/
gempy_simple.py
462 lines (314 loc) · 14.4 KB
/
gempy_simple.py
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#!/usr/bin/env python
# coding: utf-8
# In[1]:
import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import RegularGridInterpolator
def interp_with_drift(point_1_y = 2,
point_2_y = 3,
G_1_x=1):
## defining the dips position
G_1 = np.array([[0., 1.]])
G_1_x = G_1_x
G_1_y = 1
G_1_tiled = np.tile(G_1, [2, 1])
# In[3]:
def squared_euclidean_distance(x_1, x_2):
try:
sqd = np.sqrt(np.reshape(np.sum(x_1 ** 2, 1), newshape=(x_1.shape[0], 1)) + \
np.reshape(np.sum(x_2 ** 2, 1), newshape=(1, x_2.shape[0])) - \
2 * (x_1 @ x_2.T))
except RuntimeWarning:
sqd = 0.
return sqd
# In[5]:
def cartesian_dist(x_1, x_2):
return np.concatenate([
np.tile(x_1[:, 0] - np.reshape(x_2[:, 0], [x_2.shape[0], 1]), [1, 2]),
np.tile(x_1[:, 1] - np.reshape(x_2[:, 1], [x_2.shape[0], 1]), [1, 2])], axis=0)
h_u = cartesian_dist(G_1, G_1)
h_v = h_u.T
# In[5]:
a = np.concatenate([np.ones([G_1.shape[0], G_1.shape[0]]), np.zeros([G_1.shape[0], G_1.shape[0]])], axis=1)
b = np.concatenate([np.zeros([G_1.shape[0], G_1.shape[0]]), np.ones([G_1.shape[0], G_1.shape[0]])], axis=1)
perpendicularity_matrix = np.concatenate([a, b], axis=0)
# In[7]:
dist_tiled = squared_euclidean_distance(G_1_tiled, G_1_tiled)
# In[8]:
a_T = 5
c_o_T = a_T ** 2 / 14 / 3
# In[9]:
def cov_gradients(dist_tiled):
condition1 = 0
a = (h_u*h_v)
b = dist_tiled**2
t1 = np.divide(a, b, out=np.zeros_like(a), where=b!=0)
t2 = -c_o_T*((-14/a_T**2)+
105/4*dist_tiled/a_T**3 -
35/2 * dist_tiled**3 / a_T **5 +
21 /4 * dist_tiled**5/a_T**7)+\
c_o_T * 7 * (9 * dist_tiled ** 5 -
20 * a_T ** 2 * dist_tiled ** 3 +
15 * a_T ** 4 * dist_tiled -
4 * a_T ** 5) / (2 * a_T ** 7)
t3 = perpendicularity_matrix * \
c_o_T * ((-14 / a_T ** 2) + 105 / 4 * dist_tiled / a_T ** 3 -
35 / 2 * dist_tiled ** 3 / a_T ** 5 +
21 / 4 * dist_tiled ** 5 / a_T ** 7)
t4 = 1/3*np.eye(dist_tiled.shape[0])
condition2 = t1 * t2 - t3 + t4
C_G = np.where(dist_tiled==0, condition1, condition2) ## adding nugget effect
return C_G
# In[10]:
dist_tiled = dist_tiled + np.eye(dist_tiled.shape[0])
C_G = cov_gradients(dist_tiled)
C_G
# ### covariance of interface points
# In[11]:
# layer1 = np.array([[0, 0], [2, 0]])
# layer2 = np.array([[0, 2], [2, 2]])
layer1 = np.array([[0, 0], [2, 0], [3, 0], [4, 0]])
layer2 = np.array([[0, 2], [2, point_1_y], [3, point_2_y]])
number_of_layer = 2
# In[11]:
number_of_points_per_surface = np.array([layer1.shape[0], layer2.shape[0]])
# In[13]:
def set_rest_ref_matrix(number_of_points_per_surface):
ref_layer_points = np.repeat(np.stack([layer1[-1], layer2[-1]], axis=0),
repeats=number_of_points_per_surface - 1, axis=0)
rest_layer_points = np.concatenate([layer1[0:-1], layer2[0:-1]], axis=0)
return ref_layer_points, rest_layer_points
# In[14]:
ref_layer_points, rest_layer_points = set_rest_ref_matrix(number_of_points_per_surface)
