Add new tilt_angle transformation function (#486)

Add new `tilt_angle` transformation function that computes the tilt from a given a grid of potential field data. Add the function to the API index. Include tests and an example for the new function. --------- Co-authored-by: Leonardo Uieda <[email protected]> Co-authored-by: Santiago Soler <[email protected]>
fatiando · Aug 12, 2024 · c001529 · c001529
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diff --git a/doc/api/index.rst b/doc/api/index.rst
@@ -33,6 +33,7 @@ Apply well known transformations regular gridded potential fields data.
     gaussian_highpass
     reduction_to_pole
     total_gradient_amplitude
+    tilt_angle
 
 Frequency domain filters
 ------------------------

diff --git a/doc/references.rst b/doc/references.rst
@@ -11,6 +11,7 @@ References
 .. [Grombein2013] Grombein, T., Seitz, K., Heck, B. (2013), Optimized formulas for the gravitational field of a tesseroid, Journal of Geodesy. doi:`10.1007/s00190-013-0636-1 <https://doi.org/10.1007/s00190-013-0636-1>`__
 .. [Hofmann-WellenhofMoritz2006] Hofmann-Wellenhof, B., & Moritz, H. (2006). Physical Geodesy (2nd, corr. ed. 2006 edition ed.). Wien ; New York: Springer.
 .. [LiGotze2001] Li, X. and H. J. Gotze, 2001, Tutorial: Ellipsoid, geoid, gravity, geodesy, and geophysics, Geophysics, 66(6), p. 1660-1668, doi:`10.1190/1.1487109 <https://doi.org/10.1190/1.1487109>`__
+.. [MillerSingh1994] Miller, H. G., & Singh, V. (1994). Potential field tilt — A new concept for location of potential field sources. Journal of Applied Geophysics, 32(3), 213–217. doi:`10.1016/0926-9851(94)90002-7 <https://doi.org/10.1016/0926-9851(94)90002-7>`__
 .. [Nagy2000] Nagy, D., Papp, G. & Benedek, J.(2000). The gravitational potential and its derivatives for the prism. Journal of Geodesy 74: 552. doi:`10.1007/s001900000116 <https://doi.org/10.1007/s001900000116>`__
 .. [Nagy2002] Nagy, D., Papp, G. & Benedek, J.(2002). Corrections to "The gravitational potential and its derivatives for the prism". Journal of Geodesy. doi:`10.1007/s00190-002-0264-7 <https://doi.org/10.1007/s00190-002-0264-7>`__
 .. [Oliveira2021] Oliveira Jr, Vanderlei C. and Uieda, Leonardo and Barbosa, Valeria C. F.. Sketch of three coordinate systems: Geocentric Cartesian, Geocentric Geodetic, and Topocentric Cartesian. figshare. doi: `10.6084/m9.figshare.15044241.v1 <https://doi.org/10.6084/m9.figshare.15044241.v1>`__

