def _smeared_abeles_constant(q, w, resolution, parallel=True):
"""
A kernel for fast and constant dQ/Q smearing
Parameters
----------
q: np.ndarray
Q values to evaluate the reflectivity at
w: np.ndarray
Parameters for the reflectivity model
resolution: float
Percentage dq/q resolution. dq specified as FWHM of a resolution
kernel.
parallel: bool, optional
Do you want to calculate in parallel? This option is only applicable if
you are using the ``_creflect`` module. The option is ignored if using
the pure python calculator, ``_reflect``.
Returns
-------
reflectivity: np.ndarray
The resolution smeared reflectivity
"""
if resolution < 0.5:
return refcalc.abeles(q, w, parallel=parallel)
resolution /= 100
gaussnum = 51
gaussgpoint = (gaussnum - 1) / 2
gauss = lambda x, s: (1. / s / np.sqrt(2 * np.pi)
* np.exp(-0.5 * x**2 / s / s))
lowq = np.min(q)
highq = np.max(q)
if lowq <= 0:
lowq = 1e-6
start = np.log10(lowq) - 6 * resolution / _FWHM
finish = np.log10(highq * (1 + 6 * resolution / _FWHM))
interpnum = np.round(np.abs(1 * (np.abs(start - finish))
/ (1.7 * resolution / _FWHM / gaussgpoint)))
xtemp = np.linspace(start, finish, int(interpnum))
xlin = np.power(10., xtemp)
# resolution smear over [-4 sigma, 4 sigma]
gauss_x = np.linspace(-1.7 * resolution, 1.7 * resolution, gaussnum)
gauss_y = gauss(gauss_x, resolution / _FWHM)
rvals = refcalc.abeles(xlin, w, parallel=parallel)
smeared_rvals = np.convolve(rvals, gauss_y, mode='same')
interpolator = InterpolatedUnivariateSpline(xlin, smeared_rvals)
smeared_output = interpolator(q)
# smeared_output *= np.sum(gauss_y)
smeared_output *= gauss_x[1] - gauss_x[0]
return smeared_output
def test_c_abeles(self):
if TEST_C_REFLECT:
# test reflectivity calculation with values generated from Motofit
calc = _creflect.abeles(self.qvals, self.layer_format)
assert_almost_equal(calc, self.rvals)
# test for non-contiguous Q values
tempq = self.qvals[0::5]
assert_(tempq.flags['C_CONTIGUOUS'] is False)
calc = _creflect.abeles(tempq, self.layer_format)
assert_almost_equal(calc, self.rvals[0::5])
def _smearkernel(x, w, q, dq, parallel):
"""
Kernel for adaptive Gaussian quadrature integration
Parameters
----------
x : float
Independent variable for integration.
w : array-like
The uniform slab model parameters in 'layer' form.
q : float
Nominal mean Q of normal distribution
dq : float
FWHM of a normal distribution.
Returns
-------
reflectivity : float
Model reflectivity multiplied by the probability density function
evaluated at a given distance, x, away from the mean Q value.
"""
prefactor = 1 / np.sqrt(2 * np.pi)
gauss = prefactor * np.exp(-0.5 * x * x)
localq = q + x * dq / _FWHM
return refcalc.abeles(localq, w, parallel=parallel) * gauss
def _smeared_abeles_fixed(qvals, w, dqvals, quad_order=17, workers=0):
"""
Resolution smearing that uses fixed order Gaussian quadrature integration
for the convolution.
Parameters
----------
qvals : array-like
The Q values for evaluation
w : array-like
The uniform slab model parameters in 'layer' form.
dqvals : array-like
dQ values corresponding to each value in `qvals`. Each dqval is the
FWHM of a Gaussian approximation to the resolution kernel.
quad-order : int, optional
Specify the order of the Gaussian quadrature integration for the
convolution.
workers: int, optional
Specifies the number of threads for parallel calculation. This
option is only applicable if you are using the ``_creflect``
module. The option is ignored if using the pure python calculator,
``_reflect``. If `workers == 0` then all available processors are
used.
