abstract

The workers and researchers in the field of gravity exploration methods, always dream that it is possible one day, to be able to separate completely the Bouguer gravity anomalies and trace rock’ formations, and their densities distribution from a prior known control points (borehole) to any extended distance in the direction of the profile lines-it seems that day become will soon a tangible true! and it becomes possible for gravity interpretation methods to mimic to some extent the 2D seismic interpretation methods. Where, the present chapter is dealing a newly 2D semi-inversion, fast, and easily applicable gravitational technique, based on Bouguer gravity anomaly data. It now becomes possible through, Excel software, Matlab’s code, and a simple algorithm; separating the Bouguer anomaly into its corresponding rock’ formations causative sources, as well as, estimating and tracing its thicknesses (or depths) of sedimentary formations relative to the underlying basement’s structure rocks for any sedimentary basin, through using of profile(s) line(s) and previously known control points. The newly proposed method has been assessed, examine, and applied for two field cases, Abu Roash Dome Area, southwest Cairo, Egypt, and Humble Salt Dome, USA. The method has demonstrated to some extent comparable results with prior known information, for drilled boreholes.

Keywords

  • Bouguer
  • sedimentary cover
  • slab
  • relatively thicknesses
  • basement

1. Introduction

The measurements and analysis of the variation in gravity over the Earth’s surface have become powerful techniques in the investigation of the subsurface structures at various depths [1]. Where the gravity anomaly is often attributed to the lateral variation in density-contrast and therefore, one of its major applications, being is used as a reconnaissance tool for and mapping the basement rock’s morphology, and its depth below the sedimentary covering of basins. The most challenging problem of ambiguity, for interpreting the potential-field data (gravity and/or magnetic), is still facing the researchers, where the modeling of potential-field data is a non-linear problem. In general, the reference body or source body (i.e., causative body) is imported into the potential model (gravity and/or magnetic), as the initial approximation of the anomaly source, and its parameters are obtained from available geological and geophysical information [2]. Ambiguity in gravity interpretation is inevitable because of the fundamental incompleteness of real observations; it is, however, possible to provide rigorous limits on possible solutions even with incomplete data [3]. Since a unique solution cannot, in general, be recovered from a set of field measurements, geophysical interpretation is concerned either to determine properties of the subsurface that all possible solutions share or to introduce assumptions to restrict the number of admissible solutions [4].

However, a unique solution may be found, when assigning a simple geometrical shape to the causative body [5]. Also, a unique solution can be found by an attempt for treating the problem of ambiguity with a new vision for analysis of the corrected acquired data (measurements) and related it analytically, logically, or mathematically, to its causative sources, as an attempt of the present research.

The newly proposed method is an attempt to reveal and trace the concealed subsurface geological formations’ thicknesses and basement depth at each point of the profile (s), of the Bouguer gravity anomaly map, relatively to the formations’ thicknesses and basement’ rock depth of a prior known in controlling point (e.g., borehole data). Fortunately, almost most of the geological structures can be approximated, by one or more of the available simple geometrical shape models, to represent the causative sources for gravity anomalies. There are several gravity forward techniques to estimate the depth to basement based on rather different approaches that have been proposed before by many authors, such as [6, 7, 8, 9]. The forward modeling of mass distribution is a powerful tool to visualize Free Air and Bouguer gravity anomalies that result from different geological situations [10].

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2. The methodology

There is a known fact that any depositions of formations layers were deposited in a basin and were may or not subjected to tectonic, hiatus non-deposition, and/or erosions. Therefore, simply the proposed method is mainly based on that fact to calculate average vertical densities-contrasts for the formation’s layers and the basement rock in that basin, whatever the geological setting of such formation’s layers.

The method is a direct technique for interpreting the Bouguer gravity anomaly in form of profile (s), to calculate the formations’ thicknesses, and formations’ depths of the sediments relatively to the depth of basement rocks, where the deposited rock’ formations are treated as the Bouguer Horizontal Slabs (BHS) or Infinite Horizontal Slabs (IHS), which are vertically stacked in columns and does not rely on the homogeneity or inhomogeneity of densities’ distributions, but only on the average vertical densities-contrasts between each of rock’ formations slab’s in columns and the basement rocks.

Since the Bouguer gravity anomaly correlates with the lateral variation of density of the crustal rocks, a positive or a negative anomaly is created, whenever there is a change in rock density [11]. And also, the Bouguer anomaly is defined upon the datum level of gravity reduction of an arbitrary elevation [12]. Where this correction is taken into account the attraction of masses between a reference elevation often the sea level, and each of an individually measuring stations’ points. In other words, the variation of the Bouguer anomaly should reflect the lateral variation in density, such that a high-density feature in a lower-density medium should give rise to a positive Bouguer anomaly. Conversely, a low-density feature in a higher-density medium should result in a negative Bouguer anomaly [13].

2.1 Infinite horizontal slab equation (IHSE)

The gravity effect or the Bouguer gravity anomaly can be calculated at any point of Cartesian coordinates (x, z) on the surface of the earth or reference measured datum, due to BHS or IHS is given by the Eq. (1) as follows:

gBxz=2πρhGE1

where gB: is the gravity effect or Bouguer gravity anomaly due to slab in m. Gal (10−5 m/s2).

ρ: is the density of the horizontal slab in gm/cm3 (103 kg/m3).

h: is the thickness of the IHS in km (103 m).

G: is the Universal Gravitational Constant (6.67 × 10−11 Nm2/kg2), N: is referring to Newton or force unit.

The Eq. (1) can be rewritten in a modified form for the new method purpose as follows:

gBxz=2πGi=1Nρ¯iziE2

where ρ¯: is the average vertical density-contrast in gm/cm3 (103 kg/m3).

z: is the difference between the depth of top and bottom the IHS in km (103 m).

i: is the index number (i = 1, 2, 3 … N).

N: is the number of rocks ‘formations, and

ρ¯=i=1NρiNρbasementE3

where i=1NρiN: is the average density, and

ρbasement: is the density of basement rock.

