-                      r6aad2e8
+                      r5ed76f8
 ==================
 Sometimes the instrumental geometry used to acquire the experimental data has
 an impact on the clarity of features in the reduced scattering curve. For
 example, peaks or fringes might be slightly broadened. This is known as
 *Q resolution smearing*. To compensate for this effect one can either try and
 remove the resolution contribution - a process called *desmearing* - or add the
 resolution contribution into a model calculation/simulation (which by definition
 will be exact) to make it more representative of what has been measured
+Sometimes the instrumental geometry used to acquire the experimental data has
+an impact on the clarity of features in the reduced scattering curve. For
+example, peaks or fringes might be slightly broadened. This is known as
+*Q resolution smearing*. To compensate for this effect one can either try and
+remove the resolution contribution - a process called *desmearing* - or add the
+resolution contribution into a model calculation/simulation (which by definition
+will be exact) to make it more representative of what has been measured
 experimentally - a process called *smearing*. SasView will do the latter.
 Both smearing and desmearing rely on functions to describe the resolution
+Both smearing and desmearing rely on functions to describe the resolution
 effect. SasView provides three smearing algorithms:
 …
 *  *2D Smearing*
 SasView also has an option to use Q resolution data (estimated at the time of
+SasView also has an option to use $Q$ resolution data (estimated at the time of
 data reduction) supplied in a reduced data file: the *Use dQ data* radio button.
 …
 dQ Smearing
 -----------
 If this option is checked, SasView will assume that the supplied dQ values
+If this option is checked, SasView will assume that the supplied $dQ$ values
 represent the standard deviations of Gaussian functions.
 …
 **[Equation 1]**
 The functions |inlineimage004| and |inlineimage005|
 refer to the slit width weighting function and the slit height weighting
 determined at the given *q* point, respectively. It is assumed that the weighting
+The functions $W_v(v)$ and $W_u(u)$
+refer to the slit width weighting function and the slit height weighting
+determined at the given $q$ point, respectively. It is assumed that the weighting
 function is described by a rectangular function, such that
 …
 **[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.
+so that $\Delta q_\alpha = \int_0^\infty d\alpha W_\alpha(\alpha)$
+for $\alpha = v$ and $u$.
+Here $\Delta q_u$ and $\Delta q_v$ stand for
+the slit height (FWHM/2) and the slit width (FWHM/2) in $q$ space.
 This simplifies the integral in Equation 1 to
 …
 **[Equation 4]**
+which may be solved numerically, depending on the nature of |inlineimage011| and |inlineimage012| .
+which may be solved numerically, depending on the nature of
+$\Delta q_u$ and $\Delta q_v$.
 Solution 1
 ^^^^^^^^^^
 **For** |inlineimage012| **= 0 and** |inlineimage011| **= constant.**
+**For $\Delta q_v= 0$ and $\Delta q_u = \text{constant}$.**
 .. image:: sm_image016.png
 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
+For discrete $q$ values, at the $q$ values of the data points and at the $q$
+values extended up to $q_N = q_i + \Delta q_u$ the smeared
 intensity can be approximately calculated as
 …
 **[Equation 5]**
 where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *i* or *j* > *N-1*.
+where |inlineimage018| = 0 for $I_s$ when $j < i$ or $j > N-1$.
 Solution 2
 ^^^^^^^^^^
 **For** |inlineimage012| **= constant and** |inlineimage011| **= 0.**
+**For $\Delta q_v = \text{constant}$ and $\Delta q_u= 0$.**
 Similar to Case 1
 |inlineimage019| for *q*\ :sub:`p` = *q*\ :sub:`i` - |inlineimage012| and *q*\ :sub:`N` = *q*\ :sub:`i` + |inlineimage012|
+|inlineimage019| for $q_p = q_i - \Delta q_v$ and $q_N = q_i + \Delta q_v$
 **[Equation 6]**
 where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *p* or *j* > *N-1*.
+where |inlineimage018| = 0 for $I_s$ when $j < p$ or $j > N-1$.
 Solution 3
 ^^^^^^^^^^
 **For** |inlineimage011| **= constant and** |inlineimage011| **= constant.**
+**For $\Delta q_u = \text{constant}$ and $\Delta q_v = \text{constant}$.**
 In this case, the best way is to perform the integration of Equation 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*.
+for *q_p = q_i - \Delta q_v$ and $q_N = q_i + \Delta q_v$
+where |inlineimage018| = 0 for *I_s$ when $j < p$ or $j > N-1$.
 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
 …
 **[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.
+In Equation 9, $x_0 = q \cos(\theta)$, $y_0 = q \sin(\theta)$, and
+the primed axes, are all in the coordinate rotated by an angle $\theta$ about
+the z-axis (see the figure below) so that
+$x'_0 = x_0 \cos(\theta) + y_0 \sin(\theta)$ and
+$y'_0 = -x_0 \sin(\theta) + y_0 \cos(\theta)$.
+Note that the rotation angle is zero for a $xy$ symmetric
+elliptical Gaussian distribution. The $A$ is a normalization factor.
 .. image:: sm_image023.png
 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
+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 $\Delta \theta$
+and $\Delta R$, respectively, and it is further assumed that $I(x',y')$ is constant
 within the bins. Then
 …
 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).
+transform $x'y'$ back to $xy$ coordinates (by rotating it by $-\theta$ around the
+$z$ axis).
 Then, for a polar symmetric smear
 …
 .. image:: sm_image026.png
 while for a *x-y* symmetric smear
+while for a $xy$ symmetric smear
 .. image:: sm_image027.png
 …
 -------------------------
 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`.
+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 $I_s$.
 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ

Note: See TracChangeset for help on using the changeset viewer.

SasView

Changeset 5ed76f8 in sasview for src/sas/sasgui/perspectives/fitting/media/sm_help.rst

Legend:

src/sas/sasgui/perspectives/fitting/media/sm_help.rst

Download in other formats: