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# Choclo

Kernel functions for your geophysical models

**Choclo** is a Python library that hosts optimized forward modelling and
kernel functions for running geophysical forward and inverse models, intended
to be used by other libraries as the underlying layer of their computation.
"Choclo" is a term used in some countries of South America to refer to corn,
originated from the `quechua
`__
word *chuqllu*.
.. seealso::
Choclo is a part of the
`Fatiando a Terra `__ project.
Overview
--------
Choclo provides slim and optimized function to compute the gravitational and
magnetic fields of simple geometries like point masses, magnetic dipoles and
prisms. It also provides the *kernel* functions needed to run compute those
fields. The goal of Choclo is to provide developers of a simple and efficient
way to calculate these fields for a wide range of applications, like forward
modellings, sensitivity matrices calculations, equivalent sources
implementations and more.
These functions are not designed to be used by final users. Instead they are
meant to be part of the underlaying engine of a higher level codebase, like
`Harmonica `__.
All Choclo functions rely on `Numba `__ for
just-in-time compilations, meaning that there's no need to distribute
precompiled code: Choclo provides pure Python code that gets compiled during
runtime allowing to run them as fast as they were written in C.
Moreover, developers could harness the power of Numba to parallelize processes
in a quick and easy way.
Conventions
-----------
Before you jump into Choclo's functions, it's worth noting some conventions
that will be kept along its codebase:
- The functions assume a right-handed coordinate system. We avoid using names
like "x", "y" and "z" for the coordinates. Instead we use "easting",
"northing" and "upward" to make extra clear the direction of each axis.
- We use the first letter of the *easting*, *northing* and *upward* axis to
indicate direction of derivatives. For example, a function ``gravity_e`` will
compute the *easting* component of the gravitional acceleration, while the
``gravity_n`` and ``gravity_u`` will compute the *northing* and *upward*
ones, respectively.
- The arguments of the functions are always assumed in SI units. And all the
functions return results also in SI units. Choclo **doesn't** perform **unit
conversions**.
- The components of the gravitational accelerations and the magnetic fields are
computed in the same direction of the *easting*, *northing* and *upward*
axis. So ``gravity_u`` will compute the **upward** component of the
gravitational acceleration (note the difference with the **downward**
component).
The library
-----------
Choclo is divided in a few different submodules, each with different goals. The
three main modules are the ones that host the forward and kernel functions for
the different geometries supported by Choclo: ``point``, ``dipole`` and
``prism``. Inside each one of these modules we will find forward modelling
functions and potentially some kernel functions. The names of the forward
modelling functions follow a simple pattern of ``{field}_{type}``. For
example, :func:`choclo.prism.gravity_e` computes the easting component of the
gravitational acceleration of a prism, while :func:`choclo.prism.gravity_ee`
computes the easting-easting gravity tensor component.
How to use Choclo
-----------------
The simplest case
~~~~~~~~~~~~~~~~~
Using Choclo is very simple, but it requires some work from our side. Let's say
we need to compute the upward component of the gravitational acceleration that
a single rectangular prism produces on a single computation point. To do so we
can just call the :func:`choclo.prism.gravity_u` function:
.. jupyter-execute::
import numpy as np
from choclo.prism import gravity_u
# Define a single computation point
easting, northing, upward = 0.0, 0.0, 10.0
# Define the boundaries of the prism as a 1d-array
prism = np.array([-10.0, 10.0, -7.0, 7.0, -15.0, -5.0])
# And its density
density = 400.0
# Compute the upward component of the grav. acceleration
g_u = gravity_u(easting, northing, upward, prism, density)
g_u
But this case is very simple: we usually deal with multiple sources and
multiple computation points.
Multiple sources and computation points
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In case we have a collection of prisms with certain densities:
.. jupyter-execute::
prisms = np.array([
[-10.0, 0.0, -7.0, 0.0, -15.0, -10.0],
[-10.0, 0.0, 0.0, 7.0, -25.0, -15.0],
[0.0, 10.0, -7.0, 0.0, -20.0, -13.0],
[0.0, 10.0, 0.0, 7.0, -12.0, -8.0],
])
densities = np.array([200.0, 300.0, -100.0, 400.0])
And a set of observation points:
.. jupyter-execute::
easting = np.linspace(-5.0, 5.0, 21)
northing = np.linspace(-4.0, 4.0, 21)
easting, northing = tuple(a.ravel() for a in np.meshgrid(easting, northing))
upward = 10 * np.ones_like(easting)
coordinates = (easting, northing, upward)
And we want to compute the gratitational acceleration that those prisms
generate on each observation point, we need to write some kind of loop that
computes the effect of each prism on each observation point and adds it to
a running result.
