Creating a Sum Model

This tutorial demonstrates how to create and use sum models in GSTools. We’ll combine a Spherical and a Gaussian covariance model to construct a sum model, visualize its variogram, and generate spatial random fields.

Let’s start with importing GSTools and setting up the domain size.

import gstools as gs

x = y = range(100)

First, we create two individual covariance models: a Spherical model and a Gaussian model. The Spherical model with its short length scale will emphasize small-scale variability, while the Gaussian model with a larger length scale captures larger-scale patterns.

m0 = gs.Spherical(dim=2, var=2.0, len_scale=5.0)
m1 = gs.Gaussian(dim=2, var=1.0, len_scale=10.0)

Next, we create a sum model by adding these two models together. Let’s visualize the resulting variogram alongside the individual models.

model = m0 + m1
ax = model.plot(x_max=20)
m0.plot(x_max=20, ax=ax)
m1.plot(x_max=20, ax=ax)

As shown, the Spherical model controls the behavior at shorter distances, while the Gaussian model dominates at longer distances. The ratio of influence is thereby controlled by the provided variances of the individual models.

Using the sum model, we can generate a spatial random field. Let’s visualize the field created by the sum model.

srf = gs.SRF(model, seed=20250107)
srf.structured((x, y))
srf.plot()

For comparison, we generate random fields using the individual models to observe their contributions more clearly.

srf0 = gs.SRF(m0, seed=20250107)
srf0.structured((x, y))
srf0.plot()

srf1 = gs.SRF(m1, seed=20250107)
srf1.structured((x, y))
srf1.plot()

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As seen, the Gaussian model introduces large-scale structures, while the Spherical model influences the field’s roughness. The sum model combines these effects, resulting in a field that reflects multi-scale variability.