Table of ContentsList of Tables1List of Figures1Chapter 1 Introduction1Chapter 2 Literature Review1Chapter 3 Methodology1Chapter 4 Results and Discussion1References1Appendices1

List of Tables

List of FiguresFigureTitlePage3.1Input file for horizontal slab case3.2The notations used in horizontal slab case3.3Input file for dipping slab case3.4The diagram used to calculate X and Z values of each of the tablets

Chapter 1 Introduction1.1 Problem StatementMost of the valuable minerals and rocks are discovered from beneath the ground and it is located in a very deep part where people cannot see with their bare eyes. In this case, the help of the gravimeter comes in handy to search for those valuable materials. However, to search for such valuable materials in one area, a numerous numbers of computational points are needed and it is not only difficult but also time consuming to calculate for the gravity effect manually by using all these data. Therefore, this paper proposed to develop Fortran 90 code to calculate the gravity effect of each computational points to reduce the amount of effort and time that has to be used. 1.2 ObjectiveThe objectives of this paper is to develop Fortran 90 codes to calculate the gravity effect of the horizontal slap and dipping slab from a given number of computational points by using the data provided.1.3 LimitationThe bottom part of the slab is assumed to be flat in both of the cases. The slabs are assumed to have only one layer in both of the codes.

Chapter 2 Literature Review2.1 GravityDifferent materials have different density and because of which, they have different gravity. Gravity anomaly is the difference between gravity of the observed objects. There are two types of measurements of gravity: determination of the absolute magnitude of gravity at any place and measuring the change in gravity from one place to another. It is very difficult to design an instrument to measure the absolute value of gravity that has accuracy up to 0.01 mgal and that is also portable enough to be used easily in different places. Therefore, gravity surveying is carried out with a portable instrument called gravimeter, which determine the gravities relative to one or more reference locations.The regional effects has to be removed before the interpretation of the residual gravity anomaly is carried out by using anomalous density distribution. In this paper, formula determining the gravity anomalies for uniform sphere is used, which can be seen in figure 2.1 and the formula is shown in equation 2.1 (Lowrie, 2007, p.90). (2.1)

Figure 2.1: Gravity anomalies distribution for two buried sphere with same properties but in different depth (Lowrie, 2007, p.90)Since the observed slab is square shape, the formula is changed to equation 2.2 and the derivation can be seen in Appendix D.(2.2)In equation 2.2, G is gravitational constant, W is the width of the slab or tablet, L is the length of the slab or tablet, T is the thickness, is the density contrast, Z is the depth of the centroid of the slab and R is the radial distance from the computational point to the centroid of the slab.2.2 Heat FlowThere are two heat sources in the earth, external heat source and internal heat source. The external heat source is from the sun and it can only penetrate through some shallow part of the earth surface. The internal heat source is the heat from the inner core. The interior of the earth is losing heat via geothermal flux at a rate of about 4.41013 W, which amounts to 1.41021 Jyr-1 (Lowrie, 2007, p.227)In this paper, cooling of the heat oceanic lithosphere concept is used to attain the heat flow formula by using error function. The formula that is used as a reference is shown in equation 2.3. (2.3)For the error function, the one with the Taylor expansion is used, which is shown in figure 2.4.(2.3)

Chapter 3 MethodologyFortran 90 was used to calculate the gravity effect for both horizontal slab and dipping slab because the slab was divided into several tablets. The reason to do that was to achieve the more accurate data. 3.1 Gravity effect for Horizontal SlabThe slab was assumed to be horizontal (d = 0) and divided into small tablets both in X and Y directions. The thickness of all the tablets were assumed the same in this horizontal case, therefore the depths of the centroid of the tablets were the same. The thickness of the slab, density contrast, and X0, Y0 and Z0 of the origin of the slab were given. The origin point was assumed to be the bottom left hand corner of the slab. The coordinates of the computational points (station points) were also given and energy effects for each tablets was calculated using equation 2.2. Since, X0, Y0, Z0 of the origin of the slab, thickness of the slab, density contrast, and the coordinates of the computational points were given, input file was built to read the values for those data. Number of tablets for both X and Y direction and number of computational points were also included in the input file. The input file can be seen in figure 3.1. The values in the first row are X0, Y0, Z0 and density contrast of the slab respectively. Those in the second row are local dip of the slab, number of computational points, length and width of the slab respectively. The ones in the third rows are number of tablets in X and Y directions and the thickness of the slab. The rest are the X and Y coordinates of the computational points and there are eleven coordinate points for X and Y each in this case.

