From ba85299f135df9e812818c040985f0b204dedffe Mon Sep 17 00:00:00 2001 From: Duc Le Date: Wed, 8 Jun 2022 18:12:08 +0100 Subject: [PATCH 1/2] Add maths details for slice and cuts --- docs/source/cutting.rst | 42 +- docs/source/images/math_ref/rebin_cuts.png | Bin 0 -> 335042 bytes docs/source/images/math_ref/rebin_grids.svg | 725 ++++++++++++++++++++ docs/source/index.rst | 1 + docs/source/math_ref.rst | 208 ++++++ 5 files changed, 969 insertions(+), 7 deletions(-) create mode 100644 docs/source/images/math_ref/rebin_cuts.png create mode 100644 docs/source/images/math_ref/rebin_grids.svg create mode 100644 docs/source/math_ref.rst diff --git a/docs/source/cutting.rst b/docs/source/cutting.rst index 5f2812d65..b3ed7ce19 100644 --- a/docs/source/cutting.rst +++ b/docs/source/cutting.rst @@ -92,12 +92,40 @@ that tab. When MSlice is used as a Mantid interface ``MD Histo`` type workspaces can also be saved to Mantid Workbench by clicking the ``Save to Workbench`` button either on the ``MD Histo`` or the ``Cut`` tab. +.. _Cutting_Algorithms: + Cutting Algorithms ------------------ -There are two different methods used to compute cuts. ``Rebin`` uses the basic rebinning algorithms directly and effectively averages -the counts in the integration range, whilst ``Integration`` sums the counts in the integration range. -For mantid major releases from ``v6.40``, the default cutting algorithm has been changed from ``Rebin`` to ``Integration``. This change -has been made because the ``Integration`` method can be used for both absolute and non-absolute units measurements. Conversely, for -absolute units measurements the ``Rebin`` method will give incorrect and misleading values. -As a result of this change, it is expected that values calculanced henceforth will differ from those calculated historically, if the -default integration method has been used. + +There are two different methods used to compute cuts: + +- ``Integration`` sums the (signal :math:`\times` bin width) in the integration range. +- ``Rebin`` averages the signal in the integration range. + +The two methods are described in more detail in the :ref:`Mathematical_Reference`, +but in short, there is a bin-dependent conversion factor between the two types of +cuts which depends on the data coverage in the integration range of that bin. +That is, if the integration range does not include regions without data +(e.g. due to kinematic constraints), then the two cuts will be equivalent except +for a constant scaling factor (proportional to the integration range). +However, if the integration range overlaps regions without data, +then the two cuts will give markedly different results. + +The default method is ``Rebin`` and is more suitable for DOS-types cuts which +integrate over :math:`|Q|` whilst if you are interested in cross-sections and +are integrating over energy transfer, it is recommended to use ``Integration``. + +There is an option in the ``Cut`` tab to change the cut algorithm from ``Rebin`` +to ``Integration`` or vice versa and this setting will be saved for subsequent +similar cuts on the same workspace. + +You can also change the default using the ``Options`` menu, ``Cut algorithm default`` +entry. This will change the default cut algorithm *for this session of MSlice* +(the default algorithm will revert to ``Rebin`` if you restart MSlice). + +.. + For mantid major releases from ``v6.40``, the default cutting algorithm has been changed from ``Rebin`` to ``Integration``. This change + has been made because the ``Integration`` method can be used for both absolute and non-absolute units measurements. Conversely, for + absolute units measurements the ``Rebin`` method will give incorrect and misleading values. + As a result of this change, it is expected that values calculanced henceforth will differ from those calculated historically, if the + default integration method has been used. diff --git a/docs/source/images/math_ref/rebin_cuts.png b/docs/source/images/math_ref/rebin_cuts.png new file mode 100644 index 0000000000000000000000000000000000000000..94279ca61fdff99832dcb4f0c317aea9a7380940 GIT binary patch literal 335042 zcmagGWmr|+_XP?_h@^BQA|c%%5>kScq=Yn*(v8v`3W_2~2pptaR63NBMgalo25FFz zx^w&f?*F;p?gtl6PF-qkgh8gmY z%dQ45(e(elX7u0&8ig$K&m?l(pm2DPsnts1|L0>x$zQ+9TKA=OzF?B`S;>kVt@Cy+ zIKFr9-ZKWtl%mqwn61W>(xE?r=X)9W)B<5P?|dVQi;GnzJ^qbCSq5WhO?!Q}!@|S){r`HXr-^+@5q`{p zI{r6Y@;c#iGA?eGTqGeTCZ^Ab zd)?&vXho<^hW{IJ&)+VSo+A}r8-`8|`wzARj*pK|Pxty4mX<71Sk?1L85uKF8skx? z>_G=uby;r)}SLMzGGryQeb7oLy1F3cTG|<9VSld?