Note
Go to the end to download the full example code
Receiver Function#
If you are running this notebook locally, make sure you’ve followed steps here to set up the environment. (This environment.yml file specifies a list of packages required to run the notebooks)
In this notebook, we run inversion on a toy model with optimisation and parallel sampling.
0. Import modules#
# -------------------------------------------------------- #
# #
# Uncomment below to set up environment on "colab" #
# #
# -------------------------------------------------------- #
# !pip install -U cofi geo-espresso
import os
import math
import numpy as np
import matplotlib.pyplot as plt
import arviz
import emcee
import multiprocessing
import cofi
import espresso
# randomness is used to initialise emcee walkers starting points
np.random.seed(42)
# display theory on receiver function
from IPython.display import display, Markdown
with open("../../theory/geo_receiver_function.md", "r") as f:
content = f.read()
display(Markdown(content))
<IPython.core.display.Markdown object>
We are going to use the receiver function kernel wrapped in
`espresso <https://geo-espresso.readthedocs.io/en/latest/user_guide/contrib/generated/_receiver_function/index.html>`__,
with calls to Fortran routines developed by Takuo Shibutani in the
backend.
my_receiver_function = espresso.ReceiverFunctionInversionShibutani(example_number=4)
Consider a model setup of n layers described with 3 parameters for
each layer. model is a NumPy array of dimension [nlayers,3]. The
values in model[:,0] give the depths of discontinuities in the
model, while model[:,1] contains the S-wave speed above the
interface. model[:,2] is the ratio of S-wave speed to P-wave speed.
The maximum depth of discontinuity that can be considered is 60km.
In this example, we fix the ratio of S-wave speed to P-wave speed, and treat the interface depths and velocities of 3 layers as unknowns.
Here we set a starting model that is reasonably close to the true model, so that the optimisation converges.
null_model = my_receiver_function.starting_model
print(f"Number of model parameters in this example: {null_model.size}")
my_receiver_function._model_setup(null_model)
Number of model parameters in this example: 6
array([[ 9. , 3.2, 1.7],
[20. , 4. , 1.7],
[50. , 4.3, 1.7]])
Let’s plot the starting Earth model.
my_receiver_function.plot_model(null_model);
<Axes: xlabel='Vs (km/s)', ylabel='Depth (km)'>
Now we calculate the receiver function and plot it
predicted_data = my_receiver_function.forward(null_model)
observed_data = my_receiver_function.data
my_receiver_function.plot_data(
data1=predicted_data,
data2=observed_data,
label="predicted_data",
label2="observed_data",
)
plt.legend();
<matplotlib.legend.Legend object at 0x7f824bae0220>
1. Solve with an optimiser#
1.1 Define BaseProblem#
my_problem = cofi.BaseProblem()
In preparation for optimisation:
def my_misfit(model, include_uncertainty=False):
predicted_data = my_receiver_function.forward(model)
misfit_val = my_receiver_function.misfit(predicted_data, observed_data)
if math.isnan(misfit_val):
return float("inf")
return misfit_val
my_problem.set_objective(my_misfit)
my_problem.set_initial_model(null_model)
my_problem.summary()
=====================================================================
Summary for inversion problem: BaseProblem
=====================================================================
Model shape: (6,)
---------------------------------------------------------------------
List of functions/properties set by you:
['objective', 'initial_model', 'model_shape']
---------------------------------------------------------------------
List of functions/properties created based on what you have provided:
-- none --
---------------------------------------------------------------------
List of functions/properties that can be further set for the problem:
( not all of these may be relevant to your inversion workflow )
['log_posterior', 'log_posterior_with_blobs', 'log_likelihood', 'log_prior', 'gradient', 'hessian', 'hessian_times_vector', 'residual', 'jacobian', 'jacobian_times_vector', 'data_misfit', 'regularization', 'regularization_matrix', 'forward', 'data', 'data_covariance', 'data_covariance_inv', 'blobs_dtype', 'bounds', 'constraints']
1.2 Define InversionOptions#
my_options_optimiser = cofi.InversionOptions()
my_options_optimiser.set_tool("scipy.optimize.minimize")
my_options_optimiser.set_params(method="Nelder-Mead") # Nelder-Mead or COBYLA
1.3 Define Inversion and run#
inv_optimiser = cofi.Inversion(my_problem, my_options_optimiser)
my_result_optimiser = inv_optimiser.run()
my_result_optimiser.summary()
============================
Summary for inversion result
============================
SUCCESS
----------------------------
fun: 7594.579619321899
nit: 457
nfev: 787
status: 0
message: Optimization terminated successfully.