Figure 3.1: Input file for horizontal slab caseBy using those input data, X and Y coordinates for each of the tablets were calculated. In this example, there were a hundred tablets since there were ten tablets each on both X and Y direction. RX and RY are the distances from the measuring stations to the centroid of each tablets. RX were the absolute values of the subtraction of X coordinates of the tablets from X coordinates of the computational points and the same theory was applied to RY also. RH are the horizontal distances from computational points to the centroid of the tablets and they were calculated from respective RX and RY by using Pythagoras theorem. The calculated RH and the given depth of the tablets were then used to calculate the radial distance R from the computational points to the centroid of the tablets. All those calculated data were then used in equation 2.2 to get the gravity effect of each tablets. The value used for each station to plot the gravity anomalies graph was the summation of the gravity effect values of all the tablets. The notation used in this horizontal slab case were illustrated in figure 3.2 and the detail coding of this horizontal case can be seen in Appendix A.

Figure 3.2: The notations used in horizontal slab case3.2 Gravity effect for Dipping SlabMost of the assumptions and the given values were similar to that of the previous case except the dipping part. In this case, the slab is dipping in a specific angle, called local dip of the slab (d). Because the slab is not horizontal, the depths of each tablet were different and the thickness was also assumed to be different depending on their location on the slab. The input file for this dipping case is also similar to that of horizontal case except for local dip value and the thickness, where a hundred of thickness values had to input in the example for this case. The input file for the dipping can be seen in figure 3.3. The calculation procedures were the same but it was tricky to calculate the X coordinate and depth of each tablets. The aid of a diagram was needed to calculate those X and Z values and the diagram, which was used for this case is shown in figure 3.4. Three local dip values, 10, 20 and 30 degrees were used to compare and zero degree for local dip was also used to compare it with the result from horizontal slab code.(a) (b)Thickness of the tablets

Figure 3.3: Input file for the dipping slab case (a) First part of the input data (b) Input data for thicknessThe values of X and Z were different base on the thickness of each tablets. The procedures for the rest is similar to that of horizontal case. The detail coding of this dipping case can be seen in Appendix B.

Figure 3.4: The diagram used to calculate X and Z values of each of the tablets

Chapter 4 Results and Discussion4.1 Horizontal SlabAn output file was developed when the program was run by using the input file shown above. The first part of the output data for horizontal slap is shown in figure 4.1. The first eight lines in figure 4.1 are the data that are provided to calculate the gravity effect. The rest are the calculated X and Y coordinates of the tablets.

Figure 4.1: Output data developed from Fortran 90 code for horizontal slabThe summation of the gravity effect of all the tablets for each computational point is shown in table 4.1 and the gravity anomalies graph developed by using those gravity effect data is shown in figure 4.2.Computational PointsGravity Effect (delta g)

11.73E-01

23.70E-01

31.06E+00

45.97E+00

51.58E+01

61.68E+01

71.58E+01

85.97E+00

91.06E+00

103.70E-01

111.73E-01

Table 4.1: Table 4.2: Gravity anomalies for each of the computational point for horizontal slab

Figure 4.2: Gravity anomalies for horizontal slab When this gravity anomalies graph in figure 4.2 is compared with the one from figure 2.XXXX, it can be seen that the curves are similar. Therefore, this Fortran 90 code for the horizontal slab is acceptable. Further research can be done with .4.2 Dipping SlabThe first seven line of the output file of the dipping slab is similar to that of horizontal slab. Even though Y-coordinate values of the tablets are similar to that in horizontal case, there are several values for the X-coordinate for dipping case since it changes base on the thickness of each tablets. The first part of the output file can be seen in figure 4.3.

