#!/usr/bin/env python
# -*- coding: UTF-8 -*-
# Copyright (c) 2011-2023, wradlib developers.
# Distributed under the MIT License. See LICENSE.txt for more info.
"""
Attenuation Correction
^^^^^^^^^^^^^^^^^^^^^^
.. autosummary::
:nosignatures:
:toctree: generated/
{}
"""
__all__ = [
"AttenuationOverflowError",
"correct_attenuation_hb",
"constraint_dbz",
"constraint_pia",
"correct_attenuation_constrained",
"correct_radome_attenuation_empirical",
"pia_from_kdp",
"AttenMethods",
]
__doc__ = __doc__.format("\n ".join(__all__))
import logging
from functools import singledispatch
import numpy as np
from scipy import interpolate, ndimage
from xarray import DataArray, apply_ufunc
from wradlib import trafo, zr
from wradlib.util import XarrayMethods, docstring
logger = logging.getLogger("attcorr")
[docs]class AttenuationOverflowError(Exception):
"""Attenuation Overflow Error
Exception, if attenuation exceeds ``thrs`` and no handling ``mode`` is set.
"""
[docs]def correct_attenuation_hb(
gateset,
*,
coefficients=None,
mode="except",
thrs=59.0,
):
"""Gate-by-Gate attenuation correction according to \
:cite:`Hitschfeld1954`
Parameters
----------
gateset : :py:class:`numpy:numpy.ndarray`
multidimensional array. The range gates (over which iteration has to
be performed) are supposed to vary along
the *last* dimension so, e.g., for a set of `l` radar images stored in
polar form with `m` azimuths and `n` range-bins the input array's
shape can be either (l,m,n) or (m,l,n)
data has to be provided in decibel representation of reflectivity [dBZ]
coefficients : dict
- a : float
proportionality factor of the k-Z relation (:math:`k=a \\cdot Z^{b}`).
Per default set to 1.67e-4.
- b : float
exponent of the k-Z relation ( :math:`k=a \\cdot Z^{b}` ). Per default
set to 0.7.
- gate_length : float
length of a range gate [km]. Per default set to 1.0.
mode : str
controls how the function reacts, if the sum of signal and attenuation
exceeds the threshold ``thrs``
Possible values:
- 'warn' : emit a warning through the module's logger but continue
execution
- 'zero' : set offending gates to 0.0
- 'nan' : set offending gates to nan
- 'except': raise an AttenuationOverflowError exception
Any other mode will also raise the Exception.
thrs : float
threshold, for the sum of attenuation and signal, which is considered
implausible.
Returns
-------
pia : :py:class:`numpy:numpy.ndarray`
Array with the same shape as ``gateset`` containing the calculated
attenuation [dB] for each range gate.
Raises
------
:exc:`wradlib.atten.AttenuationOverflowError`
Examples
--------
See :ref:`/notebooks/attenuation/attenuation.ipynb#Hitschfeld-and-Bordan`.
"""
if coefficients is None:
coefficients = {"a": 0.000167, "b": 0.7, "gate_length": 1.0}
a = coefficients["a"]
b = coefficients["b"]
gate_length = coefficients["gate_length"]
pia = np.empty(gateset.shape)
pia[..., 0] = 0.0
ksum = 0.0
# multidimensional version
# assumes that iteration is only along the last dimension
# (i.e. range gates) all other dimensions are calculated simultaneously
# to gain some speed
for gate in range(gateset.shape[-1] - 1):
# calculate k in dB/km from k-Z relation
# c.f. Krämer2008(p. 147)
k = a * (10.0 ** ((gateset[..., gate] + ksum) / 10.0)) ** b * 2.0 * gate_length
# k = 10**(log10(a)+0.1*bin*b)
# dBkn = 10*math.log10(a) + (bin+ksum)*b + 10*math.log10(2*gate_length)
ksum += k
pia[..., gate + 1] = ksum
# stop-criterion, if corrected reflectivity is larger than 59 dBZ
overflow = (gateset[..., gate + 1] + ksum) > thrs
if np.any(overflow):
if mode == "warn":
logger.warning(f"corrected signal over threshold ({thrs:3.1f})")
elif mode == "nan":
pia[..., gate + 1][overflow] = np.nan
elif mode == "zero":
pia[..., gate + 1][overflow] = 0.0
else:
raise AttenuationOverflowError
return pia
[docs]def constraint_dbz(gateset, pia, thrs_dbz):
"""Constraint callback function for correct_attenuation_constrained.