final_simplex: (array([[11.27274409, 2.96058687, 17.33057438, 3.86879402, 46.71754491,
4.44290912],
[11.27274396, 2.96058684, 17.33057417, 3.86879402, 46.71754518,
4.44290913],
[11.2727441 , 2.96058684, 17.33057403, 3.868794 , 46.71754502,
4.44290911],
[11.27274448, 2.9605868 , 17.33057329, 3.86879393, 46.71754472,
4.44290904],
[11.27274444, 2.96058682, 17.33057346, 3.86879392, 46.71754475,
4.44290905],
[11.27274465, 2.96058681, 17.33057318, 3.86879389, 46.71754456,
4.44290902],
[11.27274386, 2.96058685, 17.33057436, 3.86879404, 46.71754524,
4.44290915]]), array([7594.57961932, 7594.57961932, 7594.57961932, 7594.57961932,
7594.57961932, 7594.57961932, 7594.57961932]))
model: [11.27274409 2.96058687 17.33057438 3.86879402 46.71754491 4.44290912]
print("Inversion result: ", my_result_optimiser.model)
print("Reference good model:", my_receiver_function.good_model)
Inversion result: [11.27274409 2.96058687 17.33057438 3.86879402 46.71754491 4.44290912]
Reference good model: [ 8. 3. 20. 3.9 45. 4.4]
1.4 Plotting#
predicted_data = my_receiver_function.forward(my_result_optimiser.model)
my_receiver_function.plot_data(
data1=predicted_data,
data2=observed_data,
label="predicted_data",
label2="observed_data",
)
plt.legend();
<matplotlib.legend.Legend object at 0x7f8249000130>
2. Solve with a sampler#
2.1 Enrich BaseProblem#
In preparation for sampling:
def my_log_likelihood(model):
data1 = my_receiver_function.data
data2 = my_receiver_function.forward(model)
log_likelihood = my_receiver_function.log_likelihood(data1, data2) / 20 # temper the likelihood
return log_likelihood
def my_log_prior(model):
log_prior = my_receiver_function.log_prior(model)
return log_prior
ndim = my_receiver_function.model_size
my_problem.set_model_shape(ndim)
my_problem.set_log_likelihood(my_log_likelihood)
my_problem.set_log_prior(my_log_prior)
my_problem.summary()
=====================================================================
Summary for inversion problem: BaseProblem
=====================================================================
Model shape: 6
---------------------------------------------------------------------
List of functions/properties set by you:
['objective', 'log_likelihood', 'log_prior', 'initial_model', 'model_shape', 'blobs_dtype']
---------------------------------------------------------------------
List of functions/properties created based on what you have provided:
['log_posterior', 'log_posterior_with_blobs']
---------------------------------------------------------------------
List of functions/properties that can be further set for the problem:
( not all of these may be relevant to your inversion workflow )
['log_posterior', 'log_posterior_with_blobs', 'gradient', 'hessian', 'hessian_times_vector', 'residual', 'jacobian', 'jacobian_times_vector', 'data_misfit', 'regularization', 'regularization_matrix', 'forward', 'data', 'data_covariance', 'data_covariance_inv', 'bounds', 'constraints']
nwalkers = 12
nsteps = 25000
walkers_start = my_result_optimiser.model + 1e-1 * np.random.randn(nwalkers, ndim)
We can run emcee in parallel. Some additional preparation:
2.2 Define InversionOptions, Inversion and run#
import warnings
warnings.filterwarnings("ignore")
my_options_sampler = cofi.InversionOptions()
my_options_sampler.set_tool("emcee")
my_options_sampler.set_params(
nwalkers=nwalkers,
nsteps=nsteps,
initial_state=walkers_start,
progress=True,
)
inv_sampler = cofi.Inversion(my_problem, my_options_sampler)
inv_result_sampler = inv_sampler.run()
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inv_result_sampler.summary()
============================
Summary for inversion result
============================
SUCCESS
----------------------------
sampler: <emcee.ensemble.EnsembleSampler object>
blob_names: ['log_likelihood', 'log_prior']
2.3 Plotting#
var_names = (
"depth1 (km)",
"velocity1 (km/s)",
"depth2 (km)",
"velocity2 (km/s)",
"depth3 (km)",
"velocity3 (km/s)",
)
az_inf_data = inv_result_sampler.to_arviz(var_names=var_names)
az_inf_data
arviz.plot_trace(az_inf_data);
plt.tight_layout();
The walkers start in small distributions around some chosen values and then they quickly wander and start exploring the full posterior distribution. In fact, after a relatively small number of steps, the samples seem pretty well “burnt-in”. That is a hard statement to make quantitatively, but we can look at an estimate of the integrated autocorrelation time (see Emcee’s package the -Autocorrelation analysis & convergence tutorial for more details):
autocorrelation time: [120.64935842 83.36629501 120.8727641 90.68159476 120.18693988
89.73412015]
Let’s discard the initial 300 steps and make a corner plot:
az_inf_data_after_300 = az_inf_data.sel(draw=slice(300,None))
arviz.plot_pair(
az_inf_data_after_300,
marginals=True,
)
print("Reference good model:", my_receiver_function.good_model)
Reference good model: [ 8. 3. 20. 3.9 45. 4.4]
true_model = my_receiver_function.good_model
mean_sample = np.array(az_inf_data["posterior"].mean().to_array())
median_sample = np.array(az_inf_data["posterior"].median().to_array())
print("Mean of samples: ", mean_sample)
print("Reference good model:", true_model)
my_receiver_function.plot_model(true_model, mean_sample, "true_model", "mean_sample")
plt.legend();
Mean of samples: [ 7.15043311 2.96042893 21.45168694 3.86795829 42.58738856 4.44218487]
Reference good model: [ 8. 3. 20. 3.9 45. 4.4]
<matplotlib.legend.Legend object at 0x7f8250e61d80>
mean_sample_predicted_data = my_receiver_function.forward(mean_sample)
my_receiver_function.plot_data(
observed_data,
mean_sample_predicted_data,
"observed_data",
"mean_sample_predicted_data",
);
plt.legend();
<matplotlib.legend.Legend object at 0x7f82507bacb0>
Watermark#
watermark_list = ["cofi", "espresso", "numpy", "matplotlib", "emcee", "arviz"]
for pkg in watermark_list:
pkg_var = __import__(pkg)
print(pkg, getattr(pkg_var, "__version__"))
cofi 0.2.7
espresso 0.3.13
numpy 1.24.4
matplotlib 3.8.3
emcee 3.1.4
arviz 0.17.0
sphinx_gallery_thumbnail_number = -1
Total running time of the script: (4 minutes 41.741 seconds)