5.1 - Probabilistic Modeling: Error Propagation

In this example we will show how easy we can propagate uncertainty from GemPy parameters to final structural models.

import sys, os
sys.path.append("../../gempy")
os.environ["THEANO_FLAGS"] = "mode=FAST_RUN,device=cuda"


import gempy as gp
from gempy.bayesian.fields import compute_prob, calculate_ie_masked
import numpy as np
import matplotlib.pyplot as plt

np.random.seed(1234)

Model definition

In the previous example we assume constant thickness to be able to reduce the problem to one dimension. This keeps the probabilistic model fairly simple since we do not need to deel with complex geometric structures. Unfortunaly, geology is all about dealing with complex three dimensional structures. In the moment data spread across the physical space, the probabilistic model will have to expand to relate data from different locations. In other words, the model will need to include either interpolations, regressions or some other sort of spatial functions. In this paper, we use an advance universal co-kriging interpolator. Further implications of using this method will be discuss below but for this lets treat is a simple spatial interpolation in order to keep the focus on the constraction of the probabilistic model.

geo_model = gp.create_model('2-layers')
gp.init_data(geo_model, extent=[0, 12e3, -2e3, 2e3, 0, 4e3], resolution=[100, 10, 200])

Out:

Active grids: ['regular']

2-layers  2020-10-27 13:23
geo_model.add_surfaces('surface 1')
geo_model.add_surfaces('surface 2')
geo_model.add_surfaces('basement')
dz = geo_model._grid.regular_grid.dz
geo_model.add_surface_values([dz, 0, 0], ['dz'])
geo_model.add_surface_values(np.array([2.6, 2.4, 3.2]), ['density'])
surface series order_surfaces color id dz density
0 surface 1 Default series 1 #015482 1 20.00 2.60
1 surface 2 Default series 2 #9f0052 2 0.00 2.40
2 basement Default series 3 #ffbe00 3 0.00 3.20


geo_model.add_surface_points(3e3, 0, 3.05e3, 'surface 1')
geo_model.add_surface_points(9e3, 0, 3.05e3, 'surface 1')

geo_model.add_surface_points(3e3, 0, 1.02e3, 'surface 2')
geo_model.add_surface_points(9e3, 0, 1.02e3, 'surface 2')

geo_model.add_orientations(6e3, 0, 4e3, 'surface 1', [0, 0, 1])
X Y Z G_x G_y G_z smooth surface
0 6000.0 0.0 4000.0 0.0 0.0 1.0 0.01 surface 1


Cell Number: mid Direction: y

Adding topography

Out:

[3200. 4000.]
Active grids: ['regular' 'topography']

Grid Object. Values:
array([[ 6.00000000e+01, -1.80000000e+03,  1.00000000e+01],
       [ 6.00000000e+01, -1.80000000e+03,  3.00000000e+01],
       [ 6.00000000e+01, -1.80000000e+03,  5.00000000e+01],
       ...,
       [ 1.20000000e+04,  1.11111111e+03,  3.38961579e+03],
       [ 1.20000000e+04,  1.55555556e+03,  3.40343266e+03],
       [ 1.20000000e+04,  2.00000000e+03,  3.41755449e+03]])

Setting up our area

Lets imagine we have two boreholes and 1 gravity device. From the boreholes we can estimate the location of the interfaces of our layers. That will be enough to create the first model.

def plot_geo_setting():
    device_loc = np.array([[6e3, 0, 3700]])
    p2d = gp.plot_2d(geo_model, show_topography=True)

    well_1 = 3.5e3
    well_2 = 3.62e3
    p2d.axes[0].scatter([3e3], [well_1], marker='^', s=400, c='#71a4b3', zorder=10)
    p2d.axes[0].scatter([9e3], [well_2], marker='^', s=400, c='#71a4b3', zorder=10)
    p2d.axes[0].scatter(device_loc[:, 0], device_loc[:, 2], marker='x', s=400, c='#DA8886', zorder=10)

    p2d.axes[0].vlines(3e3, .5e3, well_1, linewidth=4, color='gray')
    p2d.axes[0].vlines(9e3, .5e3, well_2, linewidth=4, color='gray')
    p2d.axes[0].vlines(3e3, .5e3, well_1)
    p2d.axes[0].vlines(9e3, .5e3, well_2)
    plt.savefig('model.svg')
    plt.show()
plot_geo_setting()
Cell Number: mid Direction: y

Computing model

gp.set_interpolator(geo_model)

Out:

Setting kriging parameters to their default values.
Compiling theano function...
Level of Optimization:  fast_compile
Device:  cpu
Precision:  float64
Number of faults:  0
Compilation Done!
Kriging values:
                   values
range            1.3e+04
$C_o$            4.2e+06
drift equations      [3]

<gempy.core.interpolator.InterpolatorModel object at 0x7ff9baeee050>
gp.compute_model(geo_model)
plot_geo_setting()
Cell Number: mid Direction: y

Adding Random variables

Although that can work as a good approximation, the truth is that modelling hundreds of meters underground is not specially precise. That’s why in many cases we would like to model our input data as probability distributions instead deterministic values. GemPy is specially efficiency for these type of tasks:

geo_model.modify_surface_points(2, Z=500)
gp.compute_model(geo_model)
plot_geo_setting()

Z = np.random.normal(1000, 500, size=2)
geo_model.modify_surface_points([2, 3], Z=Z)
gp.compute_model(geo_model)
plot_geo_setting()
  • Cell Number: mid Direction: y
  • Cell Number: mid Direction: y

Now we just sample from a random variable and loop it as much as we want:

prob_block = compute_prob(lith_blocks)
p2dp = gp.plot_2d(geo_model,
                  show_lith=False, show_boundaries=False, show_data=False,
                  regular_grid=prob_block[2],
                  kwargs_regular_grid={'cmap': 'viridis',
                                        'norm': None}
                  )
plt.show()
Cell Number: mid Direction: y
entropy_block = calculate_ie_masked(prob_block)

sphinx_gallery_thumbnail_number = 6

p2dp = gp.plot_2d(geo_model,
                  show_lith=False, show_boundaries=False, show_data=False,
                  regular_grid=entropy_block,
                  kwargs_regular_grid={'cmap': 'viridis',
                                        'norm': None}
                  )
Cell Number: mid Direction: y

Total running time of the script: ( 0 minutes 51.194 seconds)

Gallery generated by Sphinx-Gallery