In the proposed method, the Infinite Horizontal Slab Equation (IHSE) is used to calculate the gravity effects at the earth’s surface (or any reference datum) for each subsurface rock’ formation that, covering the basement rocks for any sedimentary basin area, and using the IHSE ability, efficiency in the estimating, tracing the formations’ thicknesses (or depth), relative to the underlying basement rocks. By using the prior information of control point (or borehole), through profile (s) line (s) of Bouguer gravity anomaly which represents the vertical cross-section (s) for the area of study.

Simply the idea of the new method is based on the assumption that: the sedimentary rock’ formations covering the basement rocks are the formations deposited individually in form of layers, or a group of HIS, of different densities distributions is being stacked in columns over the basement’s rocks (Figure 1). And hence for any point (1, 2, 3, and 4) at the earth’s surface (or datum), the total gravity effect is the summation of all gravity effects (at point 4) contributed by each individual slab (1, 2, and 3) along the vertical axis of that point at the earth’s surface, were using the average vertical density-contrast (ρ¯), between each formation’s slab individually with basement rocks at points 1, 2, 3, and at point 4. Here is the new keen point of view that is: the average vertical densities’-contrasts between the stacked vertically of the IHSs and basement rocks are used in inversion calculations instead of densities’-contrasts between IHSs and their surrounding rock materials, through the importance for this concept, it being became possible for some extent to separate of Bouguer gravity anomaly to its components that representing probably of all possible rock’ formations overlying the basement rocks, according to the prior known information about those rock’ formations (thicknesses, depths, and densities), either from subsurface (borehole, etc.) or the surface geology (field, etc.).

Figure 1.

Bouguer gravity anomalies comparable to all possible rock’ formations overlying the basement rocks.

2.2 Building two models for formations densities’ distributions

To achieve the objective of the newly proposed method, the parameters of the rock’ formations of depositions covering the basement rocks for study areas such as their thicknesses, depths, and their densities, should be prior known at least, in one controlling points (borehole data, geophysical data, etc.), and thus such parameters (formations’ thicknesses, and densities), can be probably estimated and traced through the Bouguer gravity map’s profile (s) from the known point to the other unknown points of the area of study. Hence, a two models for formations density distributions building to prove that, the heterogeneities or homogeneities of formations density distributions do not affect the resultants of depth calculation from gravity effects of the IHSs as follows.

2.2.1 Heterogeneity formations density distributions (model 1)

As shown in Figure 2, the proposed model is consisting of a number (N) of deposited layers or formations, deposited according to Walther’s Law of deposition of the heterogeneous Juxtaposition of depositions.

Figure 2.

The model consists of five formations (N = 5), and densities are heterogeneously distributed.

For simplification, assuming the model is consisting of five formations (N = 5), and densities (gm/cm3) from top to bottom are (ρ(N), ρ(N-1), ρ(N-2), ρ(N-3), and ρ(N-4)) with thicknesses (m) are (h1, h2, h3, h4, and h5), respectively. Therefore, the average vertical densities from the top will be as follows:

ρv1N=ρN/N4E4
ρv1N1=ρN+ρN1/N3E5
ρv1N2=ρN+ρN1+ρN2/N2E6
ρv1N3=ρN+ρN1+ρN2+ρN3/N1E7
ρv1N4=ρN+ρN1+ρN2+ρN3+ρN4/NE8

Therefore, the average vertical densities for the above modeling is written in form of a row matrix (for the Matlab code purpose) as follows:

ρv1¯=ρv1Nρv1N1ρv1N2ρv1N3ρv1N4E9

then the average vertical densities-contrasts are:

ρv1¯i=ρv1¯iρbasementE10

And the gravity effect for model 1, is given as follows:

gBM1:i=2πGρv1¯iziE11

so that the depths can obtained by the following Eq. (12):

ZM1:i=abs(gBM1i/2πGρv1¯E12

where gBM1:iare the gravity effects of all points x(i) i.e. all vertical points (i = 1,2,3,4,5), and ZM1:iare the inverted depths at the same vertical points, and also the thicknesses hM1:ihave obtained from the following equation:

hM1:=i1NZm1iE13

2.2.2 Homogeneity formations density distributions (model 2)

As shown in Figure 3, the proposed model is consisting of a number (N) of deposited layers or formations, deposited according to Steno’s Law of superposition or Depositional History, Principle of homogeneous Juxtaposition of depositions.

Figure 3.

The model consists of five formations (N = 5), and densities are homogeneously distributed.

For simplification, assuming the model is consisting of five formations (N = 5), and densities (gm/cm3) from top to bottom are (ρ(N), ρ(N-1), ρ(N-2), ρ(N-3), and ρ(N-4)) with thicknesses (m) are (h1, h2, h3, h4, and h5), respectively. Therefore, the average vertical densities from the top will be as follows:

ρv2N=ρN/N4E14
ρv2N1=ρN+ρN1/N3E15
ρv2N2=ρN+ρN1+ρN2/N2E16
ρv2N3=ρN+ρN1+ρN2+ρN3/N1E17
ρv2N4=ρN+ρN1+ρN2+ρN3+ρN4/NE18

Therefore, the average vertical densities for the above modeling is written in form of a row matrix (for the Matlab code purpose) as follows:

ρv2¯=ρv2Nρv2N1ρv2N2ρv2N3ρv2N4E19

then the average vertical densities-contrasts are:

ρv2¯i=ρv2¯iρbasementE20

And the gravity effect for model 1, is given as follows:

gBM2:i=2πGρv2¯iziE21

so that the depths can obtained by the following Eq. (22):

ZM2:i=abs(gBM2i/2πGρv2¯E22

where gBM2:iare the gravity effects of all points x(i) i.e. all vertical points (i = 1,2,3,4,5), and ZM2:iare the inverted depths at the same vertical points, and also the thicknesses hM2:ihave obtained from the following equation:

hM2:=i1NZm2iE23

The goal of geophysical inversion (or interpretation) is to produce models whose response matches observations with noise levels [14]. It is known that the gravity measuring tools are very sensitive only to lateral changes in the causative source, therefore there are several models, that give solutions for the observed profile (ambiguities problem). Even with, this problem the gravity anomalies often are modeled by simple geometrical shapes, (or arbitrary shapes, Talwani et al. 1960). As in all geophysical inversions, there will be ambiguities, notably between density and layer depth, and many of these were pointed out, by [15, 16]. Geophysical inversion by iterative modeling involves fitting observations by adjusting model parameters. Both seismic and potential-field model responses can be influenced by the adjustment of the parameters of rock properties [14].

