Note
Go to the end to download the full example code.
Receiver Function inversion: optimization and Bayesian sampling#
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 receiver function inversion problem with optimisation and sampling.
0. Import modules#
# -------------------------------------------------------- #
# #
# Uncomment below to set up environment on "colab" #
# #
# -------------------------------------------------------- #
# !pip install -U cofi pyrf96
# If this notebook is run locally pyrf96 needs to be installed separately by uncommenting the following lines,
# that is by removing the # and the white space between it and the exclamation mark.
# !pip install pyrf96
import os
import math
import numpy as np
import matplotlib.pyplot as plt
import arviz
import emcee
import multiprocessing
import cofi
import pyrf96
# 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 package in
`pyrf96 <inlab-geo/pyrf96>`__, with calls to
Fortran routines developed by Prof. T. Shibutani in the backend.
The function pyrf96.rfcalc() computes synthetic receiver functions
(using routines developed by Prof. T. Shibutani). It is called as
follows:
t,rfunc = pyrf96.rfcalc(model)
where model is a NumPy array of dimension [nlayers,3].
Model formats#
Three input model formats are possible depending on optional parameter
mtype.
Voronoi cell mode (mtype=0, default): In this case
model[:,0] are the depths of Voronoi cell nuclei, which each define
a layer of constant velocity. Here interfaces are defined implicitely as
the mid-points between successive nuclei. For example, the interface at
the base of the first layer lies at depth
0.5*[model[0,0]+model(1,0)] The S-wave speeds for each layer are
given by model[:,1], and model[:,2] are the ratio of P-wave
speed to S-wave speed in each layer. This format is convenient as a
parameterization for optimization, or Bayesian sampling, since nuclei
are independent and can be unordered. See Figure 1 of Bodin et
al. 2012.
Layer thickness mode (mtype=1): In this case model[:,0] are
the thicknesses of each layer. Here interfaces are defined implicitely
as the sum of successive thicknesses. For example, the interface at the
base of the second layer lies at depth model[0,0]+model(1,0). The
S-wave speeds for each layer are given by model[:,1], and
model[:,2] are the ratio of P-wave speed to S-wave speed in each
layer.
Interface depth mode (mtype=2): In this case 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 P-wave speed to S-wave speed. The maximum
depth of discontinuity that can be considered is 60km.
For full details see the docstring help(pyrf96.rfcalc)
For example, here is the Earth model in interface depth mode
(mtype=0).
good_model = np.array([[1,3.0,1.7],
[8.0,3.2,2.0],
[20, 4.0,1.7],
[45,4.2,1.7]])
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 use the model to calculate the receiver function both with and
without correlated noise added, with signal to noise parameter
sn=0.5. (The noise here is added in the frequency domain, for
details see Shibutani etal. (1996).)
# calculate and plot receiver function
t,rfunc = pyrf96.rfcalc(good_model) # Receiver function
t2,rfunc_noise = pyrf96.rfcalc(good_model,sn=0.5,seed=12345678) # Receiver function with added correlated noise
If we had field data we wanted to invert we would load here and simply
assign it to t2 and rfunc_noise that is replace our synthetic
data.
Now we plot the model and the receiver function together.
# plot RF together with noise
a,b = pyrf96.plotRFm(good_model,t2,rfunc_noise,t,rfunc,
vmin=2.8,
vmax=4.3,
dmin=-2.0,
dmax=60.0,
plotnuclei=True,
title="Observed and predicted receiver functions")
1. Solve with an optimiser#
Here we decide to only vary the layer velocities and interface depths
only, and not the Vp/Vs ratio parameters in the Earth model. So we have
to provide a subset of the model vector to the minimization routine. We
define a simple conversion between the two with routines
get_inversion_parameters() and its converse
get_model_parameters().
# conversion routines
vpvs = good_model[:,2] # fixed vpvs parameters
def get_inversion_parameters(fullmodel): # convert model to inversion parameters
return fullmodel[:,:2].flatten() # remove last column and flatten
#
def get_model_parameters(invmodel): # convert inversion parameters to fullmodel
return np.append(invmodel.reshape(len(vpvs),-1), vpvs[:,None],axis=1) # reshape and add last column
In preparation for optimisation we need to measure the fit of this observed receiver function to the receiver function predicted by any other model, \({\mathbf d}_{pred}\), we need to compute the negative log of the Likelihood
where \(C_D^{-1}\) is the inverse data covariance matrix describing
correlated noise in the receiver function. You can do this using the
supplied routine rfcalc.InvDataCov which takes arguments
(\(\sigma_t\),\(\sigma_a\),\(n\)), and builds an
\(n\times n\) inverse data covariance matrix for a time signal with
amplitude variance \(\sigma_a^2\) and time correlation length
\(\sigma_t\), for receiver length of \(n\) time samples. In this
example you can use \(\sigma_t = 2.5\), \(\sigma_a=0.01\).
