variography

Variography¶

In [1]:

%matplotlib inline

In [2]:

# loading packages
import pandas as pd
import numpy as np

geolime package can be imported as such:

In [3]:

import geolime as glm

Data¶

A dataset is available for testing. Here we load some geological data and display the first five rows.

In [4]:

assay_2d_tnb = "../../tests/data/variography/assay_2d_tnb.csv"
data = pd.read_csv(assay_2d_tnb)
data.head()

Out[4]:

	Unnamed: 0	holeid	x	y	z	from	to	ni	co	fe	mgo	sio2	domain	layer
0	0	03PD2018L370O150 ...	240242.692	383322.588	282.916	0.0	1.0	2.768	0.019	7.84	26.88	44.76	D	SAP
1	1	03PD2018L370O270 ...	240250.718	383440.213	286.957	0.0	1.0	0.699	0.021	9.73	28.16	41.88	D	SAP
2	2	03PD2018L404O430 ...	240280.549	383602.858	289.022	0.0	1.0	0.440	0.020	8.12	28.93	43.47	D	SAP
3	3	03PD2018L410O310 ...	240291.812	383480.851	305.434	0.0	1.0	1.418	0.027	7.02	29.75	47.01	D	SAP
4	4	03PD2018L410O350 ...	240288.891	383520.723	302.360	1.2	2.0	1.560	0.026	11.84	20.40	47.19	T	SAP

Experimental semivariogram¶

Prepare arrays to be used as input for experimental semivariogram computation.

In [5]:

# extract coordinates and values
data = pd.read_csv(assay_2d_tnb)
coords_df = data[['x', 'y']]
coords = coords_df.to_numpy()
values_df = data[['ni']]
values = values_df.to_numpy()

Define input lags and compute the experimental semivariogram.

In [6]:

# define input lags
input_lags = [0, 13.34, 26.68, 40.02, 53.36, 66.7,
              80.04, 93.38, 106.72, 120.06]

# compute variogram
vario_exp_az_north = glm.variogram(coords=coords, values=values, lags=input_lags,
                      tol=6.67, geographic_azimuth=0, dip=0, pitch=0,
                      atol=45, slice_width=np.nan, slice_height=np.nan,
                      method="semivariogram")

# compute variogram
vario_exp_az_east = glm.variogram(coords=coords, values=values, lags=input_lags,
                      tol=6.67, geographic_azimuth=90, dip=0, pitch=0,
                      atol=45, slice_width=np.nan, slice_height=np.nan,
                      method="semivariogram")

The resulting dataframe contains the following columns

In [7]:

vario_exp_az_north.head()

Out[7]:

	rank	npairs	lag	avgdist	vario	indices
0	0	12	0.00	5.97538	0.537738	[[17, 51], [24, 54], [32, 104], [34, 118], [35...
1	1	597	13.34	13.2226	0.506487	[[0, 259], [1, 83], [1, 92], [2, 269], [7, 268...
2	2	916	26.68	26.6512	0.555162	[[0, 255], [0, 260], [1, 76], [1, 93], [1, 98]...
3	3	999	40.02	40.2595	0.610749	[[0, 29], [0, 104], [1, 89], [1, 90], [1, 99],...
4	4	955	53.36	53.2865	0.716520	[[0, 32], [0, 91], [0, 95], [0, 242], [1, 25],...

In [8]:

vario_exp_az_east.head()

Out[8]:

	rank	npairs	lag	avgdist	vario	indices
0	0	1	0.00	6.23131	0.004050	[[203, 207]]
1	1	603	13.34	13.4262	0.475794	[[0, 254], [2, 46], [5, 266], [9, 263], [10, 2...
2	2	854	26.68	26.9149	0.595443	[[0, 249], [1, 26], [1, 96], [1, 97], [1, 251]...
3	3	1072	40.02	40.2483	0.647403	[[0, 112], [0, 241], [1, 105], [1, 244], [1, 2...
4	4	1107	53.36	53.338	0.691535	[[0, 33], [0, 111], [0, 119], [0, 235], [1, 3]...

