angular dispersity (Trac #776) #123
Description
Regarding angular dispersity, I had some discussions with a user measuring oriented samples and a theorist simulating disorder in orientation. We concluded that something like the following would be useful:
Given the orientation vector along the axis of the shape as described above, define axial dispersity as a perturbation of this axis. In the isotropic case, the dispersity weight w(x) represents the probability that the shape axis will be found at angle x with respect to the orientation direction, in any direction. For anisotropic dispersity there will be a soft axis with larger dispersity and a perpendicular hard axis with smaller dispersity. For non-rotationally symmetric particles, the rotation vector psi is likely aligned with the hard or the soft axis. There will be a separate rotational dispersity associated directly with psi. For rotationally symmetric particles, any orientational anisotropy will depend on the flow field which defined the alignment, which could be in any direction.
Translating this to SasView is a bit challenging. We could use latitude, longitude and rotation as the names of the orientation parameters. The usual rotation.width will work for rotational dispersity. Unfortunately, neither latitude.width nor longitude.width is directly tied to the hard and soft axes of the orientation dispersity. We could define latitude.width as expected, and define longitude.width as the distance along the great circle perpendicular to longitude from the given latitude. One of these is the hard axis and the other the soft axis, so all we need is another parameter to which rotates the resulting points about the orientation direction. It would be convenient if the definition we chose happened to align with the shear direction relative to the beam in shear flow experiments with the soft axis in the direction of the flow and the hard axis perpendicular to the flow.
The above definitions should work for strongly oriented samples using gaussian dispersion, which is the majority of the cases we are trying to cover.
Particle orientation can be bimodal. For example, in particles with a thick end and thin end, the thick end or the thin end may be oriented with the shear direction (with a preference for one over the the other) but very few particles will be oriented across the shear. For strongly oriented samples, such systems could be handled with the sum of two models with orientations 180 degrees apart.
Weakly oriented systems will not be well described by our existing distributions. If particles can shift as much as 90 degrees, then they are probably isotropic in that direction (including rotation about the axis). None of our distributions will exhibit this sort of behaviour. We may want to set a default maximum on the angular dispersion at 30 degrees or less, with strong warnings in the manual that we don't understand weakly oriented systems, and the simulations may not be meaningful.
It would be useful to check that our definition supports unoriented samples, ideally defining them as a rectangular distribution of width +/-90 for latitude and longitude [NOT the gaussian equivalent rectangular width as defined in SasView!], with whatever corrections are needed for the polar coordinates integration to produce the right answer. This exercise will provide hints regarding how to interpret weakly oriented samples which are not fully isotropic.
The internal implementation of polydispersity will not support the above definition. Currently we send in a set of values to evaluate for each parameter, but the definition about requires a non-linear transformation that depends on the orientation angle. Rather than centering and scaling in the distribution definition, we could send the distribution (x, w(x)) centered at 0, adding the polydispersity to the parameter value each time through the loop rather than replacing it. The distribution truncation code, which allows us to limit parameters to the min-max range will need to take this into account. Setting limits on orientation parameters should not affect the angular distribution, unlike the polydispersity on shape parameters.
We need to be careful about limits on the angles for the orientation. If they are unlimited, DREAM will happily find multiples of 360 that fit equally well. If longitude is limited to [-180, 180] and the optimal value is near 180, then DREAM will not be able to identify the proper uncertainty distribution because it will be limited to 180. In this case, the fit should be redone with limits in the range [0, 360]. There will be strange effects for latitude 90, since 91 degrees is practically equivalent to 89 degrees in the above definition. Not sure how to handle this case.