observed_data = rfunc_noise # Define observed data
Cdinv = pyrf96.InvDataCov(2.5,0.01,len(rfunc)) # Define inverse data covaraince for the noisy receiver function
def my_misfit(imodel): # misfit function to be minimized
model = get_model_parameters(imodel) # recover full model from inversion parameters
t,predicted_data = pyrf96.rfcalc(model) # predicted data from model
res = observed_data-predicted_data # residual between observed and predicted data
misfit_val = np.dot(res,np.transpose(np.dot(Cdinv, res)))/2.0 # data misfit
if math.isnan(misfit_val):
return float("inf")
return misfit_val
my_problem = cofi.BaseProblem()
Here we set a starting model that is reasonably close to the true model, so that the optimisation converges.
initialmodel = np.array([[1.1,3.0,1.7],
[7.0,3.2,2.0],
[18, 4.1,1.7],
[40,4.2,1.7]])
# problem set up in cofi
my_problem.set_objective(my_misfit)
my_problem.set_initial_model(get_inversion_parameters(initialmodel))
my_problem.summary()
=====================================================================
Summary for inversion problem: BaseProblem
=====================================================================
Model shape: (8,)
---------------------------------------------------------------------
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']
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
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: 419.71850229451877
nit: 396
nfev: 669
status: 0
message: Optimization terminated successfully.
final_simplex: (array([[ 0.87245028, 3.00727864, 7.81940018, 3.09629075, 19.61880806,
3.93653803, 40.04743158, 4.38172023],
[ 0.87245009, 3.00727871, 7.81940033, 3.09629082, 19.6188087 ,
3.9365381 , 40.04743237, 4.38172023],
[ 0.87245006, 3.00727869, 7.81940022, 3.09629084, 19.61880895,
3.9365381 , 40.04743188, 4.38172029],
[ 0.87245004, 3.00727872, 7.81940081, 3.09629083, 19.61880814,
3.93653808, 40.047433 , 4.38172023],
[ 0.87244991, 3.00727874, 7.81940057, 3.09629091, 19.61880899,
3.93653814, 40.04743282, 4.3817203 ],
[ 0.87244989, 3.00727879, 7.81940105, 3.09629088, 19.61880845,
3.93653811, 40.04743295, 4.38172025],
[ 0.8724502 , 3.00727867, 7.8194 , 3.09629083, 19.61880877,
3.93653804, 40.04743213, 4.38172021],
[ 0.87245027, 3.00727864, 7.81939976, 3.09629087, 19.61880926,
3.93653802, 40.04743142, 4.38172014],
[ 0.87245059, 3.00727854, 7.81939969, 3.09629076, 19.61880808,
3.93653793, 40.04743115, 4.38172001]]), array([419.71850229, 419.71850229, 419.71850229, 419.71850229,
419.71853557, 419.71853557, 419.7185529 , 419.7185529 ,
419.71858611]))
model: [ 0.87245028 3.00727864 7.81940018 3.09629075 19.61880806 3.93653803
40.04743158 4.38172023]
my_result_model = get_model_parameters(my_result_optimiser.model)
print("Inversion result: \n", my_result_model)
print("Reference good model:\n", good_model)
Inversion result:
[[ 0.87245028 3.00727864 1.7 ]
[ 7.81940018 3.09629075 2. ]
[19.61880806 3.93653803 1.7 ]
[40.04743158 4.38172023 1.7 ]]
Reference good model:
[[ 1. 3. 1.7]
[ 8. 3.2 2. ]
[20. 4. 1.7]
[45. 4.2 1.7]]
t,predicted_data = pyrf96.rfcalc(my_result_model) # predicted data from model
a,b = pyrf96.plotRFm(my_result_model,t,observed_data,t,predicted_data,
vmin=2.8,
vmax=4.5,
dmin=-2.0,
dmax=60.0,
title="Observed and best fit receiver functions",
velmod2=good_model,
plotnuclei = False,
modlabels=['Best fit','True'])
2. Bayesian sampling#
In preparation for sampling:
def my_log_likelihood(model): # log-likelihood of negative of our misfit
return -my_misfit(model)
def my_log_prior(model): # define a uniform prior between bounds on interface depths and S-velocity
real_model = get_model_parameters(model) # recover full model from inversion parameters
depths_in_0_60 = all([m_p < 60 and m_p > 0 for m_p in good_model[:,0]]) # are all interface depths between 0,60km?
veloc_in_range = all([m_p < 4.5 and m_p > 2 for m_p in good_model[:,1]]) # are all velocities depths between 2,4.5km/s?
