Fitting a Sum Model

In this tutorial, we demonstrate how to fit a sum model consisting of two covariance models to an empirical variogram.

We will generate synthetic data, compute an empirical variogram, and fit a sum model combining a Spherical and Gaussian model to it.

import gstools as gs

x = y = range(100)

First, we create a synthetic random field based on a known sum model. This will serve as the ground truth for fitting.

# Define the true sum model
m0 = gs.Spherical(dim=2, var=2.0, len_scale=5.0)
m1 = gs.Gaussian(dim=2, var=1.0, len_scale=10.0)
true_model = m0 + m1

# Generate synthetic field
srf = gs.SRF(true_model, seed=20250405)
field = srf.structured((x, y))

Next, we calculate the empirical variogram from the synthetic data.

# Compute empirical variogram
bin_center, gamma = gs.vario_estimate((x, y), field)

Now we define a sum model to fit to the empirical variogram. Initially, the parameters of the models are arbitrary.

A sum model can also be created by a list of model classes together with the common arguments (like dim in this case).

fit_model = gs.SumModel(gs.Spherical, gs.Gaussian, dim=2)

We fit the sum model to the empirical variogram using GSTools’ built-in fitting capabilities. We deactivate the nugget fitting to not overparameterize our model.

fit_model.fit_variogram(bin_center, gamma, nugget=False)
print(f"{true_model=}")
print(f" {fit_model=}")

true_model=SumModel(Spherical, Gaussian, dim=2, vars=[2.0, 1.0], len_scales=[5.0, 10.0])
 fit_model=SumModel(Spherical, Gaussian, dim=2, vars=[2.1, 0.923], len_scales=[3.99, 14.2])

The variance of a sum model is the sum of the sub variances (SumModel.vars) from the contained models. The length scale is a weighted sum of the sub length scales (SumModel.len_scales) where the weights are the ratios of the sub variances to the total variance of the sum model.

print(f"{true_model.var=:.2}, {true_model.len_scale=:.2}")
print(f" {fit_model.var=:.2},  {fit_model.len_scale=:.2}")

true_model.var=3.0, true_model.len_scale=6.7
 fit_model.var=3.0,  fit_model.len_scale=7.1

After fitting, we can visualize the empirical variogram alongside the fitted sum model and its components. A Sum Model is subscriptable to access the individual models its contains.

ax = fit_model.plot(x_max=max(bin_center))
ax.scatter(bin_center, gamma)
# Extract individual components
fit_model[0].plot(x_max=max(bin_center), ax=ax)
fit_model[1].plot(x_max=max(bin_center), ax=ax)
# True models
true_model.plot(x_max=max(bin_center), ax=ax, ls="--", c="C0", label="")
true_model[0].plot(x_max=max(bin_center), ax=ax, ls="--", c="C1", label="")
true_model[1].plot(x_max=max(bin_center), ax=ax, ls="--", c="C2", label="")

As we can see, the fitted sum model closely matches the empirical variogram, demonstrating its ability to capture multi-scale variability effectively. The “true” variograms are shown with dashed lines for comparison.