+SASA$?3?mt4s2<^jn^ z{)Y^;7EJYP#^jS}?p>2dH1=lPeh&%dbr(lo5f2A<%+Gf7HZ8omF9nOHL$j+^EGwkO 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z`T2R(dnd?7z}*Iz&%Th#Y~Q%<@oDXlSV5-n|U|F)?fp zBuvB0E&8DA(M(9o$!TG`70dj{#KZ&=6dji7wRg$4dOSn$SX^MD-$h3E^9l(Kd$~ni zTpY_r3dZSs_U=uXdyBDu3#AX=YHASgGXoKlvzP#%L9h3;jh1Ij)|sUuG@z==Af^)* z + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/docs/source/index.rst b/docs/source/index.rst index 7cb4b79a9..74412db5c 100644 --- a/docs/source/index.rst +++ b/docs/source/index.rst @@ -18,6 +18,7 @@ to report any bugs or suggest any improvements to the program. cutting slicing cli + math_ref * :ref:`search` diff --git a/docs/source/math_ref.rst b/docs/source/math_ref.rst new file mode 100644 index 000000000..b6e54b36f --- /dev/null +++ b/docs/source/math_ref.rst @@ -0,0 +1,208 @@ +.. _Mathematical_Reference: + +Mathematical Reference +====================== + +This page describes the mathematical operations behind the ``Slice`` and ``Cut`` operations of MSlice + +In general, MSlice handles "reduced" (processed) inelastic neutron scattering data which has been +binned (histogrammed) in energy transfer for each detector / position-sensitive-detector (PSD) element. +We will use the terminology of the `Horace `_ program and +refer to these energy-detector-element bins as "pixels". + +Since the detector elements are in the laboratory coordinates, but we often want to plot the data +in reciprocal space, a coordinate transformation is needed. +This means that the input and output binning grids of the data will not be axis aligned, and will +instead look something like this: + +.. image:: images/math_ref/rebin_grids.svg + :width: 600 + +Where the square red grid represents the target (output) bins, and the slanted green grids +(parallelograms) represents the original bins ("pixels"). +As discussed in the :ref:`PSD_and_non-PSD_modes` section, MSlice makes a distinction between +**PSD** (fine green grid) and **non-PSD** (coarser green grid) data. + +``Slice`` refers to a rebin into a two dimensional output, whilst ``Cut`` refers to a rebin _or_ +integration into a one dimensional output. +For each type of data (**PSD** and **non-PSD**) we will describe each operation in turn. + + +PSD Slice +--------- + +For **PSD** data, MSlice uses *centre-point rebinning*, treating each input bin ("pixel") as a +point and summing the full signal of each pixel whose centres lie within an output bin +(illustrated in the image by the darker green shading in top left). +Thus the output signal in the :math:`(i,j)`\ th bin, :math:`Y_{ij}`, is: + +.. math:: + Y_{ij} = \frac{1}{N_{kl}} \sum_{kl} y_{kl} + +where :math:`y_{kl}` is the input signal in the input :math:`(k,l)`\ th bin +and the sum runs over the :math:`N_{kl}` number of bins whoses centres lie within the +boundaries of the :math:`(i,j)`\ th output bin. + +The above expression uses the ``NumEventsNormalization`` convention of Mantid which is the +same as that adopted by the `Horace `_ program. +The error values are considered to be standard deviations and are summed in quadrature. + +PSD Cut +------- + +Since MSlice allows users to specify bins in the non-integrating direction which are not +necessarily aligned with respect to the original data, a rebinning step as described above +is needed for the ``Cut`` operation too. + +This leads to the two types of behaviour ("algorithms") for the ``Cut`` operation described +in the :ref:`Cutting_Algorithms` section: + +- The default ``Rebin`` method just uses the rebinning described above with one axis having + only a single bin. +- The ``Integration`` method first rebins the data as described above with integration axis + divided into :math:`N_{\mathrm{int}} =` 100 bins. + It then calls the relevant Mantid algorithm + (`IntegrateMDHistoWorkspace `_\ ) + to integrate (sums the signal) in those 100 bins. + +Taking the :math:`j` index to be over the integration axis, the integrated ``Cut`` signal +:math:`C_i` is then given by: + +.. math:: + C_i^{\mathrm{integration}} = w_i \sum_{j \in \mathcal{D}} Y_{ij} + +where the index :math:`j` only runs over regions with data :math:`\mathcal{D}`, and the width +:math:`w_i = \sum_{j \in \mathcal{D}} w_{ij}` where :math:`w_{ij}` is the width in the +:math:`j`\ th direction of the :math:`(i,j)`\ th bin. + +Note that the equivalent expression for a ``Rebin`` cut is simply: + +.. math:: + C_i^{\mathrm{rebin}} = \sum_{j \in \mathcal{F}} Y_{ij} + +where now the index :math:`j` runs over the full integration range :math:`\mathcal{F}`. +The difference is thus a coordinate-dependent