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59                7.1
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138      <li>disaggregation</li>
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152           
153  <h1>Source code for disaggregation</h1><div class="highlight"><pre>
154<span></span><span class="ch">#!/usr/bin/env python</span>
155<span class="c1"># -*- coding: utf-8 -*-</span>
156<span class="c1">#*******************************************************************************</span>
157<span class="c1"># @Author: Anne Philipp (University of Vienna)</span>
158<span class="c1">#</span>
159<span class="c1"># @Date: March 2018</span>
160<span class="c1">#</span>
161<span class="c1"># @Change History:</span>
162<span class="c1">#</span>
163<span class="c1">#    November 2015 - Leopold Haimberger (University of Vienna):</span>
164<span class="c1">#        - migration of the methods dapoly and darain from Fortran</span>
165<span class="c1">#          (flex_extract_v6 and earlier) to Python</span>
166<span class="c1">#</span>
167<span class="c1">#    April 2018 - Anne Philipp (University of Vienna):</span>
168<span class="c1">#        - applied PEP8 style guide</span>
169<span class="c1">#        - added structured documentation</span>
170<span class="c1">#        - outsourced the disaggregation functions dapoly and darain</span>
171<span class="c1">#          to a new module named disaggregation</span>
172<span class="c1">#</span>
173<span class="c1"># @License:</span>
174<span class="c1">#    (C) Copyright 2015-2018.</span>
175<span class="c1">#</span>
176<span class="c1">#    This software is licensed under the terms of the Apache Licence Version 2.0</span>
177<span class="c1">#    which can be obtained at http://www.apache.org/licenses/LICENSE-2.0.</span>
178<span class="c1">#</span>
179<span class="c1"># @Module Description:</span>
180<span class="c1">#    disaggregation of deaccumulated flux data from an ECMWF model FG field.</span>
181<span class="c1">#    Initially the flux data to be concerned are:</span>
182<span class="c1">#    - large-scale precipitation</span>
183<span class="c1">#    - convective precipitation</span>
184<span class="c1">#    - surface sensible heat flux</span>
185<span class="c1">#    - surface solar radiation</span>
186<span class="c1">#    - u stress</span>
187<span class="c1">#    - v stress</span>
188<span class="c1">#    Different versions of disaggregation is provided for rainfall</span>
189<span class="c1">#    data (darain, modified linear) and the surface fluxes and</span>
190<span class="c1">#    stress data (dapoly, cubic polynomial).</span>
191<span class="c1">#</span>
192<span class="c1"># @Module Content:</span>
193<span class="c1">#    - dapoly</span>
194<span class="c1">#    - darain</span>
195<span class="c1">#    - IA3</span>
196<span class="c1">#</span>
197<span class="c1">#*******************************************************************************</span>
198
199<span class="c1"># ------------------------------------------------------------------------------</span>
200<span class="c1"># MODULES</span>
201<span class="c1"># ------------------------------------------------------------------------------</span>
202
203<span class="c1"># ------------------------------------------------------------------------------</span>
204<span class="c1"># FUNCTIONS</span>
205<span class="c1"># ------------------------------------------------------------------------------</span>
206<div class="viewcode-block" id="dapoly"><a class="viewcode-back" href="../api.html#disaggregation.dapoly">[docs]</a><span class="k">def</span> <span class="nf">dapoly</span><span class="p">(</span><span class="n">alist</span><span class="p">):</span>
207    <span class="sd">&quot;&quot;&quot;Cubic polynomial interpolation of deaccumulated fluxes.</span>
208
209<span class="sd">    Interpolation of deaccumulated fluxes of an ECMWF model FG field</span>
210<span class="sd">    using a cubic polynomial solution which conserves the integrals</span>
211<span class="sd">    of the fluxes within each timespan.</span>
212<span class="sd">    Disaggregation is done for 4 accumluated timespans which</span>
213<span class="sd">    generates a new, disaggregated value which is output at the</span>
214<span class="sd">    central point of the 4 accumulation timespans.</span>
215<span class="sd">    This new point is used for linear interpolation of the complete</span>
216<span class="sd">    timeseries afterwards.</span>
217