# In[15]:
sed_rest_rest = squared_euclidean_distance(rest_layer_points, rest_layer_points)
sed_ref_rest = squared_euclidean_distance(ref_layer_points, rest_layer_points)
sed_rest_ref = squared_euclidean_distance(rest_layer_points, ref_layer_points)
sed_ref_ref = squared_euclidean_distance(ref_layer_points, ref_layer_points)
# In[16]:
def cov_interface(ref_layer_points, rest_layer_points):
sed_rest_rest = squared_euclidean_distance(rest_layer_points, rest_layer_points)
sed_ref_rest = squared_euclidean_distance(ref_layer_points, rest_layer_points)
sed_rest_ref = squared_euclidean_distance(rest_layer_points, ref_layer_points)
sed_ref_ref = squared_euclidean_distance(ref_layer_points, ref_layer_points)
C_I = c_o_T * ((1 - 7 * (sed_rest_rest / a_T) ** 2 + 35 / 4 * (sed_rest_rest / a_T) ** 3 - 7 / 2 * (
sed_rest_rest / a_T) ** 5 + 3 / 4 * (sed_rest_rest / a_T) ** 7) - (
1 - 7 * (sed_ref_rest / a_T) ** 2 + 35 / 4 * (sed_ref_rest / a_T) ** 3 - 7 / 2 * (
sed_ref_rest / a_T) ** 5 + 3 / 4 * (sed_ref_rest / a_T) ** 7) - (
1 - 7 * (sed_rest_ref / a_T) ** 2 + 35 / 4 * (sed_rest_ref / a_T) ** 3 - 7 / 2 * (
sed_rest_ref / a_T) ** 5 + 3 / 4 * (sed_rest_ref / a_T) ** 7) + (
1 - 7 * (sed_ref_ref / a_T) ** 2 + 35 / 4 * (sed_ref_ref / a_T) ** 3 - 7 / 2 * (
sed_ref_ref / a_T) ** 5 + 3 / 4 * (sed_ref_ref / a_T) ** 7))
return C_I
# In[17]:
C_I = cov_interface(ref_layer_points, rest_layer_points)
# In[18]:
C_I
# ### Covariance of interface points and gradients
# In[19]:
sed_dips_rest = squared_euclidean_distance(G_1_tiled, rest_layer_points)
sed_dips_ref = squared_euclidean_distance(G_1_tiled, ref_layer_points)
# In[20]:
## Cartesian distance between dips and interface points
def cartesian_dist_no_tile(x_1, x_2):
return np.concatenate([
np.transpose((x_1[:, 0] - np.reshape(x_2[:, 0], [x_2.shape[0], 1]))),
np.transpose((x_1[:, 1] - np.reshape(x_2[:, 1], [x_2.shape[0], 1])))], axis=0)
hu_rest = cartesian_dist_no_tile(G_1, rest_layer_points)
hu_ref = cartesian_dist_no_tile(G_1, ref_layer_points)
# In[21]:
def cov_interface_gradients(hu_rest, hu_ref):
C_GI = (hu_rest * (- c_o_T * ((-14 / a_T ** 2) + 105 / 4 * sed_dips_rest / a_T ** 3 -
35 / 2 * sed_dips_rest ** 3 / a_T ** 5 +
21 / 4 * sed_dips_rest ** 5 / a_T ** 7)) - \
hu_ref * (-c_o_T * ((-14 / a_T ** 2) + 105 / 4 * sed_dips_ref / a_T ** 3 -
35 / 2 * sed_dips_ref ** 3 / a_T ** 5 +
21 / 4 * sed_dips_ref ** 5 / a_T ** 7)))
return C_GI
# In[22]:
C_GI = cov_interface_gradients(hu_rest, hu_ref)
C_IG = C_GI.T
# ### Kriging Matrix
# In[23]:
K = np.concatenate([np.concatenate([C_G, C_GI], axis=1),
np.concatenate([C_IG, C_I], axis=1)], axis=0)
# In[26]:
import pandas as pd
pd.DataFrame(K)
# In[27]:
b = np.concatenate([[G_1_x, G_1_y], np.zeros(K.shape[0] - G_1.shape[0] * 2)], axis=0)
b = np.reshape(b, newshape=[b.shape[0], 1])
b
# In[28]:
w = np.linalg.solve(K, b)