The new technique in the present research is based on two synthetic models and being built first, consistent, and constrained with real data of known controlling points (or borehole), then applying the algorithm of the solved equations to determine the formations’ thicknesses, the basement rocks depth, and tracing them relatively to the formations’ thicknesses and depth of basement rocks at the point of a prior known real data, through the profile line of Bouguer anomaly map’s covering the area of sedimentary basin.

2.3 Material

The proposed new 2D semi-inversion technique is carried out for any sedimentary basin area, by using a proper corrected Bouguer gravity anomaly map, with a prior known controlling point (s) of the formation’s densities and thicknesses (borehole), in addition, using an Excel, Surfer-15 software, and Matlab software for applications the written program for the proposed technique.

2.3.1 Bouguer gravity map

A digitizing process is carried out, for Bouguer’s gravity anomaly map that covers the investigated area, processing, and re-contouring with proper contour- intervals Then re-mapping with the location of the prior known controlling point (s) or borehole (s) locations, by using the Surfer-15 software to manipulate and dealing with data easily through the Excel and Matlab software. Thus, then a profile is taken along the map, in digitized form (file with two coordinates (x, gB)) that is later used for algorithm code application in Matlab.

2.3.2 Calculation gravity effects theoretically, with heterogeneous test model 1

The previously, the hypothetical depositional basin model-1 (Figure 2) consists of five formations layered slab deposited according to Walther’s Law of deposition. Therefore, the formations are filling-basin in five-rows (N = 5), and nine- columns (juxtaposing vertical columns). The formations’ thicknesses, depths, and densities are given, as seen in Table 1. Where ∆ρv1(i), represents here; the average vertical density-contrast for formations, stacked in nine columns of rows numbers N-4, N-3, N-2, N-1, and N respectively, and symmetrically repeated around the maximum formation’s thicknesses (central basin where N = 4). By using the equations from (8) to (13) using the Matlab code, the summation values of the vertical effects for stacked slab’ columns, at each point at x(i)-coordinates (x(i) = −4, −3, −2, −1, 0, 1, 2, 3, and 4.), are calculated; as well as the formation’s thicknesses and depths, are obtained and summarized by following, Table 2, where, the formation depths’ calculated are: 0.5, 1.5, 1.8, 3.5, and 4.0 km, are corresponding to the thicknesses (h(i)) of each formation sediments in the filling-basin, densities (1.900, 2.350, 2.450, 550, and 2.75 gm/cm3), and the calculated gravity effect curve of the hypothetical sedimentary basin, representing model-1 is seen (Figure 4). The depth of the basement is assumed as 4.5 km, and its density is 2.670 gm/cm3.

FormationRow No.Z (km)h (km)P (gm/cm3)V.Av1.ρv1(i) (gm/cm3)Δρv1(i) (gm/cm3)
AN - 40.50.51.902.75000.0800
BN - 31.51.02.352.6500−0.0200
CN - 21.80.32.452.5833−0.0867
DN -13.51.72.552.5250−0.1450
EN4.00.52.752.4000−0.2700
Basement4.52.67

Table 1.

Data for hypothetical theoretical horizontal slab model (1) of heterogeneous densities distribution.

X(km)−4−3−2−101234
Av.ρv1(i) (gm/cm3)2.75002.55002.51672.52502.40002.52502.51672.55002.7500
Δρv1(i) (gm/cm3)0.0800−0.1200−0.1533−0.1450−0.2700−0.1450−0.1533−0.12000.0800
gB_M1 (m. Gal)0.001676−0.00754−0.01157−0.02127−0.04526−0.02127−0.01157−0.007540.001676
h_cal1 (km)0.51.51.83.543.51.81.50.5
z_cal1 is calculated average depth of basement ((0.5 + 1.5 + 1.8 + 3.5 + 4)/5) × 2 = 4.5200 km.

Table 2.

Theoretical calculation for infinite horizontal slab model (1) for basin filling of five-sedimentary formations overlying basement rocks.

Figure 4.

The calculated gravity effect curve of the hypothetical sedimentary basin, representing model-1.

2.3.3 Calculation gravity effects theoretically, with homogenous test model 2

The previously, hypothetical depositional basin model-2 (Figure 3), consists of the same as the previous five formations layered slab deposited according to Walther’s and Steno’s superposition or geohistory concepts. Therefore, the formations are filling-basin in five rows (N = 5), and nine-columns (juxtaposing vertical columns). The formations’ thicknesses, depths, and densities are given, as seen in Table 3. Where ∆ρv2(i), represents here; the average vertical density-contrast for formations, stacked in nine columns of rows numbers N-4, N-3, N-2, N-1, and N respectively, and symmetrically repeated around the maximum formation’s thicknesses (central basin where N = 4). By using the equations from (14) to (23) using the Matlab code, the summed values of the vertical effects for stacked slab’ columns, at each point at x(i)-coordinates (x(i) = −4, −3, −2, −1, 0, 1, 2, 3, and 4.), are calculated; as well as the formation’s thicknesses and depths, are obtained and summarized by following, Table 4, where the formations depths’ calculated are: 0.5, 1.5, 1.8, 3.5, and 4.0 km, are corresponding to the thicknesses (h(i)) of each formation sediments in the filling-basin, densities (1.900, 2.350, 2.450, 550, and 2.75 gm/cm3), and the calculated gravity effect curve of the hypothetical sedimentary basin, representing model-2 is seen (Figure 5). The depth of the basement is assumed as 4.5 km, and its density is 2.670 gm/cm3.