Returns
-------
reflectivity : np.ndarray
The smeared reflectivity
"""
# get the gauss-legendre weights and abscissae
abscissa, weights = gauss_legendre(quad_order)
# get the normal distribution at that point
prefactor = 1. / np.sqrt(2 * np.pi)
def gauss(x):
return np.exp(-0.5 * x * x)
gaussvals = prefactor * gauss(abscissa * _INTLIMIT)
# integration between -3.5 and 3.5 sigma
va = qvals - _INTLIMIT * dqvals / _FWHM
vb = qvals + _INTLIMIT * dqvals / _FWHM
va = va[:, np.newaxis]
vb = vb[:, np.newaxis]
qvals_for_res = ((np.atleast_2d(abscissa) *
(vb - va) + vb + va) / 2.)
smeared_rvals = refcalc.abeles(qvals_for_res.flatten(),
w,
workers=workers)
smeared_rvals = np.reshape(smeared_rvals,
(qvals.size, abscissa.size))
smeared_rvals *= np.atleast_2d(gaussvals * weights)
return np.sum(smeared_rvals, 1) * _INTLIMIT
def test_c_abeles_reshape(self):
# c reflectivity should be able to deal with multidimensional input
if not TEST_C_REFLECT:
return
reshaped_q = np.reshape(self.qvals, (2, 250))
reshaped_r = self.rvals.reshape(2, 250)
calc = _creflect.abeles(reshaped_q, self.layer_format)
assert_equal(reshaped_r.shape, calc.shape)
assert_almost_equal(reshaped_r, calc, 15)
def test_compare_c_py_abeles0(self):
# test two layer system
if not TEST_C_REFLECT:
return
layer0 = np.array([[0, 2.07, 0.01, 3],
[0, 6.36, 0.1, 3]])
calc1 = _reflect.abeles(self.qvals, layer0, scale=0.99, bkg=1e-8)
calc2 = _creflect.abeles(self.qvals, layer0, scale=0.99, bkg=1e-8)
assert_almost_equal(calc1, calc2)
def test_compare_c_py_abeles(self):
# test python and c are equivalent
# but not the same file
if not HAVE_CREFLECT:
return
assert_(_reflect.__file__ != _creflect.__file__)
calc1 = _reflect.abeles(self.qvals, self.layer_format)
calc2 = _creflect.abeles(self.qvals, self.layer_format)
assert_almost_equal(calc1, calc2)
calc1 = _reflect.abeles(self.qvals, self.layer_format, scale=2.)
calc2 = _creflect.abeles(self.qvals, self.layer_format, scale=2.)
assert_almost_equal(calc1, calc2)
calc1 = _reflect.abeles(self.qvals, self.layer_format, scale=0.5,
bkg=0.1)
calc2 = _creflect.abeles(self.qvals, self.layer_format, scale=0.5,
bkg=0.1)
assert_almost_equal(calc1, calc2)
def test_compare_c_py_abeles2(self):
# test two layer system
if not HAVE_CREFLECT:
return
layer2 = np.array([[0, 2.07, 0.01, 3],
[10, 3.47, 0.01, 3],
[100, 1.0, 0.01, 4],
[0, 6.36, 0.1, 3]])
calc1 = _reflect.abeles(self.qvals, layer2, scale=0.99, bkg=1e-8)
calc2 = _creflect.abeles(self.qvals, layer2, scale=0.99, bkg=1e-8)
assert_almost_equal(calc1, calc2)
def test_cabeles_parallelised(self):
# I suppose this could fail if someone doesn't have a multicore computer
if not HAVE_CREFLECT:
return
coefs = np.array([[0, 0, 0, 0],
[300, 3, 1e-3, 3],
[10, 3.47, 1e-3, 3],
[0, 6.36, 0, 3]])
x = np.linspace(0.01, 0.2, 1000000)
pstart = time.time()
_creflect.abeles(x, coefs, parallel=True)
pfinish = time.time()
sstart = time.time()
_creflect.abeles(x, coefs, parallel=False)
sfinish = time.time()
assert_(0.7 * (sfinish - sstart) > (pfinish - pstart))
def test_compare_c_py_abeles(self):
# test python and c are equivalent
# but not the same file
if not TEST_C_REFLECT:
return
assert_(_reflect.__file__ != _creflect.__file__)
calc1 = _reflect.abeles(self.qvals, self.layer_format)
calc2 = _creflect.abeles(self.qvals, self.layer_format)
assert_almost_equal(calc1, calc2)
calc1 = _reflect.abeles(self.qvals, self.layer_format, scale=2.)
calc2 = _creflect.abeles(self.qvals, self.layer_format, scale=2.)