if depths_in_0_60 and veloc_in_range: # return log of prior normalization
return (-np.log(60.)-np.log(2.5)).item()
return float("-inf")
ndim = len(get_inversion_parameters(good_model))
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: 8
---------------------------------------------------------------------
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 = 32
nsteps = 25000
walkers_start = my_result_optimiser.model + 1e-1 * np.random.randn(nwalkers, ndim)
We can run emcee in parallel. Some additional preparation:
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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8%|▊ | 1968/25000 [00:55<10:27, 36.68it/s]
8%|▊ | 1972/25000 [00:55<10:31, 36.46it/s]
8%|▊ | 1976/25000 [00:55<10:33, 36.32it/s]
8%|▊ | 1980/25000 [00:55<10:37, 36.09it/s]
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inv_result_sampler.summary()
============================
Summary for inversion result
============================
SUCCESS
----------------------------
sampler: <emcee.ensemble.EnsembleSampler object>
blob_names: ['log_likelihood', 'log_prior']
var_names = [
"depth1 (km)",
"velocity1 (km_s)",
"depth2 (km)",
"velocity2 (km_s)",
"depth3 (km)",
"velocity3 (km_s)",
"depth4 (km)",
"velocity4 (km_s)",
]
true_model = get_inversion_parameters(good_model)
az_inf_data = inv_result_sampler.to_arviz(var_names=var_names)
az_inf_data
pc = arviz.plot_trace_dist(
az_inf_data,
var_names=var_names,
visuals={"xlabel_trace": False, "trace": {"color": "C0", "lw": 0.5}, "dist": {"color": "C0", "lw": 0.5}},
figure_kwargs={"figsize": (10, 15), "constrained_layout": True},
)
n_vars = len(var_names)
data_var_names = list(az_inf_data.posterior.data_vars)
for i in range(n_vars):
ax1 = pc.iget_target(i, 0)
ax2 = pc.iget_target(i, 1)
ax1.axvline(true_model[i], linestyle="dotted", color="red")
ax1.set_title(data_var_names[i])
ax1.set_xlabel("parameter value")
ax1.set_ylabel("density value")
ax2.axhline(true_model[i], linestyle="dotted", color="red")
ax2.set_title(data_var_names[i])
ax2.set_xlabel("number of iterations")
ax2.set_ylabel("parameter value")
ax2.margins(x=0)
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). Usually for reliability we want the length of the chains to be greater that fifty times the largest autocorrelation time of any parameter.
tau = inv_result_sampler.sampler.get_autocorr_time()
print(f"autocorrelation times for each parameter:\n {tau}")
tau_ok = all([tcorr < 50*nsteps for tcorr in tau]) # were the chains long enough?
if(tau_ok):
print("All autocorrelation times OK")
else:
print("Autocorrelation unreliable. Increase length of chains and rerun")
autocorrelation times for each parameter:
[119.54823309 110.77320458 119.81772829 113.49578147 117.18562993
110.51644309 121.76554558 110.0721425 ]
All autocorrelation times OK
Let’s discard the initial 500 steps and make a corner plot:
import arviz_base
arviz_base.rcParams["plot.max_subplots"] = 80
az_inf_data_after_500 = az_inf_data.sel(draw=slice(300, None))
pm = arviz.plot_pair(
az_inf_data_after_500,
marginal=True,
triangle="lower",
figure_kwargs={"figsize": (20, 18)},
visuals={"scatter": {"s": 2}},
)
# Add reference dots for true model values
ref_values = list(true_model)
n = len(ref_values)
for i in range(n):
for j in range(n):
try:
ax = pm.iget_target(i, j)
except (ValueError, IndexError):
continue
if i == j:
ax.axvline(ref_values[i], color="yellow", linestyle="--", lw=1, alpha=0.7)
elif i > j:
ax.plot(
ref_values[j], ref_values[i], "o",
color="yellow", markeredgecolor="k", markeredgewidth=1, ms=5, zorder=5,
)
plt.tight_layout()
Now examine the mean model of the posterior ensemble.
mean_sample = np.array(az_inf_data["posterior"].ds[var_names].mean().to_array())
median_sample = np.array(az_inf_data["posterior"].ds[var_names].median().to_array())
my_mean_model = get_model_parameters(mean_sample)
t,mean_sample_predicted_rf = pyrf96.rfcalc(my_mean_model) # Receiver function
a,b = pyrf96.plotRFm(my_mean_model,t,observed_data,t,mean_sample_predicted_rf,
vmin=2.8, vmax=4.5, dmin=-2.0,dmax=48.0,
title="Observed and best fit receiver functions",
velmod2=good_model,modlabels=['Sample mean','True'])
print("Mean of samples:\n ", mean_sample)
print("Reference good model:\n", true_model)
Mean of samples:
[ 2.16212382 3.00802461 6.5372577 3.0959895 20.90090609 3.93643209
38.79678286 4.38095031]
Reference good model:
[ 1. 3. 8. 3.2 20. 4. 45. 4.2]
Watermark#
watermark_list = ["cofi", "numpy", "matplotlib", "emcee", "arviz"]
for pkg in watermark_list:
pkg_var = __import__(pkg)
print(pkg, getattr(pkg_var, "__version__"))
cofi 0.2.11+71.gb28b5b0
numpy 2.2.6
matplotlib 3.10.9
emcee 3.1.6
arviz 1.1.0
sphinx_gallery_thumbnail_number = -1
Total running time of the script: (12 minutes 11.620 seconds)