FormationRow No.Z(km)h(km)P(gm/cm3)V.Av2.ρv2(i) (gm/cm3)Δρv2(i) (gm/cm3)
AN - 40.50.51.902.75000.0800
BN - 31.51.02.352.5500−0.1200
CN - 21.80.32.452.5167−0.1533
DN -13.51.72.552.5250−0.1450
EN4.00.52.752.4000−0.2700
Basement4.52.67

Table 3.

Data for hypothetical theoretical horizontal slab model (2) of homogenous densities distribution.

X(km)−4−3−2−101234
Av.ρv2(i) (gm/cm3)2.75002.55002.51672.52502.40002.52502.51672.55002.7500
Δρv2(i) (gm/cm3)0.0800−0.1200−0.1533−0.1450−0.2700−0.1450−0.1533−0.12000.0800
gB_M2 (m. Gal)0.001676−0.00754−0.01157−0.02127−0.04526−0.02127−0.01157−0.007540.001676
h_cal2 (km)0.51.51.83.543.51.81.50.5
z_cal2 is calculated average depth of basement ((0.5 + 1.5 + 1.8 + 3.5 + 4)/5) × 2 = 4.5200 km.

Table 4.

Theoretical calculation for infinite horizontal slab model (2) for basin filling of five- sedimentary formations overlying basement rocks.

Figure 5.

The calculated gravity effect curve of the hypothetical sedimentary basin, representing model-2.

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3. Implement the method in cases of real data

3.1 Abu Roash dome area, West Cairo, Egypt (case 1)

The famous Abu Roash Area located between Latitudes 29° 58′ and 30° 03′ N, and longitudes 30° 59′ 10′′ and 31° 05′ 19″ E. The Abu Roash district is located 10 km to the southwest of Cairo and is geologically significant because of its surface exposure of Upper Cretaceous rocks [17]. Its name (Abu Roash) is derived from the neighboring village of Abu Roash. The Abu Roash Dome Area constitutes a complex Cretaceous sedimentary succession with outstanding tectonic features, as shown in Figure 6, modified after [18].

Figure 6.

The location geological setting map of Abu Roash dome area, West Cairo, Egypt.

3.1.1 Geological setting

The Abu-Roash Dome Area was formed as a result of its location crossing by the western end of the Syrian-arc folds of which extends from northern Egypt to Syria (Laramide orogeny took place in Upper Cretaceous—Lower Tertiary), where the Upper Cretaceous rock formations in the northwestern desert of Egypt had undergone several different tectonic regimes.

The interest in basement depth estimation for the Abu Roash Dome Area was made by several authors’ and researchers’ gravitational potential studies, such as [19, 20, 21]. It is worthily mentioning that, the calculated basement depths calculated by the aforementioned authors’ methods, where the depths were estimated from the used modeled body center of a sphere, an infinite long horizontal cylinder, or a semi-infinite vertical cylinder, while in the present method the basement depths are estimated from the top of an infinite horizontal slab.

3.1.2 Procedures and results

The available Bouguer gravity anomaly map (GPC, 1984), covering the Abu Roash Dome Area, was digitized and re-contouring with a proper equal contour interval of 2 m. Gal (Figure 7a). The Abu Roash-1well, after [22], was used for the interesting formations’ depths and corresponding densities were summed in Table 5 and were used for building the two hypothetical models 1 and 2 for the Abu Roash Dome Area, with heterogeneous and homogeneous formations’ densities distributions, as respectively, as shown in Figures 8 and 9. For the two models, the theoretical calculations were carried out for thicknesses, averages’ vertical densities the averages’ vertical densities-contrasts, gravity effect, and calculated thicknesses corresponding to each of the selected five formations, that consists of the Abu Roash Dome area, and were summarized in Tables 6 and 7.

Figure 7.

The bouguer gravity anomaly map of Abu Roash dome area (modified after GPC, 1984).

FormationDepth (m)Thickness (m)Density* (gm/cm3)
Pleistocene0–1610.0691.980
Cenomanian161–6074462.480
Lower Cretaceous607–7591522.610
Jurassic759–15668072.430
Paleozoic1566–9023362.380
Basement1.9022.670

Table 5.

Abu Roash-1 well data (modified after El-Malky, 1985), where the elevation = 92 m and total depth = 1918 m.

Densities calculated for lithologic compositions of each formation.


Figure 8.

The hypothetical model 1 for the Abu Roash dome area, with heterogeneous formations’ densities distributions.

Figure 9.

The hypothetical model 2 for the Abu Roash dome area, with homogeneous formations’ densities distributions.

Formationz(km)h(km)ρ (gm/cm3)ρv1 (gm/cm 3)Δρv1 (gm/cm3)gB_M1 (m. Gal)h_M1 (km)
Pleistocene0.0920.0691.9801.9800−0.6900−0.00200.0690
Cenomanian0.1610.4462.4802.2300−0.4400−0.00820.4460
Lower Cretaceous0.6070.1522.6102.3567−0.3133−0.00200.1520
Jurassic0.7590.8072.4302.3750−0.2950−0.01000.8070
Paleozoic1.5660.3362.3802.3760−0.2940−0.00410.3360
Basement1.9022.670

Table 6.

Abu Roash dome area data and theoretical calculation parameters for model (1).

Formationz(km)h(km)ρ (gm/cm3)Pv2 (gm/cm3)Δρv2 (gm/cm3)gB_M2 (m. Gal)h_M2 (km)
Pleistocene0.0920.0691.9801.9800−0.6900−0.00200.0690
Cenomanian0.1610.4462.4802.2300−0.4400−0.00820.4460
Lower Cretaceous0.6070.1522.6102.3567−0.3133−0.00200.1520
Jurassic0.7590.8072.4302.3750−0.2950−0.01000.8070
Paleozoic1.5660.3362.3802.3760−0.2940−0.00410.3360
Basement1.9022.670

Table 7.

Abu Roash dome area data and theoretical calculation parameters for model (2).