Selects beams, in which at least one pixel exceeds ``thrs_dbz`` [dBZ].
"""
return np.max(gateset + pia, axis=-1) > thrs_dbz
[docs]def constraint_pia(gateset, pia, thrs_pia):
"""Constraint callback function for correct_attenuation_constrained.
Selects beams, in which the path integrated attenuation exceeds
``thrs_pia``.
"""
return np.max(pia, axis=-1) > thrs_pia
def calc_attenuation_forward(gateset, *, a=1.67e-4, b=0.7, gate_length=1.0):
"""Gate-by-Gate forward correction as described in :cite:`Kraemer2008`
Parameters
----------
gateset : :class:`numpy:numpy.ndarray`
Multidimensional array, where the range gates (over which iteration has
to be performed) are supposed to vary along the last array-dimension.
Data has to be provided in decibel representation of reflectivity
[dBZ].
a : float
proportionality factor of the k-Z relation (:math:`k=a \\cdot Z^{b}`).
Per default set to 1.67e-4.
b : float
exponent of the k-Z relation ( :math:`k=a \\cdot Z^{b}` ). Per default
set to 0.7.
gate_length : float
length of a range gate [km]. Per default set to 1.0.
Returns
-------
pia : :class:`numpy:numpy.ndarray`
Array with the same shape as ``gateset`` containing the calculated path
integrated attenuation [dB] for each range gate.
"""
pia = np.zeros(gateset.shape)
for gate in range(gateset.shape[-1] - 1):
k = (
a
* trafo.idecibel(gateset[..., gate] + pia[..., gate]) ** b
* 2.0
* gate_length
)
pia[..., gate + 1] = pia[..., gate] + k
return pia
def bisect_reference_attenuation(
gateset,
pia_ref,
*,
a_max=1.67e-4,
a_min=2.33e-5,
b_start=0.7,
gate_length=1.0,
mode="difference",
thrs=0.25,
max_iterations=10,
):
"""Find the optimal attenuation coefficients for a gateset to achieve a \
given reference attenuation using the forward correction algorithm in \
combination with the bisection method.
Parameters
----------
gateset : :class:`numpy:numpy.ndarray`
Multidimensional array, where the range gates (over which iteration has
to be performed) are supposed to vary along the last array-dimension.
Data has to be provided in decibel representation of reflectivity
[dBZ].
pia_ref : :class:`numpy:numpy.ndarray`
Array of the same number of dimensions as ``gateset``, but the size of
the last dimension is 1, as it constitutes the reference pia [dB] of
the last range gate of every beam.
a_max : float
Upper bound of the bisection interval within the linear coefficient ``a``
of the k-Z relation has to be. ( :math:`k=a \\cdot Z^{b}` ).
Per default set to 1.67e-4.
a_min : float
Lower bound of the bisection interval within the linear coefficient ``a``
of the k-Z relation has to be. ( :math:`k=a \\cdot Z^{b}` ).
Per default set to 2.33e-5.
b_start : float
Initial value for exponential coefficient of the k-Z relation
( :math:`k=a \\cdot Z^{b}` ). This value will be lowered incremental
by 0.01 if no solution was found within the bisection interval of
``a_max`` and ``a_min`` within the number of given iterations
``max_iterations``.
Per default set to 0.7.
gate_length : float
Radial length of a range gate [km].
Per default set to 1.0.
mode : str
{‘ratio’ or ‘difference’}
Kind of tolerance of calculated pia in relation to reference pia.
Per default set to 'difference'.
thrs : float
Value of the tolerance to stop bisection iteration successful. It is
recommended to choose 0.05 for ratio ``mode`` and 0.25 for difference
``mode``, which means a deviation tolerance of 5% or 0.25 dB,
respectively.
