2.5D Forward Simulation

Author: Devin C. Cowan

Keywords: DC resistivity, forward simulation, 2.5D, apparent resistivity, tree mesh.

Summary: Here we use the module simpeg.electromagnetics.static.resistivity to simulate DC resistivity data when the local geology doesn’t change along the strike direction. Here, we use a 2.5D simulation approach to leverage the symmetry of the problem and avoid the higher computational cost associated with full 3D simulations.

Learning Objectives:

• How to define DC resistivity lines manually or by using utility functions in SimPEG.
• How to design a 2D tree mesh (or tensor mesh) for accurately simulating DC resistivity data for the 2.5D approach.
• How to define the Earth’s electrical properties according to conductivity OR resistivity.
• How to included surface topography in the forward simulation.
• How to plot simulated data in pseudosection.

Import Modules¶

Here, we import all of the functionality required to run the notebook for the tutorial exercise. All of the functionality specific to DC resistivity is imported from simpeg.electromagnetics.static.resistivity. We also import some useful utility functions from simpeg.utils. To simulate DC resistivity data, we need to define our problem geometry on a numerical grid (or mesh). To generate the mesh, we used the discretize package.

Define the Topography¶

True surface topography is defined as an (N, 3) numpy.ndarray. For use in a 2.5D simulation however, topography is defined as an (N, 2) numpy.ndarray, where the first coordinate represent along-line position and the second coordinate represents the vertical position. Here, we define the 2D topography used for the 2.5D simulation.

Define the Survey¶

DC (and IP) surveys within SimPEG require the user to create and connect three types of objects:

• receivers: which defines the locations of the potential (or MN) electrodes and the type of data; e.g. ‘volt’ for normalized voltage (V/A), ‘apparent_resistivity’ for apparent resistivity ($\Omega m$) or ‘apparent_chargeability’ for apparent chargeability (unitless). Note only M electrode locations are needed to define pole receivers.
• sources: which defines the locations of the current (or AB) electrodes, and their associated receivers. Note only A electrode locations are needed to define pole sources.
• survey: the object which stores and organizes all of the sources and receivers.

Each DC/IP datum corresponds to a unique pair of current and potential electrode locations. Within the SimPEG framework, there is a standard way in which the electrodes are organized. Because each current source (pole or dipole) is discretized to form a separate right-hand side for the DC resistivity PDE, each source object defines the electrodes for a single source. But for each source object, we can instantiate a single receiver object to store the electrode locations of all potential electrodes.

Using SimPEG, there are three approaches one might use to generate the a 2.5D resistivity survey. The final approach will be used to simulate the data for this tutorial.

Option A: Define each source and its associated receivers directly

For a 2.5D simulation, current electrode locations are defined as (2,) numpy.array and the associated set of potential electrode locations are defined as (N, 2) numpy.ndarray. We can define Pole or Dipole sources. And we can define Pole or Dipole receivers.

Here, the survey consists of an 800 m long EW dipole-dipole line with an electrode spacing of 40 m. There is a maximum of 10 potential electrodes per current electrode. And the data defined by the receivers are current-normalized voltages in V/A.

Option B: Survey from ABMN electrode locations

If we have (N, 2) numpy.ndarray for A, B, M and N electrode locations for each datum (loaded or created), we can use the generate_survey_from_abmn_locations to generate the survey automatically.

Option C: Survey from a set of survey lines

If the survey is comprised of a single DC resistivity line, we can use the generate_dcip_sources_line utility function to define the source list. I.e.:

# Generate source list for DC survey line
source_list = generate_dcip_sources_line(
    survey_type,
    data_type,
    dimension_type,
    end_locations,
    topo_2d,
    num_rx_per_src,
    station_separation,
)

# Define survey
survey = dc.survey.Survey(source_list, survey_type=survey_type)

We can extract various objects and properties from the objects used to generate the survey. As we can see, all receivers associated with each source are defined within a single object.

Here we plot the electrode locations. We use the pseudo_locations utility to extract the pseudo-locations.

Design a (Tree) Mesh¶

Meshes are designed using the discretize package. See the discretize user tutorials to learn more about creating meshes. Here, the forward simulation is computed for a tree mesh. Because of the modular nature of SimPEG, you could define a tensor mesh instead.

Standard approach for DC/IP: The electric potential produced by a current electrode falls off as $r^{-3}$. So smaller cells are needed near the current electrodes to model the fields accurately, and larger cells can be used away from the current electrodes where the fields are smooth. Tree meshes are well-suited for DC (and IP) simulation because the cell size can be increased at specified distances from the current electrodes. For DC resistivity meshing, we advise the following considerations and rules of thumb:

1. Because there are no currents in the air, we do not need to pad upwards. I.e. the top of the mesh corresponds to the top of the topography.
2. We require at least 2-3 cells between each current electrode; with more accurate results being obtained when the minimum cell size is smaller. For a 2.5D problem geometry, we can discretize much finer.
3. To be safe, the padding thickness should be at least 2-3 times the largest electrode spacing.
4. The increase in cell size at increasing distances from the current electrodes should not happen too abruptly. At each cell size, you should have a layer at least 4 cells thick before increasing the cell size.
5. Finer discretization is required when topography is significant.