A digitizing profile along line AA’ was carried out for Bouguer gravity anomaly map for Abu Roash Dome Area (Figure 7b), with equal intervals 2.09 km., then saved as Excel’s file (AbuRoash_aa_slab.xlsx), of two coordinates (x, gB). This file later will be used for data calculating, tracing the formations’ thicknesses, and depth’s basement rocks along the profile line AA’, by applying the proposed algorithm with Matlab’s codes. In the final step, it found that the calculations for the two models along the profile line AA’ are given the same results, as expected since the calculated average vertical density-contrasts are the same for the two models. The results for the two models of Abu Roash Dome Area are summarized in Tables 8 and 9 representing formations thicknesses, and depths, respectively, and represented graphically as shown in (Figures 10 and 11).

xgBh(l)h(2)h(3)h(4)h(5)H
0−8.389560.6067490.3869120.2755280.2594070.2585281.787124
2.083045−8.269290.5980510.3813660.2715780.2556880.2548221.761504
4.166091−8.152210.5895830.3759660.2677330.2520680.2512141.736565
6.249136−8.037040.5812530.3706540.2639510.2485070.2476641.71203
8.332181−7.922610.5729780.3653770.2601930.2449690.2441381.687655
10.41523−7.809920.5648280.360180.2564920.2414840.2406661.66365
12.49827−7.69720.5566760.3549820.252790.2379990.2371921.63964
14.58132−7.582060.5483480.3496710.2490080.2344390.2336441.615111
16.66436−7.466890.5400190.344360.2452260.2308780.2300951.590578
18.74741−7.344430.5311630.3387130.2412040.2270910.2263221.564493
20.83045−7.218590.5220620.3329090.2370720.22320.2224441.537686
22.9135−7.08770.5125960.3268730.2327730.2191530.218411.509806
414.526−8.80060.6364760.4058690.2890280.2721170.2711941.874683
416.6091−8.779010.6349140.4048730.2883180.2714490.2705291.870082
416.6091−8.779010.6349140.4048730.2883180.2714490.2705291.870082
0.4737420.3020970.2151290.2025420.201855
1.3878771.395366

Table 8.

The thicknesses of formation: Along profile AA’ in kilometers (H = Σh(i). i = 1, 2, 3, 4, and 5), Abu Roash dome area.

xgBz(1)z(2)z(3)z(4)z(5)Z
0−8.389561.7871241.7871241.7871241.7871241.7871241.787124
2.083045−8.269291.7615041.7615041.7615041.7615041.7615041.761504
4.166091−8.152211.7365651.7365651.7365651.7365651.7365651.736565
6.249136−8.037041.712031.712031.712031.712031.712031.71203
8.332181−7.922611.6876551.6876551.6876551.6876551.6876551.687655
10.41523−7.809921.663651.663651.663651.663651.663651.66365
12.49827−7.69721.639641.639641.639641.639641.639641.63964
14.58132−7.582061.6151111.6151111.6151111.6151111.6151111.615111
16.66436−7.466891.5905781.5905781.5905781.5905781.5905781.590578
18.74741−7.344431.5644931.5644931.5644931.5644931.5644931.564493
20.83045−7.218591.5376861.5376861.5376861.5376861.5376861.537686
416.6091−8.779011.8700821.8700821.8700821.8700821.8700821.870082
416.6091−8.779011.8700821.8700821.8700821.8700821.8700821.870082
1.3884581.3884581.3884581.3884581.388458
1.3878771.388458

Table 9.

The depths of formations thickness ‘along profile AA’ in biometers (Z = Σz(i),i = 1, 2, 3, 4 and 5), Abu Roash dome area.

Figure 10.

The resulted inversion formation’ thicknesses along profile AA’ of Bouguer map (Figure 7).

Figure 11.

The resulted inversion basement rock along profile AA’ of Bouguer map (Figure 7).

3.1.3 Interpretation of data results

From Table 8, it was found that the range of formations thicknesses’ (minimum to maximum) varying along the profile direction AA’ (Figure 10), as follows:

  • The Pleistocene formation thicknesses range (0.24145–0.71413 km).

  • The Cenomanian formation thicknesses range (0.15397–0.45539 km).

  • The Lower Cretaceous formation thicknesses range (10964–0.32429 km).

  • The Jurassic formation thicknesses range (0.10323–0.30532 km).

  • The Paleozoic formation thicknesses range (0.10288–0.30428 km).

From Table 7, the basement depth along the profile line AA’ (Figure 11), was determined as follows:

The maximum value of average depth (last column in Table 9), equal to 2.1034 km, corresponds to the Bouguer anomaly value of about −9.8743 m. Gal, and the minimum value of the last column, equal to 0.7116 km, corresponds to the Bouguer anomaly value of about −3.3385 m. Gal. Therefore, the average basement depth value is 1.40728 km, corresponds to the average Bouguer anomaly −6.6064, this is comparable with depth 1.916 km corresponds to Bouguer anomaly −5.5 m. Gal according to [23]. The calculated basement depths along profile line AA’, showed more or fewer values than actual drilled depth (1.902 km), which may be attributed to the lithologic change in the basement rocks, the above overlying sediment thicknesses, and the local faults are indicated as noses on the depths’ curve (Figure 11).

The Abu Roash Dome depth of value about 2.1 km is obtained by proposed method, that was to some extent agrees with the results obtained from drilling information (1.9 km; after [23], and a S-Curves method of depth determination (1.91 km; after [24]).

3.2 Humble salt dome in Harries County, Texas USA (case 2)

The gravimetric survey, with its sensitivity to variations in density-contrast among the subsurface structures, has been helpful in discovering and locating salt dome formations common to the Gulf Coast, of the USA.

3.2.1 Geological setting

The salt domes considering interesting as a source producing oils, minerals like Sulfur, Salts, and recently are used as burial locations for waste disposal of nuclear materials. Salt domes are common in the Gulf Coast area of Texas and Louisiana as well as in the Gulf of Mexico. The Gulf of Mexico basin began forming in the late Triassic as an intracontinental extension within the North American plate [25].

