.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/seismology/plot_03_omori_aftershock_decay.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_seismology_plot_03_omori_aftershock_decay.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_seismology_plot_03_omori_aftershock_decay.py:


Omori's aftershock decay (1894)
===============================

Fusakichi Omori counted the aftershocks of the 1891 Nobi earthquake and found
that their rate fell as 1/(t + c). Tokuji Utsu later generalized the exponent
to p, usually close to 1. On log-log axes the rate is flat for t << c and has
slope -p for t >> c. The parameters here are synthetic.

.. GENERATED FROM PYTHON SOURCE LINES 10-36



.. image-sg:: /api/gallery/seismology/images/sphx_glr_plot_03_omori_aftershock_decay_001.png
   :alt: Aftershock rate decays as a power law
   :srcset: /api/gallery/seismology/images/sphx_glr_plot_03_omori_aftershock_decay_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.seismology import omori_rate
    from earthsciencekit.units import convert

    day = convert(1, "day", "s")
    time = np.logspace(-3, 3, 300) * day
    c, k = 0.05 * day, 200.0
    fig, ax = plt.subplots()
    for p in [0.9, 1.0, 1.2]:
        rate = omori_rate(k * day ** (p - 1), c, p, time) * day
        ax.loglog(time / day, rate, label=f"p = {p}")
        late = time > 100 * c
        slope = np.polyfit(np.log(time[late] + c), np.log(rate[late]), 1)[0]
        assert np.isclose(slope, -p)

    # Omori's original law integrates to a logarithm: N(T) = K ln(1 + T / c).
    t = np.linspace(0, 30 * day, 30001)
    count = np.trapezoid(omori_rate(k, c, 1.0, t), t)
    assert np.isclose(count, k * np.log1p(30 * day / c), rtol=1e-4)
    ax.set(xlabel="time after main shock [day]", ylabel="aftershocks per day", title="Aftershock rate decays as a power law")
    ax.legend()

    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.232 seconds)


.. _sphx_glr_download_api_gallery_seismology_plot_03_omori_aftershock_decay.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../lite/lab/index.html?path=api/gallery/seismology/plot_03_omori_aftershock_decay.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_03_omori_aftershock_decay.ipynb <plot_03_omori_aftershock_decay.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_03_omori_aftershock_decay.py <plot_03_omori_aftershock_decay.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_03_omori_aftershock_decay.zip <plot_03_omori_aftershock_decay.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_