A possible solution would be to use Python for loops:
.. jupyter-execute::
def gravity_upward_slow(coordinates, prisms, densities):
"""
Compute the upward component of the acceleration of a set of prisms
"""
# Unpack coordinates of the observation points
easting, northing, upward = coordinates[:]
# Initialize a result array full of zeros
result = np.zeros_like(easting, dtype=np.float64)
# Compute the upward component that every prism generate on each
# observation point
for i in range(len(easting)):
for j in range(prisms.shape[0]):
result[i] += gravity_u(
easting[i], northing[i], upward[i], prisms[j, :], densities[j]
)
return result
g_u = gravity_upward_slow(coordinates, prisms, densities)
g_u
For loops are known to be slow, and in case we are working with very large
models and a large number of computation points these calculations could take
too long. So this solution is not recommended.
.. important::
Using Python for loops to run Choclo's functions is not advisable!
We can write a much faster and efficient solution relying on :mod:`numba`.
Since every function in Choclo is being JIT compiled, we can safely include
calls to these functions inside other JIT compiled functions. So we can write
an alterantive function by adding a `@numba.jit` decorator:
.. jupyter-execute::
import numba
@numba.jit(nopython=True)
def gravity_upward_jit(coordinates, prisms, densities):
"""
Compute the upward component of the acceleration of a set of prisms
"""
# Unpack coordinates of the observation points
easting, northing, upward = coordinates[:]
# Initialize a result array full of zeros
result = np.zeros_like(easting, dtype=np.float64)
# Compute the upward component that every prism generate on each
# observation point
for i in range(len(easting)):
for j in range(prisms.shape[0]):
result[i] += gravity_u(
easting[i], northing[i], upward[i], prisms[j, :], densities[j]
)
return result
g_u = gravity_upward_jit(coordinates, prisms, densities)
g_u
Let's benchmark these two functions to see how much faster the decorated
function runs:
.. jupyter-execute::
%timeit gravity_upward_slow(coordinates, prisms, densities)
.. jupyter-execute::
%timeit gravity_upward_jit(coordinates, prisms, densities)
From these numbers we can see that we have significantly reduced the
computation time by several factors by just decorating our function.
.. note::
The benchmarked times may vary if you run them in your system.
.. seealso::
Check `How to measure the performance of Numba?
`__
to learn more about how to properly benchmark jitted functions.
Parallelizing our runs
~~~~~~~~~~~~~~~~~~~~~~
We have already shown how we can reduce the computation times of our forward
modelling by decorating our functions with `@numba.jit(nopython=True)`. But
this is just the first step: all the computations were being run in *serial* in
a single CPU. We can harness the full power of our modern multiprocessors CPUs
by parallelizing our runs. To do so we need to use the :func:`numba.prange`
instead of the regular Python `range` function and slightly change the
decorator of our function by adding a `parallel=True`:
.. jupyter-execute::
import numba
@numba.jit(nopython=True, parallel=True)
def gravity_upward_parallel(coordinates, prisms, densities):
"""
Compute the upward component of the acceleration of a set of prisms
"""
# Unpack coordinates of the observation points
easting, northing, upward = coordinates[:]
# Initialize a result array full of zeros
result = np.zeros_like(easting, dtype=np.float64)
# Compute the upward component that every prism generate on each
# observation point
for i in numba.prange(len(easting)):
for j in range(prisms.shape[0]):
result[i] += gravity_u(
easting[i], northing[i], upward[i], prisms[j, :], densities[j]
)
return result
g_u = gravity_upward_parallel(coordinates, prisms, densities)
g_u
With :func:`numba.prange` we can specify which loop we want to run in parallel.
Since we are updating the values of ``results`` on each iteration, it's
advisable to parallelize the loop over the observation points.
By setting ``parallel=True`` in the decorator we are telling Numba to
pararellize this function, otherwise Numba will reinterpret the ``numba.prange``
function as a regular ``range`` and run this loop in serial.
.. note::
In some applications it's desirable that our forward models are run in
serial. For example, if they are part of larger problem that gets
parallelized at a higher level. The ``parallel`` parameter in the
``numba.jit`` decorator allows us to change this behaviour at will without
having to modify the function code.
Let's benchmark this function against the non-parallelized
``gravity_upward_jit``:
.. jupyter-execute::
%timeit gravity_upward_jit(coordinates, prisms, densities)
.. jupyter-execute::
%timeit gravity_upward_parallel(coordinates, prisms, densities)
By distributing the load between multiple processors we were capable of
lowering the computation time by a few more factors.
.. toctree::
:maxdepth: 1
:hidden:
:caption: Reference Documentation
api/index.rst
citing.rst
references.rst
compatibility.rst
.. toctree::
:maxdepth: 1
:hidden:
:caption: Community
Join the community
How to contribute
Code of Conduct
Source code on GitHub
The Fatiando a Terra project