Salt was accumulated in the Jurassic period and geologically identified as the Louann Salt (mother source), which is a very thick deposit of salt known as halite composed of sodium chloride but with smaller amounts of sulfate, halides, and borates. The salt was followed by carbonate deposition during the Late Jurassic and Cretaceous, and clastic deposits during the Cenozoic [26]. With the deposition of additional sediments on top of this salt, it was buried to over 20,000 feet (6.096 km) and sometimes as deep as 40,000 feet (12.192 km) in the Deep-Water Gulf of Mexico.

The Humble Salt Dome in Harris County, Texas, USA, is one of the interiors of the Gulf Coastal Plain (Figure 12), and it is more than 20,000 feet (6.096 km) in diameter and less than 2000 feet (0.6096 km) below the surface.

Figure 12.

The location of humble salt dome referred in red circle on depth map (contours in feet).

The Humble Salt Dome estimation depth has been subjected to studies from several authors such as [21, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. Also, it is worthily mentioning that the calculated salt dome depths calculated by the aforementioned authors’ methods, depths were estimated from being considering the shape modeled salt body’s center either of a sphere, an infinite long horizontal cylinder, or a semi-infinite vertical cylinder, while in the present method the salt’s depths are related to basement rocks depths’ and are being estimated from the top of an infinite horizontal slab.

3.2.2 Procedures and results

The available Bouguer gravity anomaly map of Humble Salt Dome, Harris County, Texas, USA Area (After [27]), is digitized and re-contouring with a proper equal contour interval of 2 m. Gal (Figure 13a), where the gravity anomalies range between −9 m. Gal at the northeastern part of the map and more slightly of the value −22 m. Gal, at the center of Humble Salt Dome.

Figure 13.

The bouguer gravity anomaly map of humble salt dome (modified after Nettleton 1976).

The stratigraphic of formations, depths, thicknesses, and densities, as a controlling-point are obtained (after, [39]), were summed in Table 10 and was used for building the two hypothetical models 1 and 2 for the Humble Salt Dome. model (1) with heterogeneous formations’ densities distributions and model (2) with homogeneous formations’ densities distributions, as respectively shown (Figures 14 and 15). For the two models, the theoretical calculations were carried for thicknesses, averages’ vertical densities the averages’ vertical densities-contrasts, gravity effect, and calculated thicknesses corresponding to each of the selected five formations, which consists of the Humble Salt Dome, and was summarized in Tables 10 and 11.

FormationLithologyDepth (m)Thickness (m)Density *(gm/cm3)
AClay, Shale, Silt, and Sand182.881371.62.510
BClay, Shale, and Sand1554.48274.322.670
CLimestone and Shale1828.80548.642.700
D (Salt)Rock Salt2377.44822.962.100
BasementGranitic—Dioritic3048.002.950

Table 10.

Data information for control-point of humble salt dome (modified after, Okocha, 2017).

Densities are calculated for lithologic compositions of each formation.


Figure 14.

The hypothetical model 1 for the humble salt dome, with heterogeneous formations’ densities distributions.

Figure 15.

The hypothetical model 2 for the humble salt dome, with homogeneous formations’ densities distributions.

Formationz(m)h(m)ρ (gm/cm3)ρv1 (gm/cm3)Δρv1 (gm/cm3)gB_M1 (m. Gal)h_M1 (m)
A182.88182.8802.512.5100−0.4400−0.02531.3716
B1554.481371.6002.672.5900−0.3600−0.00410.27432
C1828.8274.3202.72.4950−0.4550−0.01050.54864
D (Salt)2377.44548.6402.12.2025−0.7475−0.02580.82296
Basement30782.95

Table 11.

Humble salt dome and theoretical calculation parameters for model (1).

A digitizing profile along line AA’ was carried out for Bouguer gravity anomaly map for Humble Salt Dome (Figure 14b), with equal intervals 0.35253 km., where the profile AA’ is about 30.5 km in length. Then the digitized values are saved as Excel’s file (humble_aa_slab.xlsx), of two coordinates (x, gB), where the file was later used for calculating, tracing the formations’ thicknesses, and depth’s basement rocks along the profile line AA’, by applying the algorithm of the proposed code with Matlab’s. In the final step, it found that the calculations for the two models along the profile line AA’ are given the same results, as expected since the calculated average vertical density-contrasts is the same for the two models. The results for the two models of Humble Salt Dome are summarized in Tables 12 and 13 representing formations thicknesses, and depths, respectively, and represented graphically as shown in (Figures 16 and 17).

Formationz(m)h(m)ρ (gm/cm3)Pv2 (gm/cm3)Δρv2 (gm/cm3)gB_M2 (m. Gal)h_M2 (m)
A182.88182.8802.512.5100−0.4400−0.02531.3716
B1554.481371.6002.672.5900−0.3600−0.00410.27432
C1828.8274.3202.72.4950−0.4550−0.01050.54864
D (Salt)2377.44548.6402.12.2025−0.7475−0.02580.82296
Basement30782.95

Table 12.

Humble salt dome and theoretical calculation parameters for model (2).

xgBh(1)h(2)h(3)h(4)H
0−15.304823430.7060.5770.7301.1993.211
2.311662826−15.279753240.7040.5760.7281.1973.206
4.623325652−15.253706120.7030.5750.7271.1953.200
6.934988478−15.227565420.7020.5740.7261.1933.195
9.246651304−15.201559640.7010.5730.7251.1913.189
11.55831413−15.175571840.7000.5720.7231.1893.184
13.86997696−15.147620430.6980.5710.7221.1863.178
16.18163978−15.118222820.6970.5700.7211.1843.172
18.49330261−15.08817310.6960.5690.7191.1823.166
20.80496543−15.057244590.6940.5680.7181.1793.159
23.11662826−15.024329330.6930.5670.7161.1773.152
25.42829108−14.984680370.6910.5650.7141.1743.144
460.0209024−9.8912111150.4560.3730.4720.7752.075
462.3325652−9.8930795950.4560.3730.4720.7752.076
462.3325652−9.8930795950.4560.3730.4720.7752.076
134.925110.393139.525229.219
3.0399083.039908

Table 13.

