-                      rcfc5917
+                      rf256d9b
 .. |theta| unicode:: U+03B8
 .. |chi| unicode:: U+03C7
+.. |bigdelta| unicode:: U+0394
 .. |inlineimage004| image:: sm_image004.gif
 …
 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
+Smearing Computation
+--------------------
+The following three smearing algorithms are provided
+Smearing Functions
+==================
+Sometimes it will be necessary to correct reduced experimental data for the
+physical effects of the instrumental geometry in use. This process is called
+*desmearing*. However, calculated/simulated data - which by definition will be
+perfect/exact - can be *smeared* to make it more representative of what might
+actually be measured experimentally.
+SasView provides the following three smearing algorithms:
 *  *Slit Smearing*
 …
 Slit Smearing
+^^^^^^^^^^^^^
+The sit smeared scattering intensity for SAS is defined by
+-------------
+**This type of smearing is normally only encountered with data from X-ray Kratky**
+**cameras or X-ray/neutron Bonse-Hart USAXS/USANS instruments.**
+The slit-smeared scattering intensity is defined by
 .. image:: sm_image002.gif
 where Norm =
+where *Norm* is given by
 .. image:: sm_image003.gif
+Equation 1
+**[Equation 1]**
 The functions |inlineimage004| and |inlineimage005|
 refer to the slit width weighting function and the slit height weighting
 determined at the q point, respectively. Here, we assumes that the weighting
 function is described by a rectangular function, i.e.,
+determined at the given *q* point, respectively. It is assumed that the weighting
+function is described by a rectangular function, such that
 .. image:: sm_image006.gif
+Equation 2
+**[Equation 2]**
 and
 …
 .. image:: sm_image007.gif
+Equation 3
+so that |inlineimage008| |inlineimage009| for |inlineimage010| and u.
+The |inlineimage011| and |inlineimage012| stand for
+the slit height (FWHM/2) and the slit width (FWHM/2) in the q space. Now the
+integral of Equation 1 is simplified to
+**[Equation 3]**
+so that |inlineimage008| |inlineimage009| for |inlineimage010| and *u*\ .
+Here |inlineimage011| and |inlineimage012| stand for
+the slit height (FWHM/2) and the slit width (FWHM/2) in *q* space.
+This simplifies the integral in Equation 1 to
 .. image:: sm_image013.gif
+Equation 4
+Numerical Implementation of Equation 4: Case 1
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+For |inlineimage012| = 0 and |inlineimage011| = constant.
+**[Equation 4]**
+which may be solved numerically, depending on the nature of |inlineimage011| and |inlineimage012| .
+Solution 1
+^^^^^^^^^^
+**For** |inlineimage012| **= 0 and** |inlineimage011| **= constant.**
 .. image:: sm_image016.gif
 For discrete q values, at the q values from the data points and at the q
 values extended up to qN= qi + |inlineimage011| the smeared
 intensity can be calculated approximately
+For discrete *q* values, at the *q* values of the data points and at the *q*
+values extended up to *q*\ :sub:`N`\ = *q*\ :sub:`i` + |inlineimage011| the smeared
+intensity can be approximately calculated as
 .. image:: sm_image017.gif
+Equation 5
+|inlineimage018| = 0 for *Is* in *j* < *i* or *j* > N-1*.
+Numerical Implementation of Equation 4: Case 2
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+For |inlineimage012| = constant and |inlineimage011| = 0.
+Similarly to Case 1, we get
+|inlineimage019| for qp= qi- |inlineimage012| and qN= qi+ |inlineimage012|. |inlineimage018| = 0
+for *Is* in *j* < *p* or *j* > *N-1*.
+Numerical Implementation of Equation 4: Case 3
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+For |inlineimage011| = constant and
+|inlineimage011| = constant.
+In this case, the best way is to perform the integration, Equation 1,
+numerically for both slit height and width. However, the numerical integration
+is not correct enough unless given a large number of iteration, say at least
+by 10000 for each element of the matrix, W, which will take minutes and
+minutes to finish the calculation for a set of typical SAS data. An
+alternative way which is correct for slit width << slit hight, is used in
+SasView. This method is a mixed method that combines method 1 with the
+numerical integration for the slit width.
+**[Equation 5]**
+where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *i* or *j* > *N-1*.
+Solution 2
+^^^^^^^^^^
+**For** |inlineimage012| **= constant and** |inlineimage011| **= 0.**
+Similar to Case 1
+|inlineimage019| for *q*\ :sub:`p` = *q*\ :sub:`i` - |inlineimage012| and *q*\ :sub:`N` = *q*\ :sub:`i` + |inlineimage012|
+**[Equation 6]**
+where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *p* or *j* > *N-1*.
+Solution 3
+^^^^^^^^^^
+**For** |inlineimage011| **= constant and** |inlineimage011| **= constant.**
+In this case, the best way is to perform the integration of Equation 1
+numerically for both slit height and slit width. However, the numerical
+integration is imperfect unless a large number of iterations, say, at
+least 10000 by 10000 for each element of the matrix *W*, is performed.
+This is usually too slow for routine use.
+An alternative approach is used in SasView which assumes
+slit width << slit height. This method combines Solution 1 with the
+numerical integration for the slit width. Then
 .. image:: sm_image020.gif
+Equation 7
 for qp= qi- |inlineimage012| and
+qN= qi+ |inlineimage012|. |inlineimage018| = 0 for
 *Is* in *j* < *p* or *j* > *N-1*.
