Source code for harmonica.isostasy

# Copyright (c) 2018 The Harmonica Developers.
# Distributed under the terms of the BSD 3-Clause License.
# SPDX-License-Identifier: BSD-3-Clause
# This code is part of the Fatiando a Terra project (
Function to calculate the thickness of the roots and antiroots assuming the
Airy isostatic hypothesis.
import numpy as np
import xarray as xr

[docs]def isostasy_airy( topography, density_crust=2.8e3, density_mantle=3.3e3, density_water=1e3, reference_depth=30e3, ): r""" Calculate the isostatic Moho depth from topography using Airy's hypothesis. According to the Airy hypothesis of isostasy, topography above sea level is supported by a thickening of the crust (a root) while oceanic basins are supported by a thinning of the crust (an anti-root). This assumption is usually .. figure:: ../../_static/figures/airy-isostasy.svg :align: center :width: 400px *Schematic of isostatic compensation following the Airy hypothesis.* The relationship between the topographic/bathymetric heights (:math:`h`) and the root thickness (:math:`r`) is governed by mass balance relations and can be found in classic textbooks like [TurcotteSchubert2014]_ and [Hofmann-WellenhofMoritz2006]_. On the continents (positive topographic heights): .. math :: r = \frac{\rho_{c}}{\rho_m - \rho_{c}} h while on the oceans (negative topographic heights): .. math :: r = \frac{\rho_{c} - \rho_w}{\rho_m - \rho_{c}} h in which :math:`h` is the topography/bathymetry, :math:`\rho_m` is the density of the mantle, :math:`\rho_w` is the density of the water, and :math:`\rho_{c}` is the density of the crust. The computed root thicknesses will be added to the given reference Moho depth (:math:`H`) to arrive at the isostatic Moho depth. Use ``reference_depth=0`` if you want the values of the root thicknesses instead. Parameters ---------- topography : array or :class:`xarray.DataArray` Topography height and bathymetry depth in meters. It is usually prudent to use floating point values instead of integers to avoid integer division errors. density_crust : float Density of the crust in :math:`kg/m^3`. density_mantle : float Mantle density in :math:`kg/m^3`. density_water : float Water density in :math:`kg/m^3`. reference_depth : float The reference Moho depth (:math:`H`) in meters. Returns ------- moho_depth : array or :class:`xarray.DataArray` The isostatic Moho depth in meters. """ # Need to cast to array to make sure numpy indexing works as expected for # 1D DataArray topography oceans = np.array(topography < 0) continent = np.logical_not(oceans) scale = np.full(topography.shape, np.nan, dtype="float") scale[continent] = density_crust / (density_mantle - density_crust) scale[oceans] = (density_crust - density_water) / (density_mantle - density_crust) moho = topography * scale + reference_depth if isinstance(moho, xr.DataArray): = "moho_depth" moho.attrs["isostasy"] = "Airy" moho.attrs["density_crust"] = str(density_crust) moho.attrs["density_mantle"] = str(density_mantle) moho.attrs["density_water"] = str(density_water) moho.attrs["reference_depth"] = str(reference_depth) return moho