The thickness of formations along profile AA’ in kilometers (H = Σh(i), i = 1, 2, 3, 4, and 5), humble salt dome.

Figure 16.

The resulted inversion formation’ thicknesses along profile AA’ of bouguer map (Figure 13).

Figure 17.

The resulted inversion basement rock along profile AA’ of bouguer map (Figure 13).

3.2.3 Interpretation of data results

From Table 12, it was found that the range of formations thicknesses (minimum to maximum) varying along the profile direction AA’ (Figure 16), as follows:

  • The A-formation thicknesses range (0.453–1.035 km).

  • The B-formation thicknesses range (0.371–0.847 km).

  • The C-formation thicknesses range (0.469–1.070 km).

  • The D (Salt)-formation thicknesses range (0.770–1.758 km).

From Table 13, the basement depth along the profile line AA’ (Figure 9), was determined as follows:

The maximum value of average depth (last column in Table 13), equal to 4.70952 km, corresponds to the Bouguer anomaly value of about −22.437979 m. Gal (near the center of Salt Dome anomaly), and the minimum value of the last column, equal to 2.06245 km, corresponds to the Bouguer anomaly value of about −9.826328 m. Gal (near the edges of bounded the Salt Dom anomaly). Therefore, the average depth to the center of the Humble Salt Dome is about 3.386 km, corresponds to the Bouguer anomaly of about −16.128 m. Gal.

The Humble Salt Dome depth of value about 4.71 km is obtained by proposed method, that was agrees very well with the results obtained from drilling, seismic information (4.97 km; after [27]), and a simple method of depth determination by using shape factor (4.85 km; after [21]).

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4. Discussion and conclusions

The present research represents a new “semi-inversion” method for calculating, and tracing formations’ thicknesses and basement rocks depths, for deposition basin, relatively to a prior known control-point (s) or borehole (s), throughout profile (s) line (s) of Bouguer gravity anomaly map that connecting with the controlling-point (s). The present technique is to mimic to some extent tracing formations from borehole data to the seismic cross-section, in the seismic interpretation process. The resulting thicknesses and/or depths for profile (s) line (s) of Bouguer map, covering any being investigated area might be reused again to form grids for any interesting formation concerning the subsurface. Theoretical and field examples reveal the goodness and the efficiency of the method presented. Moreover, the method can be developed and used to help with planning seismic surveys.

4.1 The most important of advantages and disadvantages of proposed method

The new method has several advantages, more than other traditional separation methods. The most important is its capability for separation of Bouguer gravity anomaly above any depositional basin to directly its formations layers, and tracing them from a known point. But in the other methods it being separate only components of regional (basement rock or deeper) and residual (sediments rocks or shallower). On the other hand, side, the method is considered a pioneer theoretically, but still need an effort to develop and optimize of the Matlab Programming code, to be more efficient, saving time, and money in practical application.

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Acknowledgments

The author sincerely first thanks in advance to Dr. Karmen Daleta, Author Service Manager, for his kindest invitation, his follow during revision, Dr. Editor-Chief, Editors, Publishing Processor Manager, for helping, and directing during writing this paper. The gratitude extends to Prof. Dr. Khaled Essa for invitation and acceptance for sharing in IntechOpen-Chapter, and for his encouragement. Also, gratitude extended to my family especially my wife, for providing a good environment for carrying for this work.

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%Semi-Inversion_Finite_Slab_Model

% the method is depending on the concepts of Walther's Law of deposition

% and the Steno's Law of superposition of the juxtaposed columns of deposition

clc; % Clear the command window.

close all; % Close all figures (except those of in tool.)

clear; % Erase all existing variables. Or clear it if you want.

workspace; % Make sure the workspace panel is showing.

G = 6.67e-3; % universal gravitational constant (6.67e-3);

pi = (22/7); % circle D/R ratio or solid angle.

format long % for accurate decimal values

% gz in m. Gal//G = 6.67e-11//density contrast in (kg/m.^3)// (x, z, R) in (m)

%===========================================================%

%Modeling Parameters

%===========================================================%

N = 5 ;  % the number of stacked H. layers = the number of stacked H. layers

  % (The maximum columns will contain the 5-Formations)

%===========================================================%

%Reading Data File

%===========================================================%

data = xlsread('Abu Roach_aa_slab.xlsx');

xc = data(:,1);

% the observed points for digitized profile (km)

gB = data(:,2);

% digitized Bouguer anomaly data (profile) (m. Gal)

%i = 202;

x = 0: N-1;

%===========================================================%

%Borehole depths (km)

%===========================================================%

z1 =0.092; % Surface of Earth

z2 =0.161; % Cenomanian Fm.

z3 =0.607; % L. Cret. Fm.

z4 =0.759; % Jurassic Fm.

z5 =1.566; % Paleozoic Fm.

  z6 = 1.902; % T. Depth Fm.

z = [z1, z2, z3, z4 z5] ;

% measuring depths from datum sea level (L.S.(z=0) )

%===========================================================%

%Borehole thicknesses (km)

%===========================================================%

h1 = z2-z1;

h2 = z3-z2; % Cenomanian Fm.

h3 = z4-z3; % L. Cret. Fm.

h4 = z5-z4; % Jurassic Fm.

h5 = z6-z5; % Paleozoic Fm.

h= [h1, h2, h3, h4 h5] ; % ok

%===========================================================%

%Borehole vertical accumulated thickness

%===========================================================%

h_v1 = h1;

h_v2 = h1+h2;

h_v3 = h1+h2+h3;

h_v4 = h1+h2+h3+h4;

h_v5 = h1+h2+h3+h4+h5;

h_v = [h_v1 h_v2 h_v3 h_v4 h_v5];

% measuring depths from datum surface level (L.S.(z=0.092) )

% the depth to the central bottom z2(j)(km)

% h(i) are the thicknesses of formations (km)