assert_almost_equal(calc1, calc2)
calc1 = _reflect.abeles(self.qvals, self.layer_format, scale=0.5,
bkg=0.1)
# workers = 1 is a non-threaded implementation
calc2 = _creflect.abeles(self.qvals, self.layer_format, scale=0.5,
bkg=0.1, workers=1)
# workers = 2 forces the calculation to go through multithreaded calcn,
# even on single core processor
calc3 = _creflect.abeles(self.qvals, self.layer_format, scale=0.5,
bkg=0.1, workers=2)
assert_almost_equal(calc1, calc2)
assert_almost_equal(calc1, calc3)
def abeles(q, layers, scale=1, bkg=0., workers=0):
r"""
Abeles matrix formalism for calculating reflectivity from a stratified
medium.
Parameters
----------
q : array_like
the q values required for the calculation.
:math:`Q = \frac{4\pi}{\lambda}\sin(\Omega)`.
Units = Angstrom**-1
layers : np.ndarray
coefficients required for the calculation, has shape (2 + N, 4),
where N is the number of layers
* layers[0, 1] - SLD of fronting (/ 1e-6 Angstrom**-2)
* layers[0, 2] - iSLD of fronting (/ 1e-6 Angstrom**-2)
* layers[N, 0] - thickness of layer N
* layers[N, 1] - SLD of layer N (/ 1e-6 Angstrom**-2)
* layers[N, 2] - iSLD of layer N (/ 1e-6 Angstrom**-2)
* layers[N, 3] - roughness between layer N-1/N
* layers[-1, 1] - SLD of backing (/ 1e-6 Angstrom**-2)
* layers[-1, 2] - iSLD of backing (/ 1e-6 Angstrom**-2)
* layers[-1, 3] - roughness between backing and last layer
scale : float
Multiply all reflectivities by this value.
bkg : float
Linear background to be added to all reflectivities
workers : int
Specifies the number of threads for parallel calculation. This option
is only applicable if you are using the ``_creflect`` module. The
option is ignored if using the pure python calculator, ``_reflect``.
If `workers == 0` then all available processors are used.
Returns
-------
Reflectivity: np.ndarray
Calculated reflectivity values for each q value.
"""
return refcalc.abeles(q, layers, scale=scale, bkg=bkg, workers=workers)
def abeles(q, layers, scale=1, bkg=0., parallel=True):
"""
Abeles matrix formalism for calculating reflectivity from a stratified
medium.
Parameters
----------
q: array_like
the q values required for the calculation.
Q = 4 * Pi / lambda * sin(omega).
Units = Angstrom**-1
layers: np.ndarray
coefficients required for the calculation, has shape (2 + N, 4),
where N is the number of layers
* layers[0, 1] - SLD of fronting (/ 1e-6 Angstrom**-2)
* layers[0, 2] - iSLD of fronting (/ 1e-6 Angstrom**-2)
* layers[N, 0] - thickness of layer N
* layers[N, 1] - SLD of layer N (/ 1e-6 Angstrom**-2)
* layers[N, 2] - iSLD of layer N (/ 1e-6 Angstrom**-2)
* layers[N, 3] - roughness between layer N-1/N
* layers[-1, 1] - SLD of backing (/ 1e-6 Angstrom**-2)
* layers[-1, 2] - iSLD of backing (/ 1e-6 Angstrom**-2)
* layers[-1, 3] - roughness between backing and last layer
scale: float
Multiply all reflectivities by this value.
bkg: float
Linear background to be added to all reflectivities
parallel: bool
Do you want to calculate in parallel? This option is only applicable if
you are using the ``_creflect`` module. The option is ignored if using
the pure python calculator, ``_reflect``.
Returns
-------
Reflectivity: np.ndarray
Calculated reflectivity values for each q value.
"""
return refcalc.abeles(q, layers, scale=scale, bkg=bkg, parallel=parallel)
def reflectivity(q, coefs, *args, **kwds):
"""
Abeles matrix formalism for calculating reflectivity from a stratified
medium.
Parameters
----------
q : np.ndarray
The qvalues required for the calculation. Q=4*Pi/lambda * sin(omega).