Migrated from http://trac.sasview.org/ticket/776
{
"status": "closed",
"changetime": "2017-11-28T15:03:13",
"_ts": "2017-11-28 15:03:13.704159+00:00",
"description": " Regarding angular dispersity, I had some discussions with a user measuring oriented samples and a theorist simulating disorder in orientation. We concluded that something like the following would be useful:\n\n Given the orientation vector along the axis of the shape as described above, define axial dispersity as a perturbation of this axis. In the isotropic case, the dispersity weight w(x) represents the probability that the shape axis will be found at angle x with respect to the orientation direction, in any direction. For anisotropic dispersity there will be a soft axis with larger dispersity and a perpendicular hard axis with smaller dispersity. For non-rotationally symmetric particles, the rotation vector psi is likely aligned with the hard or the soft axis. There will be a separate rotational dispersity associated directly with psi. For rotationally symmetric particles, any orientational anisotropy will depend on the flow field which defined the alignment, which could be in any direction.\n\nTranslating this to SasView is a bit challenging. We could use latitude, longitude and rotation as the names of the orientation parameters. The usual rotation.width will work for rotational dispersity. Unfortunately, neither latitude.width nor longitude.width is directly tied to the hard and soft axes of the orientation dispersity. We could define latitude.width as expected, and define longitude.width as the distance along the great circle perpendicular to longitude from the given latitude. One of these is the hard axis and the other the soft axis, so all we need is another parameter to which rotates the resulting points about the orientation direction. It would be convenient if the definition we chose happened to align with the shear direction relative to the beam in shear flow experiments with the soft axis in the direction of the flow and the hard axis perpendicular to the flow.\n\nThe above definitions should work for strongly oriented samples using gaussian dispersion, which is the majority of the cases we are trying to cover.\n\nParticle orientation can be bimodal. For example, in particles with a thick end and thin end, the thick end or the thin end may be oriented with the shear direction (with a preference for one over the the other) but very few particles will be oriented across the shear. For strongly oriented samples, such systems could be handled with the sum of two models with orientations 180 degrees apart.\n\nWeakly oriented systems will not be well described by our existing distributions. If particles can shift as much as 90 degrees, then they are probably isotropic in that direction (including rotation about the axis). None of our distributions will exhibit this sort of behaviour. We may want to set a default maximum on the angular dispersion at 30 degrees or less, with strong warnings in the manual that we don't understand weakly oriented systems, and the simulations may not be meaningful.\n\nIt would be useful to check that our definition supports unoriented samples, ideally defining them as a rectangular distribution of width +/-90 for latitude and longitude [NOT the gaussian equivalent rectangular width as defined in SasView!], with whatever corrections are needed for the polar coordinates integration to produce the right answer. This exercise will provide hints regarding how to interpret weakly oriented samples which are not fully isotropic.\n\nThe internal implementation of polydispersity will not support the above definition. Currently we send in a set of values to evaluate for each parameter, but the definition about requires a non-linear transformation that depends on the orientation angle. Rather than centering and scaling in the distribution definition, we could send the distribution (x, w(x)) centered at 0, adding the polydispersity to the parameter value each time through the loop rather than replacing it. The distribution truncation code, which allows us to limit parameters to the min-max range will need to take this into account. Setting limits on orientation parameters should not affect the angular distribution, unlike the polydispersity on shape parameters.\n\nWe need to be careful about limits on the angles for the orientation. If they are unlimited, DREAM will happily find multiples of 360 that fit equally well. If longitude is limited to [-180, 180] and the optimal value is near 180, then DREAM will not be able to identify the proper uncertainty distribution because it will be limited to 180. In this case, the fit should be redone with limits in the range [0, 360]. There will be strange effects for latitude 90, since 91 degrees is practically equivalent to 89 degrees in the above definition. Not sure how to handle this case.",
"reporter": "dirk",
"cc": "",
"resolution": "fixed",
"workpackage": "SasModels Redesign",
"time": "2016-10-11T17:45:21",
"component": "sasmodels",
"summary": "angular dispersity",
"priority": "critical",
"keywords": "",
"milestone": "SasView 4.2.0",
"owner": "pkienzle",
"type": "enhancement"
}