Per default set to 0.25.
max_iterations : int
Number of bisection iteration before the exponential coefficient b of
the k-Z relation will be decreased and the bisection starts again.
Per default set to 10.
Returns
-------
pia : :class:`numpy:numpy.ndarray`
Array with the same shape as ``gateset`` containing the calculated path
integrated attenuation [dB] for each range gate.
a_mid : :class:`numpy:numpy.ndarray`
Array with the same shape as ``pia_ref`` containing the finally used
linear k-Z relation coefficient a for successful pia calculation.
b : :class:`numpy:numpy.ndarray`
Array with the same shape as ``pia_ref`` containing the finally used
exponential k-Z relation coefficient b for successful pia calculation.
"""
# Prepare arrays of initial k-Z relation coefficients for each beam.
a_hi = np.ones(pia_ref.shape) * a_max # np.repeat(a_max, pia_ref.shape)
a_lo = np.ones(pia_ref.shape) * a_min # np.repeat(a_min, pia_ref.shape)
b = np.ones(pia_ref.shape) * b_start # np.repeat(b_start, pia_ref.shape)
pia = np.empty_like(gateset)
iteration_count = 0
# Iterate until upper and lower bounds of linear k-Z relation coefficients
# for pia calculation are the same.
while not np.all(a_hi == a_lo):
a_mid = (a_hi + a_lo) / 2
pia = calc_attenuation_forward(gateset, a=a_mid, b=b, gate_length=gate_length)
# Find indices where calculated and reference pia sufficiently match
if mode == "difference":
overshoot = (pia[..., -1] - pia_ref) > thrs
undershoot = (pia[..., -1] - pia_ref) < -thrs
hit = (np.abs(pia[..., -1] - pia_ref)) < thrs
elif mode == "ratio":
overshoot = ((pia[..., -1] - pia_ref) / pia_ref) > thrs
undershoot = ((pia[..., -1] - pia_ref) / pia_ref) < -thrs
hit = (np.abs(pia[..., -1] - pia_ref) / pia_ref) < thrs
else:
raise Exception(f"Unknown mode type {mode}.")
# Define new bounds of linear k-Z relation coefficient for over- and
# undershooting pia calculations.
a_hi[overshoot] = a_mid[overshoot]
a_lo[undershoot] = a_mid[undershoot]
a_hi[hit] = a_mid[hit]
a_lo[hit] = a_mid[hit]
iteration_count += 1
# Change exponential k-Z relation coefficient in case of maximum
# iterations for linear k-Z relation coefficient are reached.
if iteration_count > max_iterations:
b[overshoot] -= 0.01
b[undershoot] += 0.01
return pia, a_mid, b
def _sector_filter(mask, min_sector_size):
"""Calculate an array of same shape as mask, which is set to 1 in case of \
at least min_sector_size adjacent values, otherwise it is set to 0.
"""
kernela = np.ones([1] * (mask.ndim - 1) + [min_sector_size])
kernelb = np.ones((min_sector_size,))
forward_origin = -(min_sector_size - (min_sector_size // 2)) + min_sector_size % 2
backward_origin = (min_sector_size - (min_sector_size // 2)) - 1
forward_sum = ndimage.correlate1d(
mask.astype(np.int_), kernelb, axis=-1, mode="wrap", origin=forward_origin
)
backward_sum = ndimage.correlate1d(
mask.astype(np.int_), kernelb, axis=-1, mode="wrap", origin=backward_origin
)
forward_corners = forward_sum == min_sector_size
backward_corners = backward_sum == min_sector_size
forward_large_sectors = np.zeros_like(mask)
backward_large_sectors = np.zeros_like(mask)
for iii in range(mask.shape[0]):
forward_large_sectors[iii] = ndimage.binary_dilation(
forward_corners[iii], kernela[0], origin=forward_origin
).astype(int)
backward_large_sectors[iii] = ndimage.binary_dilation(
backward_corners[iii], kernela[0], origin=backward_origin
).astype(int)
return forward_large_sectors | backward_large_sectors
def _interp_atten(pia, invalidbeams):
"""Interpolate reference pia of most distant rangebin of small invalid
sectors as a prerequisite for the backward calculation of attenuation.
