Changes in src/sas/sasgui/perspectives/fitting/media/sm_help.rst [27aabc1:5ed76f8] in sasview
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src/sas/sasgui/perspectives/fitting/media/sm_help.rst
r27aabc1 r5ed76f8 4 4 .. by S King, ISIS, during SasView CodeCamp-III in Feb 2015. 5 5 6 .. |inlineimage004| image:: sm_image004. gif7 .. |inlineimage005| image:: sm_image005. gif8 .. |inlineimage008| image:: sm_image008. gif9 .. |inlineimage009| image:: sm_image009. gif10 .. |inlineimage010| image:: sm_image010. gif11 .. |inlineimage011| image:: sm_image011. gif12 .. |inlineimage012| image:: sm_image012. gif13 .. |inlineimage018| image:: sm_image018. gif14 .. |inlineimage019| image:: sm_image019. gif6 .. |inlineimage004| image:: sm_image004.png 7 .. |inlineimage005| image:: sm_image005.png 8 .. |inlineimage008| image:: sm_image008.png 9 .. |inlineimage009| image:: sm_image009.png 10 .. |inlineimage010| image:: sm_image010.png 11 .. |inlineimage011| image:: sm_image011.png 12 .. |inlineimage012| image:: sm_image012.png 13 .. |inlineimage018| image:: sm_image018.png 14 .. |inlineimage019| image:: sm_image019.png 15 15 16 16 … … 20 20 ================== 21 21 22 Sometimes the instrumental geometry used to acquire the experimental data has 23 an impact on the clarity of features in the reduced scattering curve. For 24 example, peaks or fringes might be slightly broadened. This is known as 25 *Q resolution smearing*. To compensate for this effect one can either try and 26 remove the resolution contribution - a process called *desmearing* - or add the 27 resolution contribution into a model calculation/simulation (which by definition 28 will be exact) to make it more representative of what has been measured 22 Sometimes the instrumental geometry used to acquire the experimental data has 23 an impact on the clarity of features in the reduced scattering curve. For 24 example, peaks or fringes might be slightly broadened. This is known as 25 *Q resolution smearing*. To compensate for this effect one can either try and 26 remove the resolution contribution - a process called *desmearing* - or add the 27 resolution contribution into a model calculation/simulation (which by definition 28 will be exact) to make it more representative of what has been measured 29 29 experimentally - a process called *smearing*. SasView will do the latter. 30 30 31 Both smearing and desmearing rely on functions to describe the resolution 31 Both smearing and desmearing rely on functions to describe the resolution 32 32 effect. SasView provides three smearing algorithms: 33 33 … … 36 36 * *2D Smearing* 37 37 38 SasView also has an option to use Q resolution data (estimated at the time of38 SasView also has an option to use $Q$ resolution data (estimated at the time of 39 39 data reduction) supplied in a reduced data file: the *Use dQ data* radio button. 40 40 … … 43 43 dQ Smearing 44 44 ----------- 45 46 If this option is checked, SasView will assume that the supplied dQ values45 46 If this option is checked, SasView will assume that the supplied $dQ$ values 47 47 represent the standard deviations of Gaussian functions. 48 48 … … 57 57 The slit-smeared scattering intensity is defined by 58 58 59 .. image:: sm_image002. gif59 .. image:: sm_image002.png 60 60 61 61 where *Norm* is given by 62 62 63 .. image:: sm_image003. gif63 .. image:: sm_image003.png 64 64 65 65 **[Equation 1]** 66 66 67 The functions |inlineimage004| and |inlineimage005|68 refer to the slit width weighting function and the slit height weighting 69 determined at the given *q*point, respectively. It is assumed that the weighting67 The functions $W_v(v)$ and $W_u(u)$ 68 refer to the slit width weighting function and the slit height weighting 69 determined at the given $q$ point, respectively. It is assumed that the weighting 70 70 function is described by a rectangular function, such that 71 71 72 .. image:: sm_image006. gif72 .. image:: sm_image006.png 73 73 74 74 **[Equation 2]** … … 76 76 and 77 77 78 .. image:: sm_image007. gif78 .. image:: sm_image007.png 79 79 80 80 **[Equation 3]** 81 81 82 so that |inlineimage008| |inlineimage009| for |inlineimage010| and *u*\ . 