# Crosshole traveltime tomography¶

Seismic and ground penetrating radar (GPR) methods are frequently applied to image the shallow subsurface. While novel developments focus on inverting the full waveform, ray-based approximations are still widely used in practice and offer a computationally efficient alternative. Here we demonstrate the modeling of traveltimes and their inversion for the underlying slowness distribution for a crosshole scenario.

We start by importing the necessary packages.

# sphinx_gallery_thumbnail_number = 3

import matplotlib as mpl
import matplotlib.pyplot as plt
import numpy as np

import pygimli as pg
import pygimli.meshtools as mt
from pygimli.physics.traveltime import Refraction

mpl.rcParams['image.cmap'] = 'inferno_r'


Next, we build the crosshole acquisition geometry with two shallow boreholes.

# Acquisition parameters
bh_spacing = 20.0
bh_length = 25.0
sensor_spacing = 2.5

world = mt.createRectangle(start=[0, -(bh_length + 3)], end=[bh_spacing, 0.0],
marker=0)

depth = -np.arange(sensor_spacing, bh_length + sensor_spacing, sensor_spacing)

sensors = np.zeros((len(depth) * 2, 2))  # two boreholes
sensors[len(depth):, 0] = bh_spacing  # x
sensors[:, 1] = np.hstack([depth] * 2)  # y


Traveltime calculations work on unstructured meshes and structured grids. We demonstrate this here by simulating the synthetic data on an unstructured mesh and inverting it on a simple structured grid.

# Create forward model and mesh
c0 = mt.createCircle(pos=(7.0, -10.0), radius=3, segments=25, marker=1)
c1 = mt.createCircle(pos=(12.0, -18.0), radius=4, segments=25, marker=2)
geom = mt.mergePLC([world, c0, c1])
for sen in sensors:
geom.createNode(sen)

mesh_fwd = mt.createMesh(geom, quality=34, area=.25)
model = np.array([2000., 2300, 1700])[mesh_fwd.cellMarkers()]
pg.show(mesh_fwd, model, label="Velocity (m/s)", nLevs=3, logScale=False) Out:

(<matplotlib.axes._subplots.AxesSubplot object at 0x7f87ffef05c0>, <matplotlib.colorbar.Colorbar object at 0x7f8803d71f28>)


Create inversion mesh

refinement = 0.25
x = np.arange(0, bh_spacing + refinement, sensor_spacing * refinement)
y = -np.arange(0.0, bh_length + 3, sensor_spacing * refinement)
mesh = pg.createMesh2D(x, y)

ax, _ = pg.show(mesh, hold=True)
ax.plot(sensors[:, 0], sensors[:, 1], "ro") Out:

[<matplotlib.lines.Line2D object at 0x7f87ffd86710>]


Next, we create an empty DataContainer and fill it with sensor positions and all possible shot-recevier pairs for the two-borehole scenario using the product function in the itertools module (Python standard library).

from itertools import product
numbers = np.arange(len(depth))
rays = list(product(numbers, numbers + len(numbers)))

# Empty container
scheme = pg.DataContainer()

for sen in sensors:
scheme.createSensor(sen)

rays = np.array(rays)
scheme.resize(len(rays))
scheme.registerSensorIndex("s")
scheme.registerSensorIndex("g")


The forward simulation is performed with a few lines of code. We initialize an instance of the Refraction manager and call its simulate function with the mesh, the scheme and the slowness model (1 / velocity). We also add 0.1% relative and 10 microseconds of absolute noise.

Secondary nodes allow for more accurate forward simulations. Check out the paper by Giroux & Larouche (2013) to learn more about it.

tt = Refraction()
mesh_fwd.createSecondaryNodes(5)
data = tt.simulate(mesh=mesh_fwd, scheme=scheme, slowness=1. / model,
noiseLevel=0.001, noiseAbs=1e-5)


For the inversion we create a new instance of the Refraction manager to avoid confusion, since it is working on a different mesh.

ttinv = Refraction()
ttinv.setData(data)  # Set previously simulated data
ttinv.setMesh(mesh, secNodes=5)
invmodel = ttinv.invert(lam=1100, vtop=2000, vbottom=2000, zWeight=1.0)
print("chi^2 = %.2f" % ttinv.inv.getChi2())  # Look at the data fit


Out:

Data: Sensors: 20 data: 100
chi^2 = 0.88


Finally, we visualize the true model and the inversion result next to each other.

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(8, 7), sharex=True, sharey=True)
ax1.set_title("True model")
ax2.set_title("Inversion result")

pg.show(mesh_fwd, model, ax=ax1, showMesh=True, label="Velocity (m/s)",
logScale=False, nLevs=3)

for ax in (ax1, ax2):
ax.plot(sensors[:, 0], sensors[:, 1], "wo")

ttinv.showResult(ax=ax2)
ttinv.showRayPaths(ax=ax2, color="0.8", alpha=0.3)
fig.tight_layout() Note how the rays are attracted by the high velocity anomaly while circumventing the low velocity region. This is also reflected in the coverage, which can be visualized as follows:

fig, ax = plt.subplots()
ttinv.showCoverage(ax=ax, cMap="Greens")
ttinv.showRayPaths(ax=ax, color="k", alpha=0.3)
ax.plot(sensors[:, 0], sensors[:, 1], "ko") Out:

[<matplotlib.lines.Line2D object at 0x7f8818b3d3c8>]


White regions indicate the model null space, i.e. cells that are not traversed by any ray.

Total running time of the script: ( 1 minutes 6.918 seconds)

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Created using Bootstrap, Sphinx and pyGIMLi 1.0.12+20.g65e71e67 on Mar 10, 2020.