"""
# Build a spatial equidistant array for interpolation of the ahead and
# behind extended temporary pia-array for handling invalid sectors
# overlapping the seam of the radarcircle.
x = np.arange(3 * pia.shape[1])
for i in range(pia.shape[0]):
sub_invalid = invalidbeams[i, :]
sub_pia = pia[i, :, -1]
# Build the extended bool-array with the invalid sectors.
extended_invalid = np.concatenate([sub_invalid] * 3)
# Build the extended pia-array.
extended_pia = np.concatenate([sub_pia] * 3)
# Build interpolation class.
intp = interpolate.interp1d(
x[~extended_invalid], extended_pia[~extended_invalid], kind="linear"
)
# Interpolate where sectors are invalid.
pia[i, sub_invalid, -1] = intp(x[pia.shape[1] : 2 * pia.shape[1]][sub_invalid])
[docs]@singledispatch
def correct_attenuation_constrained(
gateset,
*,
a_max=1.67e-4,
a_min=2.33e-5,
n_a=4,
b_max=0.7,
b_min=0.65,
n_b=6,
gate_length=1.0,
constraints=None,
constraint_args=None,
sector_thr=10,
):
"""Gate-by-Gate attenuation correction based on the iterative approach of \
:cite:`Kraemer2008` and :cite:`Jacobi2016` with a generalized and \
scalable number of constraints.
Differing from the original approach, the method for addressing
small sectors which conflict with the constraints is based on a bisection
forward calculating method, and not on backwards attenuation calculation.
Parameters
----------
gateset : :class:`numpy:numpy.ndarray`
Multidimensional array, where the range gates (over which iteration has
to be performed) are supposed to vary along the last array-dimension
and the azimuths are supposed to vary along the next to last
array-dimension.
Data has to be provided in decibel representation of reflectivity
[dBZ].
a_max : float
Initial value for linear coefficient of the k-Z relation
( :math:`k=a \\cdot Z^{b}` ).
Per default set to 1.67e-4.
a_min : float
Minimal allowed linear coefficient of the k-Z relation
( :math:`k=a \\cdot Z^{b}` ) in the downwards iteration of 'a' in case
of breaching one of thresholds ``constr_args`` of the optional
conditions ``constraints``.
Per default set to 2.33e-5.
n_a : int
Number of iterations from ``a_max`` to ``a_min``.
Per default set to 4.
b_max : float
Initial value for exponential coefficient of the k-Z relation
( :math:`k=a \\cdot Z^{b}` ).
Per default set to 0.7.
b_min : float
Minimal allowed exponential coefficient of the k-Z relation
( :math:`k=a \\cdot Z^{b}` ) in the downwards iteration of 'b' in case
of breaching one of thresholds ``constr_args`` of the optional
conditions ``constraints`` and the linear coefficient 'a' has already
reached the lower limit ``a_min``.
Per default set to 0.65.
n_b : int
Number of iterations from ``b_max`` to ``b_min``.
Per default set to 6.
gate_length : float
Radial length of a range gate [km].
Per default set to 1.0.
constraints : list
List of constraint functions. The signature of these functions has to
be constraint_function(`gateset`, `k`, `*constr_args`). Their return
value must be a boolean array of shape `gateset.shape[:-1]` set to True
for beams, which do not fulfill the constraint.
constraint_args : list
List of lists, which are to be passed to the individual constraint
functions using the `*args` mechanism
(len(constr_args) == len(constraints)).
sector_thr : int
Number of adjacent beams, for which in case of breaching the
constraints the attenuation with downward iterated ``a`` and ``b`` -
parameters is recalculated. For more narrow sectors the integrated
attenuation of the last gate is interpolated and used as reference
for the recalculation.
Returns
-------
pia : :class:`numpy:numpy.ndarray`
Array with the same shape as ``gateset`` containing the calculated path
integrated attenuation [dB] for each range gate.