83 84 Here |inlineimage011| and |inlineimage012| stand for 85 the slit height (FWHM/2) and the slit width (FWHM/2) in *q* space. 82 so that $\Delta q_\alpha = \int_0^\infty d\alpha W_\alpha(\alpha)$ 83 for $\alpha = v$ and $u$. 84 85 Here $\Delta q_u$ and $\Delta q_v$ stand for 86 the slit height (FWHM/2) and the slit width (FWHM/2) in $q$ space. 86 87 87 88 This simplifies the integral in Equation 1 to 88 89 89 .. image:: sm_image013. gif90 .. image:: sm_image013.png 90 91 91 92 **[Equation 4]** 92 93 93 which may be solved numerically, depending on the nature of |inlineimage011| and |inlineimage012| . 94 which may be solved numerically, depending on the nature of 95 $\Delta q_u$ and $\Delta q_v$. 94 96 95 97 Solution 1 96 98 ^^^^^^^^^^ 97 99 98 **For ** |inlineimage012| **= 0 and** |inlineimage011| **= constant.**99 100 .. image:: sm_image016. gif101 102 For discrete *q* values, at the *q* values of the data points and at the *q*103 values extended up to *q*\ :sub:`N`\ = *q*\ :sub:`i` + |inlineimage011|the smeared100 **For $\Delta q_v= 0$ and $\Delta q_u = \text{constant}$.** 101 102 .. image:: sm_image016.png 103 104 For discrete $q$ values, at the $q$ values of the data points and at the $q$ 105 values extended up to $q_N = q_i + \Delta q_u$ the smeared 104 106 intensity can be approximately calculated as 105 107 106 .. image:: sm_image017. gif108 .. image:: sm_image017.png 107 109 108 110 **[Equation 5]** 109 111 110 where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *i* or *j* > *N-1*.112 where |inlineimage018| = 0 for $I_s$ when $j < i$ or $j > N-1$. 111 113 112 114 Solution 2 113 115 ^^^^^^^^^^ 114 116 115 **For ** |inlineimage012| **= constant and** |inlineimage011| **= 0.**117 **For $\Delta q_v = \text{constant}$ and $\Delta q_u= 0$.** 116 118 117 119 Similar to Case 1 118 120 119 |inlineimage019| for *q*\ :sub:`p` = *q*\ :sub:`i` - |inlineimage012| and *q*\ :sub:`N` = *q*\ :sub:`i` + |inlineimage012|121 |inlineimage019| for $q_p = q_i - \Delta q_v$ and $q_N = q_i + \Delta q_v$ 120 122 121 123 **[Equation 6]** 122 124 123 where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *p* or *j* > *N-1*.125 where |inlineimage018| = 0 for $I_s$ when $j < p$ or $j > N-1$. 124 126 125 127 Solution 3 126 128 ^^^^^^^^^^ 127 129 128 **For ** |inlineimage011| **= constant and** |inlineimage011| **= constant.**130 **For $\Delta q_u = \text{constant}$ and $\Delta q_v = \text{constant}$.** 129 131 130 132 In this case, the best way is to perform the integration of Equation 1 … … 138 140 numerical integration for the slit width. Then 139 141 140 .. image:: sm_image020. gif142 .. image:: sm_image020.png 141 143 142 144 **[Equation 7]** 143 145 144 for *q*\ :sub:`p` = *q*\ :sub:`i` - |inlineimage012| and *q*\ :sub:`N` = *q*\ :sub:`i` + |inlineimage012| 145 146 where |inlineimage018| = 0 for *I*\ :sub:`s` when *j* < *p* or *j* > *N-1*. 146 for *q_p = q_i - \Delta q_v$ and $q_N = q_i + \Delta q_v$ 147 where |inlineimage018| = 0 for *I_s$ when $j < p$ or $j > N-1$. 