# Geostatistical regularization¶

In this example we illustrate the use of geostatistical constraints on irregular meshes as presented by [JDGunther+18], compared to classical smoothness operators of first or second kind.

The elements of the covariance matrix $$\textbf{C}_{\text{M}}$$ are defined by the distances H between the model cells i and j into the three directions

$\textbf{C}_{\text{M},ij}=\sigma^{2}\exp{\left( -3\sqrt{\left(\frac{\textbf{H}^x_{ij}}{I_{x}}\right)^{2}+ \left(\frac{\textbf{H}^y_{ij}}{I_{y}}\right)^{2}+ \left(\frac{\textbf{H}^z_{ij}}{I_{z}}\right)^{2}}\right)}.$

It defines the correlation between model cells as a function of correlation lenghts (ranges) $$I_x$$, $$I_y$$, and $$I_z$$. Of course, the orientation of the coordinate axes is arbitrary and can be chosen by rotation. Let us illustrate this by a simple mesh:

# Computing covariance and constraint matrices¶

We create a simple mesh using a box geometry

import pygimli as pg
import pygimli.meshtools as mt

# We create a rectangular domain and mesh it with small triangles
rect = mt.createRectangle(start=[0, -10], end=[10, 0])
mesh = mt.createMesh(rect, quality=34.5, area=0.1)


We compute such a covariance matrix by calling

CM = pg.utils.covarianceMatrix(mesh, I=5)  # I taken for both x and y
# We search for the cell where the midpoint (5, -5) is located in
ind = mesh.findCell([5, -5]).id()
# and plot the according column using index access (numpy)
ax, cb = pg.show(mesh, CM[:, ind], cMap="magma_r") According to inverse theory, we use the square root of the covariance matrix as single-side regularization matrix C. It is computed by using an eigenvalue decomposition

$\textbf{C}_\text{M} = \textbf{Q}\textbf{D}\textbf{Q}^{T}$

based on LAPACK (numpy.linalg). The inverse square root is defined by

$\textbf{C}_\text{M}^{-0.5} = \textbf{Q}\textbf{D}^{-0.5}\textbf{Q}^{T}$

In order to avoid a matrix inverse (square root), a special matrix is derived that does the decomposition and stores the eigenvectors and eigenvalues values. A multiplication is done by multiplying with Q and scaling with the diagonal D. This matrix is implemented in the pygimli.matrix module by the class pg.matrix.Cm05Matrix

Cm05 = pg.matrix.Cm05Matrix(CM)


However, this matrix does not return a zero vector for a constant vector

out = Cm05 * pg.Vector(mesh.cellCount(), 1.0)
print(min(out), max(out))


Out:

0.021592434321538633 0.20503355104238002


as desired for a roughness operator. Therefore, an additional matrix called pg.matrix.GeostatisticalConstraintsMatrix was implemented where this spur is corrected for. It is, like the correlation matrix, created by a mesh, a list of correlation lengths I, a dip angle# that distorts the x/y plane and a strike angle towards the third direction.

C = pg.matrix.GeostatisticConstraintsMatrix(mesh=mesh, I=5)


In order to extract a certain column, we generate a vector with a single 1

vec = pg.Vector(mesh.cellCount())
vec[ind] = 1.0
ax, cb = pg.show(mesh, pg.log10(pg.abs(C*vec)), cMin=-6, cMax=0, cMap="magma_r") The constraints have a rather small footprint compared to the correlation (note the logarithmic scale) but still to the whole mesh unlike the classical constraint matrices that only include relations to neighboring cells.

Such a matrix can also be defined for different ranges and dip angles, e.g.