Examples
--------
Implementing the original Hitschfeld & Bordan (1954) algorithm with
otherwise default parameters::
from wradlib.atten import *
pia = correct_attenuation_constrained(gateset, a_max=8.e-5,
b_max=0.731, n_a=1, n_b=1,
gate_length=1.0)
Implementing the basic Kraemer algorithm::
pia = atten.correct_attenuation_constrained(gateset, a_max=1.67e-4,
a_min=2.33e-5, n_a=100,
b_max=0.7, b_min=0.65,
n_b=6, gate_length=1.,
constraints=
[wrl.atten.constraint_dbz],
constraint_args=[[59.0]])
Implementing the PIA algorithm by Jacobi et al.::
pia = atten.correct_attenuation_constrained(gateset, a_max=1.67e-4,
a_min=2.33e-5, n_a=100,
b_max=0.7, b_min=0.65,
n_b=6, gate_length=1.,
constraints=
[wrl.atten.constraint_dbz,
wrl.atten.constraint_pia],
constraint_args=
[[59.0],[20.0]])
"""
if constraints is None:
constraints = []
if constraint_args is None:
constraint_args = []
n_az = gateset.shape[-2]
n_rng = gateset.shape[-1]
tmp_gateset = gateset.reshape((-1, n_az, n_rng))
pia = np.zeros_like(tmp_gateset)
a_used = np.empty(tmp_gateset.shape[:-1])
b_used = np.empty(tmp_gateset.shape[:-1])
# Calculate attenuation forward.
# Indexing all rows of last dimension (radarbeams)
beams2correct = np.where(np.ones(tmp_gateset.shape[:-1], dtype=np.bool_))
small_sectors = np.zeros(tmp_gateset.shape[:-1], dtype=np.bool_)
if n_a != 1:
delta_a = (a_max - a_min) / (n_a - 1)
else:
delta_a = 0.0
if n_b != 1:
delta_b = (b_max - b_min) / (n_b - 1)
else:
delta_b = 0.0
# Iterate over possible b-parameters
for j in range(n_b):
b = b_max - delta_b * j
# Iterate over possible a-parameters
for i in range(n_a):
a = a_max - delta_a * i
# Generate subset of beams that have to be corrected
sub_gateset = tmp_gateset[beams2correct]
sub_pia = calc_attenuation_forward(
sub_gateset, a=a, b=b, gate_length=gate_length
)
pia[beams2correct] = sub_pia
a_used[beams2correct] = a
b_used[beams2correct] = b
# Indexing threshold exceeding beams
incorrectbeams = np.zeros(tmp_gateset.shape[:-1], dtype=np.bool_)
for constraint, constraint_arg in zip(constraints, constraint_args):
incorrectbeams = np.logical_or(
incorrectbeams, constraint(tmp_gateset, pia, *constraint_arg)
)
# Determine incorrect sectors larger than sector_thr
large_sectors = _sector_filter(incorrectbeams, sector_thr)
# Determine incorrect sectors smaller than sector_thr
small_sectors = np.logical_or(
small_sectors, (incorrectbeams & ~large_sectors)
)
beams2correct = np.where(large_sectors)
if len(pia[beams2correct]) == 0:
break
if len(pia[beams2correct]) == 0:
break
if np.any(small_sectors):
# Interpolate reference pia of most distant
# rangebin of invalid sectors.
_interp_atten(pia, small_sectors)
# Calculate attenuation forward by achieving reference
# attenuation based on bisection-method.
tmp_pia, tmp_a, tmp_b = bisect_reference_attenuation(
tmp_gateset[small_sectors, :],
pia[small_sectors, -1],
a_max=a_max,
a_min=a_min,
b_start=b_max,
gate_length=gate_length,
mode="difference",
thrs=0.25,
max_iterations=10,
)
pia[small_sectors, :] = tmp_pia
a_used[small_sectors] = tmp_a
b_used[small_sectors] = tmp_b
return pia.reshape(gateset.shape)
@correct_attenuation_constrained.register(DataArray)
def _correct_attenuation_constrained_xarray(obj, **kwargs):
dim0 = obj.wrl.util.dim0()
out = apply_ufunc(
correct_attenuation_constrained,
obj,
input_core_dims=[[dim0, "range"]],
output_core_dims=[[dim0, "range"]],
dask="parallelized",
kwargs=kwargs,
dask_gufunc_kwargs=dict(allow_rechunk=True),
)
out.name = "correct_attenuation_constrained"
return out
[docs]def correct_radome_attenuation_empirical(
gateset, *, frequency=5.64, hydrophobicity=0.165, n_r=2, stat=np.mean
):
"""Estimate two-way wet radome losses.
Empirical function of frequency and rainfall rate for both standard and
hydrophobic radomes based on the approach of :cite:`Merceret2000`.
Parameters
----------
gateset : :class:`numpy:numpy.ndarray`
Multidimensional array, where the range gates (over which
iteration has to be performed) are supposed to vary along the
last array-dimension and the azimuths are supposed to vary
along the next to last array-dimension. Data has to be provided
in decibel representation of reflectivity [dBZ].
frequency : float
Radar-frequency [GHz]:
Standard frequencies in X-band range between 8.0 and 12.0 GHz,
Standard frequencies in C-band range between 4.0 and 8.0 GHz,
Standard frequencies in S-band range between 2.0 and 4.0 GHz.
Be aware that the empirical fit of the formula was just
done for C- and S-band. The use for X-band is probably an
undue extrapolation.
Per default set to 5.64 as used by the German Weather
Service radars.
hydrophobicity : float
Empirical parameter based on the hydrophobicity of the radome
material.
- 0.165 for standard radomes,
- 0.0575 for hydrophobic radomes.
Per default set to 0.165.
n_r : int
The radius of rangebins within the rain-intensity is
statistically evaluated as the representative rain-intensity
over radome.
stat : callable
A numpy function for statistical aggregation of the
central rangebins defined by n_r.
Potential options: :func:`numpy:numpy.mean`, :func:`numpy:numpy.median`,
:func:`numpy:numpy.max`, :func:`numpy:numpy.min`.
Returns
-------
k : :class:`numpy:numpy.ndarray`
Array with the same shape as ``gateset`` containing the
calculated two-way transmission loss [dB] for each range gate.
In case the input array (gateset) contains NaNs the
corresponding beams of the output array (k) will be set as NaN,
too.
"""
# Select rangebins inside the defined center-range n_r.
center = gateset[..., :n_r].reshape(-1, n_r * gateset.shape[-2])
center_m = np.ma.masked_array(center, np.isnan(center))
# Calculate rainrate in the center-range based on statistical method stat
# and with standard ZR-relation.
rain_over_radome = zr.z_to_r(trafo.idecibel(stat(center_m, axis=-1)))
# Estimate the empirical two-way transmission loss due to
# radome-attenuation.
k = 2 * hydrophobicity * rain_over_radome * np.tanh(frequency / 10.0) ** 2
# Reshape the result to gateset-shape.
k = np.repeat(k, gateset.shape[-1] * gateset.shape[-2]).reshape(gateset.shape)
return k.filled(fill_value=np.nan)
[docs]def pia_from_kdp(kdp, dr, *, gamma=0.08):
"""Retrieving path integrated attenuation from specific differential \
phase (Kdp).
The default value of gamma is based on :cite:`Carey2000`.
Parameters
----------
kdp : :class:`numpy:numpy.ndarray`
array specific differential phase
Range dimension must be the last dimension.
dr : float
gate length (km)
gamma : float
linear coefficient (default value: 0.08) in the relation between
Kdp phase and specific attenuation (alpha)
Returns
-------
output : :class:`numpy:numpy.ndarray`
array of same shape as kdp containing the path integrated attenuation
"""
alpha = gamma * kdp
return 2 * np.cumsum(alpha, axis=-1) * dr
[docs]class AttenMethods(XarrayMethods):
"""wradlib xarray SubAccessor methods for Atten Methods."""
[docs] @docstring(correct_attenuation_constrained)
def correct_attenuation_constrained(self, *args, **kwargs):
if not isinstance(self, AttenMethods):
return correct_attenuation_constrained(self, *args, **kwargs)
else:
return correct_attenuation_constrained(self._obj, *args, **kwargs)
if __name__ == "__main__":
print("wradlib: Calling module <atten> as main...")