147 148 148 149 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 158 159 Equation 6 becomes 159 160 160 .. image:: sm_image021. gif161 .. image:: sm_image021.png 161 162 162 163 **[Equation 8]** … … 171 172 Thus 172 173 173 .. image:: sm_image022. gif174 .. image:: sm_image022.png 174 175 175 176 **[Equation 9]** 176 177 177 In Equation 9, *x*\ :sub:`0` = *q* cos(|theta|), *y*\ :sub:`0` = *q* sin(|theta|), and 178 the primed axes, are all in the coordinate rotated by an angle |theta| about 179 the z-axis (see the figure below) so that *x'*\ :sub:`0` = *x*\ :sub:`0` cos(|theta|) + 180 *y*\ :sub:`0` sin(|theta|) and *y'*\ :sub:`0` = -*x*\ :sub:`0` sin(|theta|) + 181 *y*\ :sub:`0` cos(|theta|). Note that the rotation angle is zero for a x-y symmetric 182 elliptical Gaussian distribution. The *A* is a normalization factor. 183 184 .. image:: sm_image023.gif 185 186 Now we consider a numerical integration where each of the bins in |theta| and *R* are 187 *evenly* (this is to simplify the equation below) distributed by |bigdelta|\ |theta| 188 and |bigdelta|\ R, respectively, and it is further assumed that *I(x',y')* is constant 178 In Equation 9, $x_0 = q \cos(\theta)$, $y_0 = q \sin(\theta)$, and 179 the primed axes, are all in the coordinate rotated by an angle $\theta$ about 180 the z-axis (see the figure below) so that 181 $x'_0 = x_0 \cos(\theta) + y_0 \sin(\theta)$ and 182 $y'_0 = -x_0 \sin(\theta) + y_0 \cos(\theta)$. 183 Note that the rotation angle is zero for a $xy$ symmetric 184 elliptical Gaussian distribution. The $A$ is a normalization factor. 185 186 .. image:: sm_image023.png 187 188 Now we consider a numerical integration where each of the bins in $\theta$ and $R$ are 189 *evenly* (this is to simplify the equation below) distributed by $\Delta \theta$ 190 and $\Delta R$, respectively, and it is further assumed that $I(x',y')$ is constant 189 191 within the bins. Then 190 192 191 .. image:: sm_image024. gif193 .. image:: sm_image024.png 192 194 193 195 **[Equation 10]** 194 196 195 197 Since the weighting factor on each of the bins is known, it is convenient to 196 transform *x'-y'* back to *x-y* coordinates (by rotating it by -|theta|around the197 *z*axis).198 transform $x'y'$ back to $xy$ coordinates (by rotating it by $-\theta$ around the 199 $z$ axis). 198 200 199 201 Then, for a polar symmetric smear 200 202 201 .. image:: sm_image025. gif203 .. image:: sm_image025.png 202 204 203 205 **[Equation 11]** … … 205 207 where 206 208 207 .. image:: sm_image026. gif208 209 while for a *x-y*symmetric smear210 211 .. image:: sm_image027. gif209 .. image:: sm_image026.png 210 211 while for a $xy$ symmetric smear 212 213 .. image:: sm_image027.png 212 214 213 215 **[Equation 12]** … … 215 217 where 216 218 217 .. image:: sm_image028. gif219 .. image:: sm_image028.png 218 220 219 221 The current version of the SasView uses Equation 11 for 2D smearing, assuming … … 225 227 ------------------------- 226 228 227 In all the cases above, the weighting matrix *W*is calculated on the first call228 to a smearing function, and includes ~60 *q*values (finely and evenly binned)229 below (>0) and above the *q*range of data in order to smear all data points for230 a given model and slit/pinhole size. The *Norm*factor is found numerically with the231 weighting matrix and applied on the computation of *I*\ :sub:`s`.229 In all the cases above, the weighting matrix $W$ is calculated on the first call 230 to a smearing function, and includes ~60 $q$ values (finely and evenly binned) 231 below (>0) and above the $q$ range of data in order to smear all data points for 232 a given model and slit/pinhole size. The $Norm$ factor is found numerically with the 233 weighting matrix and applied on the computation of $I_s$. 232 234 233 235 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
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