+**[Equation 7]**
+for *q*\ :sub:`p` = *q*\ :sub:`i` - |inlineimage012| and *q*\ :sub:`N` = *q*\ :sub:`i` + |inlineimage012|
+where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *p* or *j* > *N-1*.
 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
 Pinhole Smearing
+^^^^^^^^^^^^^^^^
+The pinhole smearing computation is done similar to the case above except
+that the weight function used is the Gaussian function, so that the Equation 6
+for this case becomes
+----------------
+**This is the type of smearing normally encountered with data from synchrotron**
+**SAXS cameras and SANS instruments.**
+The pinhole smearing computation is performed in a similar fashion to the slit-
+smeared case above except that the weight function used is a Gaussian. Thus
+Equation 6 becomes
 .. image:: sm_image021.gif
+Equation 8
+For all the cases above, the weighting matrix *W* is calculated when the
+smearing is called at the first time, and it includes the ~ 60 q values
+(finely binned evenly) below (\>0) and above the q range of data in order
+to cover all data points of the smearing computation for a given model and
+for a given slit size. The *Norm*  factor is found numerically with the
+weighting matrix, and considered on *Is* computation.
+**[Equation 8]**
 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
 D Smearing
+^^^^^^^^^^^
+The 2D smearing computation is done similar to the 1D pinhole smearing above
+except that the weight function used was the 2D elliptical Gaussian function
+-----------
+The 2D smearing computation is performed in a similar fashion to the 1D pinhole
+smearing above except that the weight function used is a 2D elliptical Gaussian.
+Thus
 .. image:: sm_image022.gif
+Equation 9
 In Equation 9, x0 = qcos/theta/ and y0 = qsin/theta/, and the primed axes
+are in the coordinate rotated by an angle /theta/ around the z-axis (below)
+so that xÃ¢â¬â¢0= x0cos/theta/+y0sin/theta/ and yÃ¢â¬â¢0= -x0sin/theta/+y0cos/theta/.
+Note that the rotation angle is zero for x-y symmetric elliptical Gaussian
 distribution. The A is a normalization factor.
+**[Equation 9]**
+In Equation 9, *x*\ :sub:`0` = *q* cos(|theta|), *y*\ :sub:`0` = *q* sin(|theta|), and
+the primed axes, are all in the coordinate rotated by an angle |theta| about
+the z-axis (see the figure below) so that *x'*\ :sub:`0` = *x*\ :sub:`0` cos(|theta|) +
+*y*\ :sub:`0` sin(|theta|) and *y'*\ :sub:`0` = -*x*\ :sub:`0` sin(|theta|) +
+*y*\ :sub:`0` cos(|theta|). Note that the rotation angle is zero for a x-y symmetric
+elliptical Gaussian distribution. The *A* is a normalization factor.
 .. image:: sm_image023.gif
 Now we consider a numerical integration where each bins in /theta/ and R are
 *evenly* (this is to simplify the equation below) distributed by /delta//theta/
 and /delta/R, respectively, and it is assumed that I(xÃ¢â¬â¢, yÃ¢â¬â¢) is constant
 within the bins which in turn becomes
+Now we consider a numerical integration where each of the bins in |theta| and *R* are
+*evenly* (this is to simplify the equation below) distributed by |bigdelta|\ |theta|
+and |bigdelta|\ R, respectively, and it is further assumed that *I(x',y')* is constant
+within the bins. Then
 .. image:: sm_image024.gif
+Equation 10
+Since we have found the weighting factor on each bin points, it is convenient
+to transform xÃ¢â¬â¢-yÃ¢â¬â¢ back to x-y coordinate (rotating it by -/theta/ around z
+axis). Then, for the polar symmetric smear
+**[Equation 10]**
+Since the weighting factor on each of the bins is known, it is convenient to
+transform *x'-y'* back to *x-y* coordinates (by rotating it by -|theta| around the
+*z* axis).
+Then, for a polar symmetric smear
 .. image:: sm_image025.gif
+Equation 11
+**[Equation 11]**
 where
 …
 .. image:: sm_image026.gif
 while for the x-y symmetric smear
+while for a *x-y* symmetric smear
 .. image:: sm_image027.gif
+Equation 12
+**[Equation 12]**
 where
 …
 .. image:: sm_image028.gif
+Here, the current version of the SasView uses Equation 11 for 2D smearing
+assuming that all the Gaussian weighting functions are aligned in the polar
+coordinate.
+In the control panel, the higher accuracy indicates more and finer binnng
+points so that it costs more in time.
+.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
+The current version of the SasView uses Equation 11 for 2D smearing, assuming
+that all the Gaussian weighting functions are aligned in the polar coordinate.
+.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
+Weighting & Normalization
+-------------------------
+In all the cases above, the weighting matrix *W* is calculated on the first call
+to a smearing function, and includes ~60 *q* values (finely and evenly binned)
+below (>0) and above the *q* range of data in order to smear all data points for
+a given model and slit/pinhole size. The *Norm*  factor is found numerically with the
+weighting matrix and applied on the computation of *I*\ :sub:`s`.
+.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
+.. note::  This help document was last changed by Steve King, 01May2015

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