w
# ### Interpolation without Universal term
# In[29]:
## create grid
xx = np.arange(-.5, 4.5, 0.1)
yy = np.arange(-.5, 4.5, 0.1)
XX, YY = np.meshgrid(xx, yy)
X = (np.reshape(XX, [-1])).T
Y = (np.reshape(YY, [-1])).T
grid = np.stack([X, Y], axis=1)
# In[30]:
hu_Simpoints = cartesian_dist_no_tile(G_1, grid)
# In[31]:
sed_dips_SimPoint = squared_euclidean_distance(G_1_tiled, grid)
# In[32]:
# gradient contribution
sigma_0_grad = w[:G_1.shape[0] * 2] * (
-hu_Simpoints * (- c_o_T * ((-14 / a_T ** 2) + 105 / 4 * sed_dips_SimPoint / a_T ** 3 -
35 / 2 * sed_dips_SimPoint ** 3 / a_T ** 5 +
21 / 4 * sed_dips_SimPoint ** 5 / a_T ** 7)))
sigma_0_grad = np.sum(sigma_0_grad, axis=0)
# In[33]:
sed_rest_SimPoint = squared_euclidean_distance(rest_layer_points, grid)
sed_ref_SimPoint = squared_euclidean_distance(ref_layer_points, grid)
# In[34]:
# surface point contribution
sigma_0_interf = -w[G_1.shape[0] * 2:] * (c_o_T * ((1 - 7 * (sed_rest_SimPoint / a_T) ** 2 +
35 / 4 * (sed_rest_SimPoint / a_T) ** 3 -
7 / 2 * (sed_rest_SimPoint / a_T) ** 5 +
3 / 4 * (sed_rest_SimPoint / a_T) ** 7) -
(1 - 7 * (sed_ref_SimPoint / a_T) ** 2 +
35 / 4 * (sed_ref_SimPoint / a_T) ** 3 -
7 / 2 * (sed_ref_SimPoint / a_T) ** 5 +
3 / 4 * (sed_ref_SimPoint / a_T) ** 7)))
sigma_0_interf = np.sum(sigma_0_interf, axis=0)
# In[35]:
interpolate_result = sigma_0_grad + sigma_0_interf
intp = np.reshape(interpolate_result, [50, 50])
# In[36]:
#
# plt.contour(XX,YY,intp)
# plt.colorbar()
# plt.plot(layer1[:,0], layer1[:,1], 'ro')
# plt.plot(layer2[:,0], layer2[:,1], 'bo')
#
# plt.plot(G_1[0,0], G_1[0,1], 'go')
# ## add drift term
# In[37]:
U = np.concatenate([np.eye(number_of_layer),
np.stack([ref_layer_points[:, 0] - rest_layer_points[:, 0],
ref_layer_points[:, 1] - rest_layer_points[:, 1]])], axis=1)
U_T = U.T
# In[38]:
### new K matrix
zero_matrix = np.zeros([2, 2])
K_U = np.concatenate([np.concatenate([K, U_T], axis=1), np.concatenate([U, zero_matrix], axis=1)], axis=0)
# In[39]:
pd.DataFrame(K_U)
# In[40]:
### universal contribution
sigma_0_1st_drift = np.sum(grid * (w[-2:]).T, axis=1)
sigma_0_1st_drift
# In[41]:
# rhs b-vector
b = np.concatenate([[G_1_x, G_1_y], np.zeros(K_U.shape[0] - G_1.shape[0] * 2)], axis=0)
b = np.reshape(b, newshape=[b.shape[0], 1])
# In[42]:
w = np.linalg.solve(K_U, b)
w
# In[43]:
# gradient contribution
sigma_0_grad = w[:G_1.shape[0] * 2] * (
-hu_Simpoints * (- c_o_T * ((-14 / a_T ** 2) + 105 / 4 * sed_dips_SimPoint / a_T ** 3 -
35 / 2 * sed_dips_SimPoint ** 3 / a_T ** 5 +
21 / 4 * sed_dips_SimPoint ** 5 / a_T ** 7)))
sigma_0_grad = np.sum(sigma_0_grad, axis=0)
# surface point contribution
sigma_0_interf = -w[G_1.shape[0] * 2:-2] * (c_o_T * ((1 - 7 * (sed_rest_SimPoint / a_T) ** 2 +
35 / 4 * (sed_rest_SimPoint / a_T) ** 3 -
7 / 2 * (sed_rest_SimPoint / a_T) ** 5 +
3 / 4 * (sed_rest_SimPoint / a_T) ** 7) -