Units = Angstrom**-1
coefs : np.ndarray
* coefs[0] = number of layers, N
* coefs[1] = scale factor
* coefs[2] = SLD of fronting (/1e-6 Angstrom**-2)
* coefs[3] = iSLD of fronting (/1e-6 Angstrom**-2)
* coefs[4] = SLD of backing
* coefs[5] = iSLD of backing
* coefs[6] = background
* coefs[7] = roughness between backing and layer N
* coefs[4 * (N - 1) + 8] = thickness of layer N in Angstrom (layer 1 is
closest to fronting)
* coefs[4 * (N - 1) + 9] = SLD of layer N (/ 1e-6 Angstrom**-2)
* coefs[4 * (N - 1) + 10] = iSLD of layer N (/ 1e-6 Angstrom**-2)
* coefs[4 * (N - 1) + 11] = roughness between layer N and N-1.
kwds : dict, optional
The following keys are used:
'dqvals' - float or np.ndarray, optional
If dqvals is a float, then a constant dQ/Q resolution smearing is
employed. For 5% resolution smearing supply 5.
If `dqvals` is the same shape as q, then the array contains the
FWHM of a Gaussian approximated resolution kernel. Point by point
resolution smearing is employed. Use this option if dQ/Q varies
across your dataset.
If `dqvals.ndim == q.ndim + 2` and
`q.shape == dqvals[..., -3].shape` then an individual resolution
kernel is applied to each measurement point. This resolution kernel
is a probability distribution function (PDF). `dqvals` will have the
shape (qvals.shape, M, 2). There are `M` points in the kernel.
`dqvals[..., 0]` holds the q values for the kernel, `dqvals[..., 1]`
gives the corresponding probability.
'quad_order' - int, optional
the order of the Gaussian quadrature polynomial for doing the
resolution smearing. default = 17. Don't choose less than 13. If
quad_order == 'ultimate' then adaptive quadrature is used. Adaptive
quadrature will always work, but takes a _long_ time (2 or 3 orders
of magnitude longer). Fixed quadrature will always take a lot less
time. BUT it won't necessarily work across all samples. For example,
13 points may be fine for a thin layer, but will be atrocious at
describing a multilayer with bragg peaks.
'parallel': bool, optional
Do you want to calculate in parallel? This option is only
applicable if you are using the ``_creflect`` module. The option is
ignored if using the pure python calculator, ``_reflect``. The
default is `True`.
"""
parallel=True
if 'parallel' in kwds:
parallel = kwds['parallel']
qvals = q
quad_order = 17
scale = coefs[1]
bkg = coefs[6]
if not is_proper_abeles_input(coefs):
raise ValueError('The size of the parameter array passed to reflectivity'
' should be 4 * coefs[0] + 8')
# make into form suitable for reflection calculation
w = coefs_to_layer(coefs)
if 'quad_order' in kwds:
quad_order = kwds['quad_order']
if 'dqvals' in kwds and kwds['dqvals'] is not None:
dqvals = kwds['dqvals']
# constant dq/q smearing
if isinstance(dqvals, numbers.Real):
dqvals = float(dqvals)
return (scale * _smeared_abeles_constant(qvals,
w,
dqvals,
parallel=parallel)) + bkg
# point by point resolution smearing
if dqvals.size == qvals.size:
dqvals_flat = dqvals.flatten()
qvals_flat = q.flatten()
# adaptive quadrature
if quad_order == 'ultimate':
smeared_rvals = (scale *
_smeared_abeles_adaptive(qvals_flat,
w,
dqvals_flat,
parallel=parallel) + bkg)
return smeared_rvals.reshape(q.shape)
# fixed order quadrature
else:
smeared_rvals = (scale * _smeared_abeles_fixed(
qvals_flat,
w,
dqvals_flat,
quad_order=quad_order,
parallel=parallel)
+ bkg)
return np.reshape(smeared_rvals, q.shape)
# resolution kernel smearing
elif (dqvals.ndim == qvals.ndim + 2
and dqvals.shape[0: qvals.ndim] == qvals.shape):
# TODO may not work yet.
qvals_for_res = dqvals[..., 0]
# work out the reflectivity at the kernel evaluation points
smeared_rvals = refcalc.abeles(qvals_for_res,
w,
scale=coefs[1],
bkg=coefs[6],
parallel=parallel)
# multiply by probability
smeared_rvals *= dqvals[..., 1]
# now do simpson integration
return scipy.integrate.simps(smeared_rvals, x=dqvals[..., 0])
# no smearing
return refcalc.abeles(q,
w,
scale=coefs[1],
bkg=coefs[6],
parallel=parallel)