Figure 4.3: Output data for dipping slab The gravity anomalies values for the 0, 10, 20 and 30 degrees dipping slab are shown in table 4.2 and the graph for those are shown in figure 4.4.

Computational PointsGravity Effect (delta g) 0 degreeGravity Effect (delta g) 10 degreeGravity Effect (delta g) 20 degreeGravity Effect (delta g) 30 degree

12.67E-014.12E-015.61E-017.17E-01

25.75E-018.46E-011.12E+001.41E+00

31.67E+002.27E+002.86E+003.45E+00

41.04E+011.16E+011.26E+011.36E+01

52.96E+012.73E+012.49E+012.26E+01

63.10E+012.49E+012.00E+011.62E+01

72.96E+011.93E+011.36E+011.01E+01

81.04E+019.13E+006.96E+005.36E+00

91.67E+002.89E+002.96E+002.63E+00

105.75E-011.12E+001.34E+001.34E+00

112.67E-015.32E-016.90E-017.43E-01

Table 4.2: Gravity anomalies for each of the computational point dipping slab

Figure 4.4: Gravity anomalies for dipping slab As can be seen in figure 4.4, the maximum gravity anomalies of the slab is smaller when the local dip of the slab is larger. The gravity anomaly line for the zero degree in figure 4.4 is similar to that in the figure 4.2, therefore the code for the dipping case is acceptable.

References

Lowrie, W. (2007).Fundamentals of geophysics. Cambridge University Press.