218<span class="sd">    Parameters</span>
219<span class="sd">    ----------</span>
220<span class="sd">    alist : :obj:`list` of :obj:`array` of :obj:`float`</span>
221<span class="sd">        List of 4 timespans as 2-dimensional, horizontal fields.</span>
222<span class="sd">        E.g. [[array_t1], [array_t2], [array_t3], [array_t4]]</span>
223
224<span class="sd">    Return</span>
225<span class="sd">    ------</span>
226<span class="sd">    nfield : :obj:`array` of :obj:`float`</span>
227<span class="sd">        Interpolated flux at central point of accumulation timespan.</span>
228
229<span class="sd">    Note</span>
230<span class="sd">    ----</span>
231<span class="sd">    March 2000    : P. JAMES</span>
232<span class="sd">        Original author</span>
233
234<span class="sd">    June 2003     : A. BECK</span>
235<span class="sd">        Adaptations</span>
236
237<span class="sd">    November 2015 : Leopold Haimberger (University of Vienna)</span>
238<span class="sd">        Migration from Fortran to Python</span>
239
240<span class="sd">    &quot;&quot;&quot;</span>
241
242    <span class="n">pya</span> <span class="o">=</span> <span class="p">(</span><span class="n">alist</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">-</span> <span class="n">alist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mf">3.</span> <span class="o">*</span> <span class="p">(</span><span class="n">alist</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">alist</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span> <span class="o">/</span> <span class="mf">6.</span>
243    <span class="n">pyb</span> <span class="o">=</span> <span class="p">(</span><span class="n">alist</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="n">alist</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">/</span> <span class="mf">2.</span> <span class="o">-</span> <span class="n">alist</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mf">9.</span> <span class="o">*</span> <span class="n">pya</span> <span class="o">/</span> <span class="mf">2.</span>
244    <span class="n">pyc</span> <span class="o">=</span> <span class="n">alist</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">alist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mf">7.</span> <span class="o">*</span> <span class="n">pya</span> <span class="o">/</span> <span class="mf">2.</span> <span class="o">-</span> <span class="mf">2.</span> <span class="o">*</span> <span class="n">pyb</span>
245    <span class="n">pyd</span> <span class="o">=</span> <span class="n">alist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">pya</span> <span class="o">/</span> <span class="mf">4.</span> <span class="o">-</span> <span class="n">pyb</span> <span class="o">/</span> <span class="mf">3.</span> <span class="o">-</span> <span class="n">pyc</span> <span class="o">/</span> <span class="mf">2.</span>
246    <span class="n">nfield</span> <span class="o">=</span> <span class="mf">8.</span> <span class="o">*</span> <span class="n">pya</span> <span class="o">+</span> <span class="mf">4.</span> <span class="o">*</span> <span class="n">pyb</span> <span class="o">+</span> <span class="mf">2.</span> <span class="o">*</span> <span class="n">pyc</span> <span class="o">+</span> <span class="n">pyd</span>
247
248    <span class="k">return</span> <span class="n">nfield</span></div>
249
250
251<div class="viewcode-block" id="darain"><a class="viewcode-back" href="../api.html#disaggregation.darain">[docs]</a><span class="k">def</span> <span class="nf">darain</span><span class="p">(</span><span class="n">alist</span><span class="p">):</span>
252    <span class="sd">&quot;&quot;&quot;Linear interpolation of deaccumulated fluxes.</span>
253
254<span class="sd">    Interpolation of deaccumulated fluxes of an ECMWF model FG rainfall</span>
255<span class="sd">    field using a modified linear solution which conserves the integrals</span>
256<span class="sd">    of the fluxes within each timespan.</span>
257<span class="sd">    Disaggregation is done for 4 accumluated timespans which generates</span>
258<span class="sd">    a new, disaggregated value which is output at the central point</span>
259<span class="sd">    of the 4 accumulation timespans. This new point is used for linear</span>
260<span class="sd">    interpolation of the complete timeseries afterwards.</span>
261
262<span class="sd">    Parameters</span>
263<span class="sd">    ----------</span>
264<span class="sd">    alist : :obj:`list` of :obj:`array` of :obj:`float`</span>