%===========================================================%

%Borehole Densities (gm/cm^3)

%===========================================================%

rho(N-4)= 1.980; % Pleistocene Fm. 1 (gm/cm^3)

rho(N-3)= 2.480; % Cenomanian Fm. 2 (gm/cm^3)

rho(N-2)= 2.610; % L.Cret. Fm. 3 (gm/cm^3)

rho(N-1)= 2.430; % Jurassic Fm. 4 (gm/cm^3)

rho(N) = 2.380; % Paleozoic Fm. 5 (gm/cm^3)

rho_basement = 2.67; % basement rock 6 (gm/cm^3)

%===========================================================%

rho_v1(N-4)= (rho(N-4))/(N-4);

rho_v1(N-3)= (rho(N-4)+ rho(N-3))/(N-3);

rho_v1(N-2)= (rho(N-4)+ rho(N-3)+ rho(N-2))/(N-2);

rho_v1(N-1)= (rho(N-4)+ rho(N-3)+ rho(N-2)+ rho(N-1))/(N-1);

rho_v1(N) = (rho(N-4)+ rho(N-3)+ rho(N-2)+ rho(N-1)+ rho(N))/(N);

rho_v1 = [rho_v1(N-4) rho_v1(N-3) rho_v1(N-2) rho_v1(N-1) rho_v1(N)];%ok

delta_rho1 = rho_v1 - rho_basement; %ok

%===========================================================%

rho_v2(N-4)= (rho(N-4))/(N-4);

rho_v2(N-3)= (rho(N-4)+ rho(N-3))/(N-3);

rho_v2(N-2)= (rho(N-4)+ rho(N-3)+ rho(N-2))/(N-2);

rho_v2(N-1)= (rho(N-4)+ rho(N-3)+ rho(N-2)+ rho(N-1))/(N-1);

rho_v2(N) = (rho(N-4)+ rho(N-3)+ rho(N-2)+ rho(N-1)+ rho(N))/(N);

%=============================================%

rho_v2 = [rho_v2(N-4) rho_v2(N-3) rho_v2(N-2) rho_v2(N-1) rho_v2(N)];%ok

delta_rho2 = rho_v2 - rho_basement; %ok

%===========================================================%

%conditions for calculations

%===========================================================%

%Theoritical_Salb calculations

%===========================================================%

i = zeros();

for i = 1: N;

%===========================================================%

%Model (1) Historical Concept

%===========================================================%

gB_M1(:,i)= 2*pi()*G*delta_rho1(i)*h(i);

% gravity effect of slab_model (1) ok

z_M1(i)= abs(gB_M1(i)/(2*pi()*G*delta_rho1(i)));

% thicknesses of formation (km) ok

h_calM1(:,i) = sum(z_M1(i)); % depth of formation

delt_gB1 = 2*pi()*G*delta_rho1(i); % rate of anomaly change with thickness

%===========================================================%

%Model (2) Bouguer Concept

%===========================================================%

gB_M2(:,i)= 2*pi()*G*delta_rho2(i)*h(i);

% gravity effect of slab_model (2) ok

z_M2(i)= abs(gB_M2(i)/(2*pi()*G*delta_rho2(i)));

% thicknesses of formation (km) ok

h_calM2(:,i) = sum(z_M2(i)); % depth of formation

delt_gB2 = 2*pi()*G*delta_rho2(i); % rate of anomaly change with thickness

%===========================================================%

%Profile Calculations

%===========================================================%

% Model (1)

h_v_cal1= (abs((gB/2*pi()*G*delta_rho1))*10);

% thicknesses of formation (km)

z_cal1(:,i) = (sum(h_v_cal1, 2));

% maximum depth (depth to the basement) (km)

%===========================================================%

% Model (2)

h_v_cal2= (abs((gB/2*pi()*G*delta_rho2))*10);

% thicknesses of formation (km)

z_cal2(:,i) = (sum(h_v_cal2, 2));

% maximum depth (depth to the basement) (km)

%===========================================================%

end

%===========================================================%

%Graphical Representations Model (1)& Model (2)

%===========================================================%

figure(1)

plot(x,gB_M1,'k-')

hold on

grid on

set(gca, 'YDir','reverse')

xlabel('xc -axis of measured Bouguer (km)');

ylabel('gravity anomaly gB_Model(1) in (m.Gals)');

title('Gravity anomaly over horizontal slabs')

%===========================================================%

figure(2)

plot(x,z_M1,'k-')

hold on

grid on

set(gca, 'YDir','reverse')

xlabel('xc -axis of measuered Bouguer (km)');

ylabel(' calculated h_M1 Model (1) depth in (km)');

title('calculated thickness using slab model')

%===========================================================%

figure(3)

plot(x,h_calM1,'k-')

hold on

grid on

set(gca, 'YDir','reverse')

xlabel('xc -axis of measuered Bouguer (km)');

ylabel(' calculated z_calM1 Model (1) thickness in (km)');

title('calculated depth using slab model')

%===========================================================%

figure(4)

plot(x,gB_M2,'k-')

hold on

grid on

set(gca, 'YDir','reverse')

xlabel('xc -axis of measuered Bouguer (km)');

ylabel('gravity anomaly gB_Model(2) in (m.Gals)');

title('Gravity anomaly over horizontal salbs')

%===========================================================%

figure(5)

plot(x,z_M2,'k-')

hold on

grid on

set(gca, 'YDir','reverse')

xlabel('xc -axis of measuered Bouguer (km)');

ylabel(' calculated h_M1 Model (2) depth in (km)');

title('calculated thickness using slab model')

%===========================================================%

figure(6)

plot(x,h_calM2,'k-')

hold on

grid on

set(gca, 'YDir','reverse')

xlabel('xc -axis of measuered Bouguer (km)');

ylabel(' calculated z_calM2 Model (2) thickness in (km)');

title('calculated depth using slab model')

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Written By

Abdel Fattah

Submitted: June 12th, 2021Reviewed: November 11th, 2021Published: May 10th, 2022