diff --git a/examples/transformations/tilt.py b/examples/transformations/tilt.py
@@ -0,0 +1,136 @@
+# 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 (https://www.fatiando.org)
+#
+"""
+Tilt of a regular grid
+======================
+"""
+import ensaio
+import pygmt
+import verde as vd
+import xarray as xr
+import xrft
+
+import harmonica as hm
+
+# Fetch magnetic grid over the Lightning Creek Sill Complex, Australia using
+# Ensaio and load it with Xarray
+fname = ensaio.fetch_lightning_creek_magnetic(version=1)
+magnetic_grid = xr.load_dataarray(fname)
+
+# Pad the grid to increase accuracy of the FFT filter
+pad_width = {
+    "easting": magnetic_grid.easting.size // 3,
+    "northing": magnetic_grid.northing.size // 3,
+}
+# drop the extra height coordinate
+magnetic_grid_no_height = magnetic_grid.drop_vars("height")
+magnetic_grid_padded = xrft.pad(magnetic_grid_no_height, pad_width)
+
+# Compute the tilt of the grid
+tilt_grid = hm.tilt_angle(magnetic_grid_padded)
+
+# Unpad the tilt grid
+tilt_grid = xrft.unpad(tilt_grid, pad_width)
+
+# Show the tilt
+print("\nTilt:\n", tilt_grid)
+
+# Define the inclination and declination of the region by the time of the data
+# acquisition (1990).
+inclination, declination = -52.98, 6.51
+
+# Apply a reduction to the pole over the magnetic anomaly grid. We will assume
+# that the sources share the same inclination and declination as the
+# geomagnetic field.
+rtp_grid_padded = hm.reduction_to_pole(
+    magnetic_grid_padded, inclination=inclination, declination=declination
+)
+
+# Unpad the reduced to the pole grid
+rtp_grid = xrft.unpad(rtp_grid_padded, pad_width)
+
+# Compute the tilt of the padded rtp grid
+tilt_rtp_grid = hm.tilt_angle(rtp_grid_padded)
+
+# Unpad the tilt grid
+tilt_rtp_grid = xrft.unpad(tilt_rtp_grid, pad_width)
+
+# Show the tilt from RTP
+print("\nTilt from RTP:\n", tilt_rtp_grid)
+
+# Plot original magnetic anomaly, its RTP, and the tilt of both
+region = (
+    magnetic_grid.easting.values.min(),
+    magnetic_grid.easting.values.max(),
+    magnetic_grid.northing.values.min(),
+    magnetic_grid.northing.values.max(),
+)
+fig = pygmt.Figure()
+with fig.subplot(
+    nrows=2,
+    ncols=2,
+    subsize=("20c", "20c"),
+    sharex="b",
+    sharey="l",
+    margins=["1c", "1c"],
+):
+    scale = 0.5 * vd.maxabs(magnetic_grid, rtp_grid)
+    with fig.set_panel(panel=0):
+        # Make colormap of data
+        pygmt.makecpt(cmap="polar+h", series=[-scale, scale], background=True)
+        # Plot magnetic anomaly grid
+        fig.grdimage(
+            grid=magnetic_grid,
+            projection="X?",
+            cmap=True,
+            frame=["a", "+tTotal field anomaly grid"],
+        )
+    with fig.set_panel(panel=1):
+        # Make colormap of data
+        pygmt.makecpt(cmap="polar+h", series=[-scale, scale], background=True)
+        # Plot reduced to the pole magnetic anomaly grid
+        fig.grdimage(
+            grid=rtp_grid,
+            projection="X?",
+            cmap=True,
+            frame=["a", "+tReduced to the pole (RTP)"],
+        )
+        # Add colorbar
+        label = "nT"
+        fig.colorbar(
+            frame=f"af+l{label}",
+            position="JMR+o1/-0.25c+e",
+        )
+
+    scale = 0.6 * vd.maxabs(tilt_grid, tilt_rtp_grid)
+    with fig.set_panel(panel=2):
+        # Make colormap for tilt (saturate it a little bit)
+        pygmt.makecpt(cmap="polar+h", series=[-scale, scale], background=True)
+        # Plot tilt
+        fig.grdimage(
+            grid=tilt_grid,
+            projection="X?",
+            cmap=True,
+            frame=["a", "+tTilt of total field anomaly grid"],
+        )
+    with fig.set_panel(panel=3):
+        # Make colormap for tilt rtp (saturate it a little bit)
+        pygmt.makecpt(cmap="polar+h", series=[-scale, scale], background=True)
+        # Plot tilt
+        fig.grdimage(
+            grid=tilt_rtp_grid,
+            projection="X?",
+            cmap=True,
+            frame=["a", "+tTilt of RTP grid"],
+        )
+        # Add colorbar
+        label = "rad"
+        fig.colorbar(
+            frame=f"af+l{label}",
+            position="JMR+o1/-0.25c+e",
+        )
+fig.show()
diff --git a/harmonica/__init__.py b/harmonica/__init__.py
@@ -28,6 +28,7 @@
     gaussian_highpass,
     gaussian_lowpass,
     reduction_to_pole,
+    tilt_angle,
     total_gradient_amplitude,
     upward_continuation,
 )

diff --git a/harmonica/_transformations.py b/harmonica/_transformations.py
@@ -343,7 +343,7 @@ def reduction_to_pole(
 
 def total_gradient_amplitude(grid):
     r"""
-    Calculate the total gradient amplitude a magnetic field grid
+    Calculates the total gradient amplitude of a potential field grid
 
     Compute the total gradient amplitude of a regular gridded potential field
     `M`. The horizontal derivatives are calculated though finite-differences
@@ -393,6 +393,68 @@ def total_gradient_amplitude(grid):
     return np.sqrt(gradient[0] ** 2 + gradient[1] ** 2 + gradient[2] ** 2)
 