In [9]:

vario_exp = [vario_exp_az_north, vario_exp_az_east]

It also contains useful metadata

In [10]:

vario_exp_az_north.attrs

Out[10]:

{'method': 'semivariogram',
 'geographic_azimuth': 0,
 'dip': 0,
 'pitch': 0,
 'tol': 6.67,
 'atol': 45.0,
 'trend': 90.0,
 'plunge': 0.0,
 'variable_mean': 1.8995185185185182,
 'variable_var': 0.7367737459533608,
 'number_of_lags': 10,
 'input_lags': [0,
  13.34,
  26.68,
  40.02,
  53.36,
  66.7,
  80.04,
  93.38,
  106.72,
  120.06],
 'slice_width': nan,
 'slice_height': nan,
 'dim': 2,
 'directional_vector': array([6.123234e-17, 1.000000e+00])}

We can plot the experimental semivariogram dataframe with the following command:

In [11]:

glm.plot_semivariogram(variograms=vario_exp, model=None, display_sill=True)

Out[11]:

(<Figure size 432x288 with 1 Axes>, <AxesSubplot:>)

Variogram model¶

Experimental semivariogram can be fitted the following way. We first need to define a target grid.

In [12]:

# create a grid
target_grid = glm.create_grid_2d(origin=[0, 0], n_dim=[30, 30], cell_size=[5, 5], angle=[0.0, 0.0, 0.0])

We also need to define a covariance model for fitting

In [13]:

# define a covariance model
cov = glm.Nugget() + 0.5 * glm.Spherical(2)

In [14]:

cov.print()

Model with  2  components
--------------------------------------------------
Component  1  :
Variance : 1
Covariance type :  Nugget
---------
Component  2  :
Variance : 0.5
Covariance type :  Spherical
Scales :  [1. 1.]
Angles :  0.0
Rotation matrix
 [[ 1. -0.]
 [ 0.  1.]]
---------
Total sill =  1.5

We can fit the experimental variogram

In [15]:

# fit model
cov_model = glm.model_fit(variograms=vario_exp, cov=cov, constraint=None)

In [16]:

glm.plot_semivariogram(variograms=[vario_exp_az_north], model=cov_model)

Out[16]:

(<Figure size 432x288 with 1 Axes>, <AxesSubplot:>)

In [17]:

cov_model.print()

Model with  2  components
--------------------------------------------------
Component  1  :
Variance : 0.40751929468954295
Covariance type :  Nugget
---------
Component  2  :
Variance : 0.3432202933243947
Covariance type :  Spherical
Scales :  [79.52391655 83.24973102]
Angles :  14.651468428093645
Rotation matrix
 [[ 0.96748235 -0.25293854]
 [ 0.25293854  0.96748235]]
---------
Total sill =  0.7507395880139376

We apply model regularization with the following call:

In [18]:

pattern_generator = glm.GridPatternGenerator(target_grid, [5, 5])
mr = glm.model_regularize(vario_exp_az_north, cov, pattern_generator, nlag=10, border_ratio=1, normalize=True)
mr

Out[18]:

	lag	vario	npairs
0	0.00	0.000000	1.0
1	13.34	0.203300	1.0
2	26.68	0.440564	1.0
3	40.02	0.651105	1.0
4	53.36	0.823227	1.0
5	66.70	0.943547	1.0
6	80.04	0.997068	1.0
7	93.38	1.000000	1.0
8	106.72	1.000000	1.0
9	120.06	1.000000	1.0

Finally, the experimental variogram can be fitted and plotted with the following call:

In [19]:

# fit regularized model
rmodel = glm.model_fit(variograms=[mr], cov=cov, constraint=None)

In [20]:

glm.plot_semivariogram(variograms=[mr], model=rmodel)

Out[20]:

(<Figure size 432x288 with 1 Axes>, <AxesSubplot:>)