(1 - 7 * (sed_ref_SimPoint / a_T) ** 2 +
35 / 4 * (sed_ref_SimPoint / a_T) ** 3 -
7 / 2 * (sed_ref_SimPoint / a_T) ** 5 +
3 / 4 * (sed_ref_SimPoint / a_T) ** 7)))
sigma_0_interf = np.sum(sigma_0_interf, axis=0)
# In[44]:
sigma_0_1st_drift = np.sum(grid * (w[-2:]).T, axis=1)
sigma_0_1st_drift
# In[45]:
interpolate_result = sigma_0_grad + sigma_0_interf + sigma_0_1st_drift
intp = np.reshape(interpolate_result, [50, 50]) # reshape the result to matrix shape
return intp
def plot_interp_with_drift(point_1_y = 2.,
point_2_y = 3.,
G_1_x=1.,
show_contours=False,
show_field=False,
pick_contour=False,
contour_val=2):
intp = interp_with_drift(point_1_y=point_1_y,
point_2_y=point_2_y,
G_1_x=G_1_x)
plt.figure(figsize=(14, 8))
## defining the dips position
G_1 = np.array([[0., 1.]])
G_1_x = 1
G_1_y = 1
G_1_tiled = np.tile(G_1, [2, 1])
# layer1 = np.array([[0, 0], [2, 0]])
# layer2 = np.array([[0, 2], [2, 2]])
layer1 = np.array([[0, 0], [2, 0], [3, 0], [4, 0]])
layer2 = np.array([[0, 2], [2, point_1_y], [3, point_2_y]])
number_of_layer = 2
## create grid
xx = np.arange(-.5, 4.5, 0.1)
yy = np.arange(-.5, 4.5, 0.1)
XX, YY = np.meshgrid(xx, yy)
X = (np.reshape(XX, [-1])).T
Y = (np.reshape(YY, [-1])).T
grid = np.stack([X, Y], axis=1)
if show_contours:
plt.contour(XX, YY, intp, 20)
if show_field:
plt.pcolor(XX, YY, intp, alpha=0.6)
if show_contours or show_field:
try:
plt.colorbar()
except IndexError:
pass
if pick_contour:
plt.contour(XX, YY, intp,
levels=np.array([contour_val]),
linewidths=3)
plt.plot(layer1[:, 0], layer1[:, 1], 'ro')
plt.plot(layer2[:, 0], layer2[:, 1], 'bo')
plt.xlim([0,4.])
plt.ylim([-.5,3.5])
plt.plot(G_1[0, 0], G_1[0, 1], 'go')
def get_lithology_from_interp_with_drift(point_1_y = 2.,
point_2_y = 3.,
G_1_x = 1.):
# similar to interp_with_drift, but not generating a plot, only returing the interpolated and mapped lithologies
# for example for uncertainty quantification
interp = interp_with_drift(point_1_y=point_1_y,
point_2_y=point_2_y,
G_1_x=G_1_x)
xx = np.arange(-.5, 4.5, 0.1)
yy = np.arange(-.5, 4.5, 0.1)
XX, YY = np.meshgrid(xx, yy)
X = (np.reshape(XX, [-1])).T
Y = (np.reshape(YY, [-1])).T
grid = np.stack([X, Y], axis=1)
# perform regular grid interpolation
my_interpolating_function = RegularGridInterpolator((xx, yy), interp.T)
# extract lithologies
layer_1_interp = np.ones((50, 50))
layer_2_interp = np.ones((50, 50)) * 2
l1_bool = interp < my_interpolating_function([0, 0])
l2_bool = interp < my_interpolating_function([0, 2])
layer_1_interp = l1_bool * layer_1_interp
layer_2_interp = l2_bool * layer_2_interp
return layer_1_interp, layer_2_interp
def create_litho_plot(layer_interp, cmap='viridis', colorbar=False):
plt.figure(figsize=(14, 8))
xx = np.arange(-.5, 4.5, 0.1)
yy = np.arange(-.5, 4.5, 0.1)
XX, YY = np.meshgrid(xx, yy)
plt.pcolor(XX, YY, layer_interp, alpha=0.6, cmap=cmap)
if colorbar:
plt.colorbar()