University:
California State University, Northridge
Course:
MATH 331 | Mathematical Explorations
Academic year:

2023
Views:

179

Pages:

58
Author:

Taha H.

The Three-Point Problem N E Describing a plane in 3-D space - Graphical - Cramer's Rule (2-D and 3-D) The Three-Point Problem Given the elevation of 3 points on a geologic surface  What is the attitude (strike and dip) of that surface ?  The Three-Point Problem Given the water level in 3 wells  What is the gradient of the potentiometric surface ?  The Three-Point Problem The 3 point problem is also a gateway to useful mathematics !  We will study 2 solutions to this problem using Cramer's Rule  The Three-Point Problem The figure above represents an unconformity surface  We want to find the strike and dip of the unconformity  The Three-Point Problem: Graphical Solution How would you do it ? What are the sequence of elevations ? The elevation at B is between elevations of A and C The Three-Point Problem: Graphical Solution A contour line passing through B must cross the line segment AC By the definition of strike, the direction of this contour is the strike of the unconformity surface The Three-Point Problem: Graphical Solution - Strike 700 ft 300 ft Locate the contour: by dividing segment AC into increments This unconformity drops 1000 ft between A and C Therefore B' is 70% of the distance from A to C We can measure the azimuth of the strike with a protractor The Three-Point Problem: Graphical Solution - Dip Draw a cross-section perpendicular to BB' The Three-Point Problem: Graphical Solution - Dip Draw a cross-section perpendicular to BB' Then use a vertical scale = horizontal scale Plot known elevations Connect-the-dots to draw the line of dip. The Three-Point Problem: Graphical Solution - Dip Because the cross-section is perpendicular to strike The included angle is the true dip. You can measure the dip angle with a protractor (32o) The Three-Point Problem: Graphical Solution - Modified Drawing parallels and perpendiculars with triangles First draw 2 lines which are perpendicular to the strike line (AA' and CC') The Three-Point Problem: Graphical Solution - Modified Drawing parallels and perpendiculars with triangles First draw 2 lines which are perpendicular to the strike line (AA' and CC') Second, draw the right triangle, BDC'. Measure distances BD, DC', AA', and CC'. The Three-Point Problem: Graphical Solution - Modified The azimuth of strike strike is: strike = arctan (DC' / BD) The angle of dip dip is: dip = arctan (hA – hA' / AA') where hA, hA', hC, and hC' are the elevation at each location The Three-Point Problem: Graphical Solutions The limitations of the graphical approach are that errors can be made in measurements What if you had 50 well logs to use ? You may need a bigger desk! The Three-Point Problem: Computational Solutions There are several ways to calculate the strike and dip of a surface (for a 3 point problem) without measuring anything. With these techniques, you can solve 50 or more 3 point problems in the time it takes you to enter the data.....but it will require a little math... The Three-Point Problem: Computational Solutions Let's look at the original problem in a Cartesian reference frame (x,y,z). If you are not given this info in the original format, you can easily convert using sines and cosines. The Three-Point Problem: Computational Solutions What is the target, what is the question ? We are looking for the set of parallel lines which define the plane of interest. The Three-Point Problem: Computational Solutions In 3-D space, the plane may look like this Triangles for the plane can be projected onto each axes The Three-Point Problem: Start with the 2 Point Problem Given 2 points, how to find the slope of a line ? y = yo + mx m = slope = (y1 – y2) / (x1 - x2) The Three-Point Problem: Start with the 2 Point Problem The slope can also be obtained by differentiating the equation d dy = (y + mx) = m o dx dx The Three-Point Problem: Start with the 2 Point Problem You can find the equation for the slope and y intercept also Consider an arbitrary point (x,y) on the line The slope, m is Slope = m = (y – y1) / (x - x1) The Three-Point Problem: Start with the 2 Point Problem The slope between either of 3 points on this line will be the same m = (y – y1) / (x – x1) = (y2 – y1) / (x2 – x1) Solve for y and identify the slope and y intercept: y = [ y1 - (y2 – y1) / (x2 – x1)*x1 ] + (y2 – y1) / (x2 – x1)*x The Three-Point Problem: Start with the 2 Point Problem The slope between either of 3 points on this line will be the same m = (y – y1) / (x – x1) = (y2 – y1) / (x2 – x1) Solve for y and identify the slope and y intercept: y = [ y1 - (y2 – y1) / (x2 – x1)*x1 ] + (y2 – y1) / (x2 – x1)*x The Three-Point Problem: The 2 Point Problem: yet Another Aproach Write an equation for this same line with linear-coefficients ax + by + c = 0 Rearrange and solve for y y = -c/b - (a/b)x + c What is the slope and intercept ? ....the ratio of coefficients The Three-Point Problem: The 2 Point Problem – 3 Point Problem The 3-D analog of the line (from last figure) is shown here ax + by + cz + d = 0 The Three-Point Problem: The 2 Point Problem – 3 Point Problem The three intercepts (x,y,z axes) can be obtained by setting 2 of the 3 variables (x,y,z) to zero. xo = -d/a yo = -d/b zo = -d/c The Three-Point Problem: The 2 Point Problem – 3 Point Problem By setting 1 of the 3 variables (x,y,z) to zero one at a time you can obtain the slope: for the xz-plane, y = 0 ax + cz + d = 0 z = -d/c - (a/c)x dz/dx = -a/c = slope in that plane (partial derivative) The Three-Point Problem: The 2 Point Problem – 3 Point Problem The xy plane is horizontal. The equation where in this plane, z = 0, is a line of strike ax + by + d = 0 strike = arctan (dx/dy) = arctan (-b/a) dip = arctan (dy/dx) = arctan (-a/b) The Three-Point Problem N E Describing a plane in 3-D space - Graphical - Cramer's Rule (2-D and 3-D) 1 The Three-Point Problem   Given the elevation of 3 points on a geologic surface What is the attitude (strike and dip) of that surface ? 2 The Three-Point Problem   Given the water level in 3 wells What is the gradient of the potentiometric surface ? 3 The Three-Point Problem   The 3 point problem is also a gateway to useful mathematics ! We will study 2 solutions to this problem using Cramer's Rule 4 The Three-Point Problem   The figure above represents an unconformity surface We want to find the strike and dip of the unconformity 5 The Three-Point Problem: Graphical Solution How would you do it ? What are the sequence of elevations ? The elevation at B is between elevations of A and C 6 The Three-Point Problem: Graphical Solution A contour line passing through B must cross the line segment AC By the definition of strike, the direction of this contour is the strike of the unconformity surface 7 The Three-Point Problem: Graphical Solution - Strike 700 ft 300 ft Locate the contour: by dividing segment AC into increments This unconformity drops 1000 ft between A and C Therefore B' is 70% of the distance from A to C We can measure the azimuth of the strike with a protractor 8 The Three-Point Problem: Graphical Solution - Dip Draw a cross-section perpendicular to BB' 9 The Three-Point Problem: Graphical Solution - Dip Draw a cross-section perpendicular to BB' Then use a vertical scale = horizontal scale Plot known elevations Connect-the-dots to draw the line of dip. 10 The Three-Point Problem: Graphical Solution - Dip Because the cross-section is perpendicular to strike The included angle is the true dip. You can measure the dip angle with a protractor (32o) 11 The Three-Point Problem: Graphical Solution - Modified Drawing parallels and perpendiculars with triangles First draw 2 lines which are perpendicular to the strike line (AA' and CC') 12 The Three-Point Problem: Graphical Solution - Modified Drawing parallels and perpendiculars with triangles First draw 2 lines which are perpendicular to the strike line (AA' and CC') Second, draw the right triangle, BDC'. Measure distances BD, DC', AA', and CC'. 13 The Three-Point Problem: Graphical Solution - Modified The azimuth of strike strike is: strike = arctan (DC' / BD) The angle of dip dip is: dip = arctan (hA – hA' / AA') where hA, hA', hC, and hC' are the elevation at each location 14 The Three-Point Problem: Graphical Solutions The limitations of the graphical approach are that errors can be made in measurements What if you had 50 well logs to use ? You may need a bigger desk! 15 The Three-Point Problem: Computational Solutions There are several ways to calculate the strike and dip of a surface (for a 3 point problem) without measuring anything. With these techniques, you can solve 50 or more 3 point problems in the time it takes you to enter the data.....but it will require a little math... 16 The Three-Point Problem: Computational Solutions Let's look at the original problem in a Cartesian reference frame (x,y,z). If you are not given this info in the original format, you can easily convert using sines and cosines. 17 The Three-Point Problem: Computational Solutions What is the target, what is the question ? We are looking for the set of parallel lines which define the plane of interest. 18 The Three-Point Problem: Computational Solutions In 3-D space, the plane may look like this Triangles for the plane can be projected onto each axes 19 The Three-Point Problem: Start with the 2 Point Problem Given 2 points, how to find the slope of a line ? y = yo + mx m = slope = (y1 – y2) / (x1 - x2) 20 The Three-Point Problem: Start with the 2 Point Problem The slope can also be obtained by differentiating the equation d dy = dx (yo + mx) = m dx 21 The Three-Point Problem: Start with the 2 Point Problem You can find the equation for the slope and y intercept also Consider an arbitrary point (x,y) on the line The slope, m is Slope = m = (y – y1) / (x - x1) 22 The Three-Point Problem: Start with the 2 Point Problem The slope between either of 3 points on this line will be the same m = (y – y1) / (x – x1) = (y2 – y1) / (x2 – x1) Solve for y and identify the slope and y intercept: y = [ y1 - (y2 – y1) / (x2 – x1)*x1 ] + (y2 – y1) / (x2 – x1)*x 23 The Three-Point Problem: Start with the 2 Point Problem The slope between either of 3 points on this line will be the same m = (y – y1) / (x – x1) = (y2 – y1) / (x2 – x1) Solve for y and identify the slope and y intercept: y = [ y1 - (y2 – y1) / (x2 – x1)*x1 ] + (y2 – y1) / (x2 – x1)*x 24 The Three-Point Problem: The 2 Point Problem: yet Another Aproach Write an equation for this same line with linear-coefficients ax + by + c = 0 Rearrange and solve for y y = -c/b - (a/b)x + c What is the slope and intercept ? ....the ratio of coefficients 25 The Three-Point Problem: The 2 Point Problem – 3 Point Problem The 3-D analog of the line (from last figure) is shown here ax + by + cz + d = 0 26 The Three-Point Problem: The 2 Point Problem – 3 Point Problem The three intercepts (x,y,z axes) can be obtained by setting 2 of the 3 variables (x,y,z) to zero. xo = -d/a yo = -d/b zo = -d/c 27 The Three-Point Problem: The 2 Point Problem – 3 Point Problem By setting 1 of the 3 variables (x,y,z) to zero one at a time you can obtain the slope: for the xz-plane, y = 0 ax + cz + d = 0 z = -d/c - (a/c)x dz/dx = -a/c = slope in that plane 28 (partial derivative) The Three-Point Problem: The 2 Point Problem – 3 Point Problem The xy plane is horizontal. The equation where in this plane, z = 0, is a line of strike ax + by + d = 0 strike = arctan (dx/dy) = arctan (-b/a) dip = arctan (dy/dx) = arctan (-a/b) 29