weighting factor :math:`w_i`. +If the integration range *does not include regions without data* (e.g. :math:`\mathcal{D} \equiv \mathcal{F}`) +then all the :math:`w_i` will be equal to the full integration width and the difference between +:math:`C_i^{\mathrm{integration}}` and :math:`C_i^{\mathrm{rebin}}` is a constant. + +However, if the integration range covers region with no data (e.g. beyond the kinematic limits) +then the two cuts *may* look very different because :math:`C_i^{\mathrm{integration}}` will weight +regions with more data more heavily than regions without data. + + +Non-PSD Slice +------------- + +For **non-PSD** data, MSlice uses *fractional rebinning*, where it first calculates the +overlap area between the input and output bins, and then sums only the fractions of the +signal of the input bins within the output bin (darker green shading in bottom right). + +The output signal is computed as: + +.. math:: + Y_{ij} = \left. \sum_{kl} y_{kl} f_{kl} \right/ \sum_{kl} f_{kl} + +and the output uncertainty as: + +.. math:: + E_{ij} = \left. \sqrt{\sum_{kl} e^2_{kl} f_{kl}} \right/ \sum_{kl} f_{kl} + +where :math:`f_{kl}` is the fractional overlap of the input :math:`(k,l)`\ th bin with +the output :math:`(i,j)`\ th bin. + +This is illustrated in the image at the start of the page by the square on the right +hand side with blue triangular and orange quadrilateral shaded regions. +The blue and orange shading illustrates the fractional overlap areas which weights +the signal in the top left and top right input bins (large parallelograms). + +Non-PSD Cuts +------------ + +Like for **PSD** data, there are two ``Cut`` "algorithms" for **non-PSD** data also. + +The ``Rebin`` cut algorithm performs the same operation described in the previous section +but with a single bin in the integration axis, yielding + +.. math:: + C_i^{\mathrm{rebin}} = \left. \sum_{j \in \mathcal{F}} Y_{ij} \right/ \sum_{j \in \mathcal{F}} F_{ij} + +where :math:`F_{ij} = \sum_{kl} f_{kl}`, and :math:`\mathcal{F}` indicates the full integration range. + +In order to support rebinning in the non-integration axis, the ``Integration`` algorithm +first rebins the data into the desired bins in the non-integration axis, +and 100 bins in the integration axis and then sums them as: + +.. math:: + C_i^{\mathrm{integration}} = \left. N_i \sum_{j \in \mathcal{F}} Y_{ij} w_{ij} \right/ \sum_{j \in \mathcal{F}} F_{ij} + +where :math:`N_i = \sum_{j \in \mathcal{D}} 1` is the number of :math:`j` bins at a given +:math:`i` with non-zero fraction (e.g. if the integration contains only regions with data +then :math:`N_i` = 100, otherwise :math:`N_i` will be less), +and :math:`w_{ij}` is the width along the :math:`j`\ th axis of the :math:`(i,j)`\ th bin. +The :math:`N_i` normalisation is needed because in the limiting case where all the fractions +:math:`F_i` are unity, the denominator would be :math:`N_i`, so we recover the usual +expression for integrating over a distribution. +Note that as previously, :math:`\mathcal{D}` indicates the region within the integration +range with non-zero fractions. + +Like with the **PSD** case there is thus an :math:`i` dependent scaling factor :math:`N_i w` +(assuming all the bins have the same width) between ``Cuts`` computed using the ``Rebin`` or +``Integration`` algorithm. +This scaling factor is a constant if the integration range includes only regions with data +(e.g. :math:`\mathcal{D} \equiv \mathcal{F}`), but will not be constant if the integration +overlaps regions without data. + +The difference is illustrated below: + +.. image:: images/math_ref/rebin_cuts.png + +The cuts have been normalised to the peak intensity so that the constant scaling factor between +the two algorithms factorises out. +In the top cut, integrating over :math:`6 \leq |Q| < 8 \mathrm{\AA}^{-1}` there are no regions +without data so the two cuts are equivalent except for a constant scaling factor. +In the bottom cut, integrating over :math:`8 \leq |Q| < 10 \mathrm{\AA}^{-1}` there is a large +region with no data, so now cuts from the two techniques differ markedly. +At :math:`E<0` meV Where the data covers the full integration range, we have :math:`N_i` = 100 +and the two cuts are equivalent. +As :math:`E` increases, :math:`N_i` decreases until at around 20 meV, it is :math:`N_i` = 50, +and we see that at that point the (normalised) ``Integration`` cut is half the intensity of +the (normalised) ``Rebin`` cut. + +A note on units +--------------- + +One advantage of inelastic neutron scattering over other techniques is that it is (relatively) +easy to normalise the measured data to absolute units. +At the ISIS Neutron and Muon Source if this normalisation is done, then the signal will be in +units of [milibarns per steradian per meV per formula unit] or [mb/sr/meV/f.u.]