AppendicesAppendix A

program GP_SU ! Heading: Gravity Effect for horizontal slab ! Student Name: SU PYAE SONE SOE ! Student ID: st20000150 ! Email: sharonsoe.19@gmail.com ! Program Discription: This program calculate the gravity anomalies for each station or computational points by using data in the input file. implicit none !!! !Variable declaration integer :: NX, NY, m !no: of grid in X and Y direction, no: of computational points and no: of layers in Z direction character(len=80):: InputFile !Input File character(len=80):: OutputFile !Output File doubleprecision:: X, Y, Z, delD, G = 6.674D-11 !X and Y coordinate of the origin and Depth of the COGs, density contrast and gravitational constant(m3/kgs2) doubleprecision:: L, W, d, TABTHK, LS, WS, ZS !length and width of the tablet, local dip and thickness of the subducting slab doubleprecision, allocatable:: R(:,:), RH(:,:),RX(:), RY(:) !Radial and horizontal distance from the computational point to COG and horizontal distance in X and Y coordinate doubleprecision, allocatable:: GE(:,:), GES(:) !Gravity effect doubleprecision, allocatable:: XP(:), YP(:), XS(:), YS(:) !X and Y coordinate of the computational points and the slab integer:: i, j, k !temporary parameter !Open Files for Input and Output write(*,*) 'Please enter input filename (Max = 80 character):' read(*,*) InputFile write(*,*) 'Please enter output filename (Max = 80 character):' read(*,*) OutputFile !!!!!!!!!!!! open(5,file=InputFile) open(6,file=OutputFile) !!!!!!!!!!!! !Read & write the input data read(5,*)X , Y, Z, delD read(5,*)d, m, WS, LS read(5,*)NX, NY, TABTHK write(6,'(A,F20.3)') 'Density Contrast:',delD write(6,'(A,F20.3)') 'X-Coordinate of the origin:',X write(6,'(A,F20.3)') 'Y-Coordinate of the origin:',Y write(6,'(A,F20.3)') 'Thickness of the crust:', TABTHK write(6,'(A,F20.3)') 'Local dip on the slab:',d write(6,'(A,F20.3)') 'Width of the slab:',WS write(6,'(A,F20.3)') 'Length of the slab:',LS write(6,'(A,F20.3)') 'Depth of the slab:',Z allocate(XP(m), YP(m), XS(NX), YS(NY)) do j= 1, m read(5,*)XP(j), YP(j) end do L = LS/NX W = WS/NY ZS = Z-(TABTHK/2) XS(1) = X+(L/2) write(6,*) 'X-Coordinate of the tablets:' write(6,'(F20.3)') XS(1) do j= 2, NX XS(j) = XS(j-1)+L write(6,'(F20.3)') XS(j) end do YS(1) = Y+(W/2) write(6,*) 'Y-Coordinate of the tablets:' write(6,'(F20.3)') YS(1) do j= 2, NY YS(j) = YS(j-1)+W write(6,'(F20.3)') YS(j) end do write(6,'(A,F20.3)') 'Depth of the cells to their centroid:', ZS allocate(RX(NX), RY(NY), RH(NX,NY), R(NX,NY), GE(NX,NY), GES(m)) do i = 1, m do j= 1, NX RX(j) = abs(XS(j)-XP(i)) end do do j= 1, NY RY(j) = abs(YS(j)-YP(i)) end do do j= 1, NX do k=1, NY RH(j,k)= sqrt((RX(j)*RX(j))+(RY(k)*RY(k))) R(j,k)= sqrt((RH(j,k)*RH(j,k))+(ZS*ZS)) GE(j,k) = (G*W*L*delD*TABTHK*ZS/(R(j,k)*R(j,k)*R(j,k)))*1.0E5 GES(i)=GES(i)+GE(j,k) end do end do write(6,*) 'Gravity Effect from station', i do j= 1, NX write(6,'(10E20.5)') GE(j,1), GE(j,2), GE(j,3), GE(j,4), GE(j,5), GE(j,6), GE(j,7), GE(j,8), GE(j,9), GE(j,10) end do end do write(6,*) 'Gravity effect from stations' do i = 1, m write(6,'(E20.5)') GES(i) end doend program

Appendix B

program GP_SU ! Heading: Gravity Effect for dipping slab ! Student Name: SU PYAE SONE SOE ! Student ID: st20000150 ! Email: sharonsoe.19@gmail.com ! Program Discription: This program calculate the gravity anomalies for each station or computational ! points by using data in the input file. ! ! implicit none !!! !Variable declaration integer :: m, NX, NY !No: of stations and no: of grid in the slab character(len=80) :: InputFile !Input File character(len=80) :: OutputFile !Output File doubleprecision:: X0, Y0, Z0, delD, L, W, Lc, Wc, d !X and Y coordinate of the origin, density contrast and length and width of the slabs and cells and local dip doubleprecision:: G = 6.674D-11, pi = 3.141592654 !Gravitational Constant (m3/kgs2) and PI value doubleprecision, allocatable:: R(:,:), RH(:,:), RX(:,:), RY(:) !Radial and horizontal distance from the computational point to COG and horizontal distance in X and Y coordinate and depth of bottom of COG doubleprecision, allocatable:: GE(:,:), TABTHK(:,:),GES(:) !Gravity effect and thickness of the slab and Summation of all gravity effect doubleprecision, allocatable:: XP(:), YP(:), XS(:,:), YS(:), ZS(:,:) !X and Y coordinate of the stations and cells, depth of the slabs doubleprecision, allocatable:: X1(:,:), Z1(:,:) !temporary parameters doubleprecision:: i, j, k, drad !temporary parameters and local dip in radian !Open Files for Input and Output write(*,*) 'Please enter input filename (Max = 80 character):' read(*,*) InputFile write(*,*) 'Please enter output filename (Max = 80 character):' read(*,*) OutputFile !!!!!!!!!!!! open(5,file=InputFile) open(6,file=OutputFile) !!!!!!!!!!!! !Read & write the input data read(5,*)X0, Y0, Z0, delD read(5,*)d, m, W, L read(5,*)NX, NY write(6,'(A,F20.3)') 'Density Contrast:',delD write(6,'(A,F20.3)') 'X-Coordinate of the origin:',X0 write(6,'(A,F20.3)') 'Y-Coordinate of the origin:',Y0 write(6,'(A,F20.3)') 'Depth of the origin:',Z0 write(6,'(A,F20.3)') 'Width of the slab:',W write(6,'(A,F20.3)') 'Length of the slab:',L write(6,'(A,F20.3)') 'Local dip on the slab:',d allocate(XP(m), YP(m), X1(NX,NY), Z1(NX,NY), TABTHK(NX,NY)) allocate(XS(NX,NY), YS(NY), ZS(NX,NY)) do i= 1, m read(5,*)XP(i), YP(i) end do do i=1, NX do j=1, NY read(5,*)TABTHK(i,j) end do end do