Cdip = pg.matrix.GeostatisticConstraintsMatrix(mesh=mesh, I=[10, 3], dip=-25)
ax, cb = pg.show(mesh, pg.log10(pg.abs(Cdip*vec)), cMin=-6, cMax=0, cMap="magma_r") In order to illustrate the role of the constraints, we use a very simple mapping forward operator that retrieves the values in the mesh at some given positions. The constraints are therefore used as interpolation operators. Note that the mapping forward operator can also be used for defining prior knowledge if combined with another forward operator in a classical joint inversion framework. In the initialization, the indices are stored and a mapping matrix is created that projects the model vector to the forward response. This matrix is also the Jacobian matrix for the inversion.

class PriorFOP(pg.core.ModellingBase):
"""Forward operator for grabbing values."""

def __init__(self, mesh, pos, verbose=False):
"""Init with mesh and some positions that are converted into ids."""
super().__init__(self, verbose)
self.setMesh(mesh)
self.ind = [mesh.findCell(po).id() for po in pos]
self.J = pg.SparseMapMatrix()
self.J.resize(len(self.ind), mesh.cellCount())
for i, ii in enumerate(self.ind):
self.J.setVal(i, ii, 1.0)

self.setJacobian(self.J)

def response(self, model):
"""Return values at the indexed cells."""
return model[self.ind]

def createJacobian(self, model):
"""Do nothing (linear)."""
pass


# Inversion with geostatistical constraints¶

We choose some positions and initialize the forward operator

pos = [[2, -2], [8, -2], [5, -5], [2, -8], [8, -8]]
fop = PriorFOP(mesh, pos)
# For plotting the results, we create a figure and define some plotting options
fig, ax = pg.plt.subplots(nrows=2, ncols=2, sharex=True, sharey=True)
kw = dict(
colorBar=True,
cMin=30,
cMax=300,
orientation='vertical',
cMap='Spectral_r',
logScale=True)
# We want to use a homogenenous starting model
startModel = pg.Vector(mesh.cellCount(), 30)
tLog = pg.core.TransLog()
vals = [30, 50, 300, 100, 200]
inv = pg.core.Inversion(vals, fop, tLog, tLog)
inv.setRelativeError(0.05)  # 5 % error
inv.setModel(startModel)
inv.setLambda(200)
# first we use the second order (curvature) constraint type
fop.regionManager().setConstraintType(2)
res = inv.run()
print(('{:.1f} ' * 5).format(*fop(res)), inv.chi2())
pg.show(mesh, res, ax=ax[0, 1], **kw)
# Next, we use first-order constraints
fop.regionManager().setConstraintType(1)
res = inv.run()
print(('{:.1f} ' * 5).format(*fop(res)), inv.chi2())
pg.show(mesh, res, ax=ax[0, 0], **kw)
# Now we set the geostatistic isotropic operator with 5m correlation length
fop.setConstraints(C)
inv.setModel(startModel)
inv.setLambda(30)
res = inv.run()
print(('{:.1f} ' * 5).format(*fop(res)), inv.chi2())
pg.show(mesh, res, ax=ax[1, 0], **kw)
ax[0, 0].set_title("1st order")
ax[0, 1].set_title("2nd order")
ax[1, 0].set_title("I=5")
# and finally we use the dipping constraint matrix
fop.setConstraints(Cdip)
inv.setLambda(20)
inv.setModel(startModel)
res = inv.run()
print(('{:.1f} ' * 5).format(*fop(res)), inv.chi2())
pg.show(mesh, res, ax=ax[1, 1], **kw)
ax[1, 1].set_title("I=[10/3], dip=25")
# plot the position of the priors
for ai in ax.flat:
for po in pos:
ai.plot(*po, marker='o', markersize=10, color='k', fillstyle='none')
# Out:

30.3 49.1 277.9 95.7 189.2  0.8988703345983048
30.1 49.8 282.7 97.4 182.7  0.9905993445704484
31.5 51.8 267.0 102.6 201.6  1.4240540668902233
31.6 49.0 280.5 97.9 198.3  0.6654150178516384


Note that all four regularization operators fit the data equivalently but the images (i.e. how the gaps between the data points are filled) are quite different. This is something we should have in mind using regularization. %% Generating geostatistical media ——————————- For generating geostatistical media, one can use the function generateGeostatisticalModel. It computes a correlation matrix and multiplies it with a pseudo-random (randn) series. The arguments are the same as for the correlation or constraint matrices.

model = pg.utils.generateGeostatisticalModel(mesh, I=[20, 4])
ax, cb = pg.show(mesh, model) Total running time of the script: ( 0 minutes 45.092 seconds)

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