. + +An ``Integration`` over energy would then yield a differential cross-section in [mb/sr/f.u.], +whereas a ``Rebin`` over energy would leave the units unchanged at [mb/sr/meV/f.u.]. + +However, an ``Integration`` over :math:`|Q|` instead of energy will yield units of +[mb/Å/sr/meV/f.u.] rather [mb/meV/f.u.] and as such it may be more useful to perform an +average ``Rebin`` which will leave the units unchanged. + +Unfortunately, the input files read by MSlice do not indicate if the signal values saved +are in absolute units or not, so MSlice cannot automatically write the correct units to axes +- this is left to the user. + +A note on the regions of validity of the two algorithms +------------------------------------------------------- + +As can be seen in the example above, the ``Integration`` cut algorithm will produce low signals +where there is less data, whereas the ``Rebin`` cut algorithm will amplify the signals in such +regions - effectively assuming that the signal is constant across the integration range and +thus extrapolating over regions without data (so the only manifestation of the lack of data +are larger errorbars associated with these bins). + +This assumption *may* be valid for density-of-states (DOS) type cuts where one would expect +that the signal is approximately constant over :math:`|Q|` and only varies in energy. +Thus for these applications, it may be suitable to chose the ``Rebin`` algorithm, and to +extrapolate the high-energy, high-:math:`|Q|` regions which are kinematically inaccessible. + +Conversely, for integration over energy - for example over the elastic line to compute a +differential cross-section or over a finite energy crystal field excitation to obtain a +magnetic cross-section, the ``Integration`` algorithm should be chosen else the signal +in the cut will vary with the integration range and would not be a cross-section. From 061fff2415b341f4a1fb75f8efc9dae4e5f28e40 Mon Sep 17 00:00:00 2001 From: Duc Le Date: Thu, 9 Jun 2022 10:40:28 +0100 Subject: [PATCH 2/2] Remove note on default cut algo; clean up text --- docs/source/cutting.rst | 14 ++------- docs/source/images/math_ref/rebin_grids.svg | 2 +- docs/source/math_ref.rst | 32 +++++++++++---------- 3 files changed, 21 insertions(+), 27 deletions(-) diff --git a/docs/source/cutting.rst b/docs/source/cutting.rst index b3ed7ce19..3f55640bd 100644 --- a/docs/source/cutting.rst +++ b/docs/source/cutting.rst @@ -30,10 +30,9 @@ integrate over [0,3], [3,6], [6,9] and [9,10] respectively. Cuts with the same range from multiple datasets can be plotted by first selecting multiple workspaces in the left panel. -There are two different methods to compute cuts: ``Rebin`` and ``Integration``. -**NOTE: for mantid major releases from** ``v6.40``\ **, the default cutting algorithm has been changed from** ``Rebin`` -**to** ``Integration``\ **. For more detail on this change, and cutting algorithms in general, see the** *Cutting Algorithms* -**section below**. +There are two different methods to compute cuts: ``Rebin`` and ``Integration``, which can be selected from the +``Cut Algorithm`` drop down menu. The difference between these methods are described in the :ref:`Cutting_Algorithms` +section below and in more detail in the :ref:`Mathematical_Reference`. Clicking on the ``Norm to 1`` check box will cause the resulting cut data to be normalised such that the maximum of the data of each cut is unity. @@ -122,10 +121,3 @@ similar cuts on the same workspace. You can also change the default using the ``Options`` menu, ``Cut algorithm default`` entry. This will change the default cut algorithm *for this session of MSlice* (the default algorithm will revert to ``Rebin`` if you restart MSlice). - -.. - For mantid major releases from ``v6.40``, the default cutting algorithm has been changed from ``Rebin`` to ``Integration``. This change - has been made because the ``Integration`` method can be used for both absolute and non-absolute units measurements. Conversely, for - absolute units measurements the ``Rebin`` method will give incorrect and misleading values. - As a result of this change, it is expected that values calculanced henceforth will differ from those calculated historically, if the - default integration method has been used. diff --git a/docs/source/images/math_ref/rebin_grids.svg b/docs/source/images/math_ref/rebin_grids.svg index 24ca6f0e4..a0b72faa1 100644 --- a/docs/source/images/math_ref/rebin_grids.svg +++ b/docs/source/images/math_ref/rebin_grids.svg @@ -420,7 +420,7 @@ style="stroke-width:0.105833;stroke-miterlimit:4;stroke-dasharray:none" /> `_\ ) @@ -88,15 +90,15 @@ then all the :math:`w_i` will be equal to the full integration width and the dif However, if the integration range covers region with no data (e.g. beyond the kinematic limits) then the two cuts *may* look very different because :math:`C_i^{\mathrm{integration}}` will weight -regions with more data more heavily than regions without data. +regions with data more heavily than regions without data. Non-PSD Slice ------------- For **non-PSD** data, MSlice uses *fractional rebinning*, where it first calculates the -overlap area between the input and output bins, and then sums only the fractions of the -signal of the input bins within the output bin (darker green shading in bottom right). +overlap area between the input and output bins, and then sums only the fraction of the +signal of the input bins which overlaps the output bin. The output signal is computed as: @@ -111,10 +113,10 @@ and the output uncertainty as: where :math:`f_{kl}` is the fractional overlap of the input :math:`(k,l)`\ th bin with the output :math:`(i,j)`\ th bin. -This is illustrated in the image at the start of the page by the square on the right +This is illustrated in the :ref:`figure ` at the start of the page by the square on the right hand side with blue triangular and orange quadrilateral shaded regions. The blue and orange shading illustrates the fractional overlap areas which weights -the signal in the top left and top right input bins (large parallelograms). +the signal in the top left and top right input bins (large parallelograms) respectively. Non-PSD Cuts ------------ @@ -137,14 +139,14 @@ and 100 bins in the integration axis and then sums them as: C_i^{\mathrm{integration}} = \left. N_i \sum_{j \in \mathcal{F}} Y_{ij} w_{ij} \right/ \sum_{j \in \mathcal{F}} F_{ij} where :math:`N_i = \sum_{j \in \mathcal{D}} 1` is the number of :math:`j` bins at a given -:math:`i` with non-zero fraction (e.g. if the integration contains only regions with data +:math:`i` *with non-zero fraction* (e.g. if the integration contains only regions with data then :math:`N_i` = 100, otherwise :math:`N_i` will be less), and :math:`w_{ij}` is the width along the :math:`j`\ th axis of the :math:`(i,j)`\ th bin. The :math:`N_i` normalisation is needed because in the limiting case where all the fractions -:math:`F_i` are unity, the denominator would be :math:`N_i`, so we recover the usual +:math:`F_{ij}` are unity, the denominator would be :math:`N_i`, so we recover the usual expression for integrating over a distribution. Note that as previously, :math:`\mathcal{D}` indicates the region within the integration -range with non-zero fractions. +range with data (in this case equivalent to regions with non-zero fractions). Like with the **PSD** case there is thus an :math:`i` dependent scaling factor :math:`N_i w` (assuming all the bins have the same width) between ``Cuts`` computed using the ``Rebin`` or @@ -185,7 +187,7 @@ However, an ``Integration`` over :math:`|Q|` instead of energy will yield units average ``Rebin`` which will leave the units unchanged. Unfortunately, the input files read by MSlice do not indicate if the signal values saved -are in absolute units or not, so MSlice cannot automatically write the correct units to axes +are in absolute units or not, so MSlice cannot automatically display the correct units on plots - this is left to the user. A note on the regions of validity of the two algorithms @@ -193,8 +195,8 @@ A note on the regions of validity of the two algorithms As can be seen in the example above, the ``Integration`` cut algorithm will produce low signals where there is less data, whereas the ``Rebin`` cut algorithm will amplify the signals in such -regions - effectively assuming that the signal is constant across the integration range and -thus extrapolating over regions without data (so the only manifestation of the lack of data +regions. In effect, it assumes that the signal is constant across the integration range and +can be extrapolated over regions without data (so the only manifestation of the lack of data are larger errorbars associated with these bins). This assumption *may* be valid for density-of-states (DOS) type cuts where one would expect