 
+def tilt_angle(grid):
+    r"""
+    Calculates the tilt angle of a potential field grid
+
+    Compute the tilt of a regular gridded potential field
+    :math:`M`. The horizontal derivatives are calculated
+    through finite-differences while the upward derivative
+    is calculated using FFT.
+
+    Parameters
+    ----------
+    grid : :class:`xarray.DataArray`
+        A two dimensional :class:`xarray.DataArray` whose coordinates are
+        evenly spaced (regular grid). Its dimensions should be in the following
+        order: *northing*, *easting*. Its coordinates should be defined in the
+        same units.
+
+    Returns
+    -------
+    tilt_grid : :class:`xarray.DataArray`
+        A :class:`xarray.DataArray` with the calculated tilt
+        in radians.
+
+    Notes
+    -----
+    The tilt is calculated as:
+
+    .. math::
+
+        \text{tilt}(f) = \tan^{-1} \left(
+            \frac{
+                \frac{\partial M}{\partial z}
+            }{
+                \sqrt{
+                    \left( \frac{\partial M}{\partial x} \right)^2
+                    +
+                    \left( \frac{\partial M}{\partial y} \right)^2
+                }
+            }
+        \right)
+
+    where :math:`M` is the regularly gridded potential field.
+
+    References
+    ----------
+    [Blakely1995]_
+    [MillerSingh1994]_
+    """
+    # Run sanity checks on the grid
+    grid_sanity_checks(grid)
+    # Calculate the gradients of the grid
+    gradient = (
+        derivative_easting(grid, order=1),
+        derivative_northing(grid, order=1),
+        derivative_upward(grid, order=1),
+    )
+    # Calculate and return the tilt
+    horiz_deriv = np.sqrt(gradient[0] ** 2 + gradient[1] ** 2)
+    tilt = np.arctan2(gradient[2], horiz_deriv)
+    return tilt
+
+
 def _get_dataarray_coordinate(grid, dimension_index):
     """
     Return the name of the easting or northing coordinate in the grid

diff --git a/harmonica/tests/test_transformations.py b/harmonica/tests/test_transformations.py
@@ -25,6 +25,7 @@
     gaussian_highpass,
     gaussian_lowpass,
     reduction_to_pole,
+    tilt_angle,
     total_gradient_amplitude,
     upward_continuation,
 )
@@ -601,6 +602,77 @@ def test_invalid_grid_with_nans(self, sample_potential):
             total_gradient_amplitude(sample_potential)
 
 
+class TestTilt:
+    """
+    Test tilt function
+    """
+
+    def test_against_synthetic(
+        self, sample_potential, sample_g_n, sample_g_e, sample_g_z
+    ):
+        """
+        Test tilt function against the synthetic model
+        """
+        # Pad the potential field grid to improve accuracy
+        pad_width = {
+            "easting": sample_potential.easting.size // 3,
+            "northing": sample_potential.northing.size // 3,
+        }
+        # need to drop upward coordinate (bug in xrft)
+        potential_padded = xrft.pad(
+            sample_potential.drop_vars("upward"),
+            pad_width=pad_width,
+        )
+        # Calculate the tilt and unpad it
+        tilt_grid = tilt_angle(potential_padded)
+        tilt_grid = xrft.unpad(tilt_grid, pad_width)
+        # Compare against g_tilt (trim the borders to ignore boundary effects)
+        trim = 6
+        tilt_grid = tilt_grid[trim:-trim, trim:-trim]
+        g_e = sample_g_e[trim:-trim, trim:-trim]
+        g_n = sample_g_n[trim:-trim, trim:-trim]
+        g_z = sample_g_z[trim:-trim, trim:-trim]
+        g_horiz_deriv = np.sqrt(g_e**2 + g_n**2)
+        g_tilt = np.arctan2(
+            -g_z, g_horiz_deriv
+        )  # use -g_z to use the _upward_ derivative
+        rms = root_mean_square_error(tilt_grid, g_tilt)
+        assert rms / np.abs(tilt_grid).max() < 0.1
+
+    def test_invalid_grid_single_dimension(self):
+        """
+        Check if tilt raises error on grid with single
+        dimension
+        """
+        x = np.linspace(0, 10, 11)
+        y = x**2
+        grid = xr.DataArray(y, coords={"x": x}, dims=("x",))
+        with pytest.raises(ValueError, match="Invalid grid with 1 dimensions."):
+            tilt_angle(grid)
+
+    def test_invalid_grid_three_dimensions(self):
+        """
+        Check if tilt raises error on grid with three
+        dimensions
+        """
+        x = np.linspace(0, 10, 11)
+        y = np.linspace(-4, 4, 9)
+        z = np.linspace(20, 30, 5)
+        xx, yy, zz = np.meshgrid(x, y, z)
+        data = xx + yy + zz
+        grid = xr.DataArray(data, coords={"x": x, "y": y, "z": z}, dims=("y", "x", "z"))
+        with pytest.raises(ValueError, match="Invalid grid with 3 dimensions."):
+            tilt_angle(grid)
+
+    def test_invalid_grid_with_nans(self, sample_potential):
+        """
+        Check if tilt raises error if grid contains nans
+        """
+        sample_potential.values[0, 0] = np.nan
+        with pytest.raises(ValueError, match="Found nan"):
+            tilt_angle(sample_potential)
+
+
 class Testfilter:
     """
     Test filter result against the output from oasis montaj