Wc = W/NY Lc = L/NX drad = d*(pi/180) do i= 1, NY X1(1,i) = ((Lc/2)+(TABTHK(1,i)*tan(drad)/2))*cos(drad) Z1(1,i) = Z0 +((Lc/2)+(TABTHK(1,i)*tan(drad)/2))*sin(drad) do j= 2, NX X1(j,i) = (((j-1)*Lc)+(Lc/2)+(TABTHK(j,i)*tan(drad)/2))*cos(drad) Z1(j,i) = Z0 +(((j-1)*Lc)+(Lc/2)+(TABTHK(j,i)*tan(drad)/2))*sin(drad) end do end do write(6,*) 'X-Coordinate of the cells:' do j= 1, NX do i= 1, NY XS(j,i) = X0+X1(j,i) end do write(6,'(10F20.3)') XS(j,1),XS(j,2),XS(j,3),XS(j,4),XS(j,5),XS(j,6),XS(j,7),XS(j,8),XS(j,9),XS(j,10) end do YS(1) = Y0+(Wc/2) write(6,*) 'Y-Coordinate of the cells:' write(6,'(F20.3)') YS(1) do j= 2, NY YS(j) = YS(j-1)+Wc write(6,'(F20.3)') YS(j) end do

write(6,*) 'Depth of the cells to their centroid:' do j= 1, NX do i= 1, NY ZS(j,i) = Z1(j,i)-(TABTHK(j,i)/(2*cos(drad))) end do write(6,'(10F20.3)') ZS(j,1), ZS(j,2), ZS(j,3), ZS(j,4), ZS(j,5), ZS(j,6), ZS(j,7), ZS(j,8), ZS(j,9), ZS(j,10) end do allocate(RX(NX,NY), RY(NY), RH(NX,NY), R(NX,NY), GE(NX,NY), GES(m)) do i = 1, m do j= 1, NX do k = 1,NY RX(j,k) = abs(XS(j,k)-XP(i)) end do end do do j= 1, NY RY(j) = abs(YS(j)-YP(i)) end do do j= 1, NX do k=1, NY RH(j,k)= sqrt((RX(j,k)*RX(j,k))+(RY(k)*RY(k))) R(j,k)= sqrt((RH(j,k)*RH(j,k))+(ZS(j,k)*ZS(j,k))) GE(j,k) = (G*Wc*Lc*delD*TABTHK(j,k)*ZS(j,k)/(R(j,k)*R(j,k)*R(j,k)))*1.0E5 GES(i)=GES(i)+GE(j,k) end do end do

write(6,*) 'Gravity Effect from station', i do j= 1, NX write(6,'(10E20.5)') GE(j,1), GE(j,2), GE(j,3), GE(j,4), GE(j,5), GE(j,6), GE(j,7), GE(j,8), GE(j,9), GE(j,10) end do end do write(6,*) 'Summation of Gravity effects from each station' do i = 1, m write(6,'(E20.5)') GES(i) end doend program