Tutorial mesh: Here, a minimum cell width of 4 m (or 1/10 the minimum electrode spacing) is used within our survey region. The largest electrode spacing was 400 m, so a the padding was extended at least 1200 m from the survey region. Using the refine_surface method, we refine the tree mesh where there is significant topography. And using the refine_points methods, we refine based on electrodes locations. Visit the tree mesh API to see additional refinement methods.

If desired, we can extract various properties of the mesh. E.g.

And we can plot the mesh and electrode locations.

Define the Active Cells¶

Simulated geophysical data are dependent on the subsurface distribution of physical property values. As a result, the cells lying below the surface topography are commonly referred to as ‘active cells’. And air cells, whose physical property values are fixed, are commonly referred to as ‘inactive cells’. Here, the discretize active_from_xyz utility function is used to find the indices of the active cells using the mesh and surface topography. The output quantity is a bool array.

Models and Mappings¶

In SimPEG, the term ‘model’ is not necessarily synonymous with the set of physical property values defined on the mesh. For example, the model may be defined as the logarithms of the physical property values, or be the parameters defining a layered Earth geometry. Models in SimPEG are 1D numpy.ndarray whose lengths are equal to the number of model parameters.

Classes within the simpeg.maps module are used to define the mapping that connects the model to the physical property values used in the DC resistivity simulation. Sophisticated mappings can be defined by combining multiple mapping objects. But in the simplest case, the mapping is an identity map and the model consists of the conductivity/resistivity values for all mesh cells (including air).

When simulating DC resistivity data, we have the choice of using resistivity or conductivity to define the Earth’s electrical properties. Here, we define the model and its associate mapping for two cases:

1. The model consists of the conductivity values for all active cells
2. The model consists of the log-resistivity values for all active cells

Define the Model¶

The units for resistivity are $\Omega m$ and the units for conductivity are $S/m$.

Mapping from the Model to the Mesh¶

For our first case, we use the simpeg.maps.InjectActiveCells mapping. This mapping projects quantities defined on the active cells to the entire mesh, and sets a constant value for all inactive cells. Important: we set all inactive (air) cells to 1e-8 S/m instead of 0. This is done to ensure that the linear system constructed to solve the PDE for the DC resistivity problem is well-conditioned.

For the second case, we both the simpeg.maps.InjectActiveCells and simpeg.maps.ExpMap mappings; the latter of which takes the natural exponential. Important: we set all inactive (air) cells to 1e8 $\Omega m$ instead of ∞. Once again, this is done to ensure that the linear system constructed to solve the PDE for the DC resistivity problem is well-conditioned.

Plot the Model¶

To show the geometry of the problem, we plot the conductivity model using the plot_slice method.

Project Electrodes to Discretized Topography¶

Surface DC resistivity data will not be modeled accurately if the electrodes are modeled as living above or below the surface. It is especially problematic when electrodes are modeled as living in the air. Prior to simulating surface DC resistivity data, we must project the electrodes from their true elevation to the surface of the discretized topography. This is done using the drape_electrodes_on_topography method.

Define the Forward Simulation¶

In SimPEG, the physics of the forward simulation is defined by creating an instance of an appropriate simulation class. There are two simulation classes which may be used to simulate 2.5D DC resistivity data:

• Simulation2DNodel, which defines the discrete electric potentials on mesh nodes.
• Simulation2DCellCentered, which defines the discrete electric potentials at cell centers.

For surface DC resistivity data, the nodal formulation is more well-suited and will be used here. The cell-centered formulation works well for simulating borehole DC resistivity data. To fully define the forward simulation, we need to connect the simulation object to:

• the survey
• the mesh
• the mapping from the model to the mesh

This is accomplished by setting each one of the aforementioned items as a property of the simulation object. Here, we define two simulation objects, one where the model defines the subsurface conductivities, and one where the model defines subsurface log-resistivities. When our model is used to define subsurface electric conductivity, the mapping is set using the sigmaMap keyword argument. However when our model is used to define subsurface electric resistivity, the mapping must be set using the rhoMap keyword argument

Predict DC Resistivity Data¶

Once any simulation within SimPEG has been properly constructed, simulated data for a given model vector can be computed using the dpred method. Note that despite the difference in how we defined the models representing the Earth’s electrical properties, the data predicted by both simulations is equivalent.

Plot Data in Pseudosection¶

Here we use the plot_pseudosection utility function to represent the predicted data on a pseudosection plot as apparent conductivities. Since our receivers were defined to simulate data as normalized voltages, we use the apparent_resistivity_from_voltage utility function to convert the data to apparent resistivities, then take the reciprocal to obtain apparent conductivities.

Optional: Write DC resistivity data and topography.