265<span class="sd">        List of 4 timespans as 2-dimensional, horizontal fields.</span>
266<span class="sd">        E.g. [[array_t1], [array_t2], [array_t3], [array_t4]]</span>
267
268<span class="sd">    Return</span>
269<span class="sd">    ------</span>
270<span class="sd">    nfield : :obj:`array` of :obj:`float`</span>
271<span class="sd">        Interpolated flux at central point of accumulation timespan.</span>
272
273<span class="sd">    Note</span>
274<span class="sd">    ----</span>
275<span class="sd">    March 2000    : P. JAMES</span>
276<span class="sd">        Original author</span>
277
278<span class="sd">    June 2003     : A. BECK</span>
279<span class="sd">        Adaptations</span>
280
281<span class="sd">    November 2015 : Leopold Haimberger (University of Vienna)</span>
282<span class="sd">        Migration from Fortran to Python</span>
283<span class="sd">    &quot;&quot;&quot;</span>
284
285    <span class="n">xa</span> <span class="o">=</span> <span class="n">alist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
286    <span class="n">xb</span> <span class="o">=</span> <span class="n">alist</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
287    <span class="n">xc</span> <span class="o">=</span> <span class="n">alist</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
288    <span class="n">xd</span> <span class="o">=</span> <span class="n">alist</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
289    <span class="n">xa</span><span class="p">[</span><span class="n">xa</span> <span class="o">&lt;</span> <span class="mf">0.</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.</span>
290    <span class="n">xb</span><span class="p">[</span><span class="n">xb</span> <span class="o">&lt;</span> <span class="mf">0.</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.</span>
291    <span class="n">xc</span><span class="p">[</span><span class="n">xc</span> <span class="o">&lt;</span> <span class="mf">0.</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.</span>
292    <span class="n">xd</span><span class="p">[</span><span class="n">xd</span> <span class="o">&lt;</span> <span class="mf">0.</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.</span>
293
294    <span class="n">xac</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">xb</span>
295    <span class="n">mask</span> <span class="o">=</span> <span class="n">xa</span> <span class="o">+</span> <span class="n">xc</span> <span class="o">&gt;</span> <span class="mf">0.</span>
296    <span class="n">xac</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">=</span> <span class="n">xb</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">*</span> <span class="n">xc</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">/</span> <span class="p">(</span><span class="n">xa</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">+</span> <span class="n">xc</span><span class="p">[</span><span class="n">mask</span><span class="p">])</span>
297    <span class="n">xbd</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">xc</span>
298    <span class="n">mask</span> <span class="o">=</span> <span class="n">xb</span> <span class="o">+</span> <span class="n">xd</span> <span class="o">&gt;</span> <span class="mf">0.</span>
299    <span class="n">xbd</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">=</span> <span class="n">xb</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">*</span> <span class="n">xc</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">/</span> <span class="p">(</span><span class="n">xb</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">+</span> <span class="n">xd</span><span class="p">[</span><span class="n">mask</span><span class="p">])</span>
300    <span class="n">nfield</span> <span class="o">=</span> <span class="n">xac</span> <span class="o">+</span> <span class="n">xbd</span>
301
302    <span class="k">return</span> <span class="n">nfield</span></div>
303
304<div class="viewcode-block" id="IA3"><a class="viewcode-back" href="../api.html#disaggregation.IA3">[docs]</a><span class="k">def</span> <span class="nf">IA3</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
305    <span class="sd">&quot;&quot;&quot; Interpolation with a non-negative geometric mean based algorithm.</span>
306
307<span class="sd">    The original grid is reconstructed by adding two sampling points in each</span>
308<span class="sd">    data series interval. This subgrid is used to keep all information during</span>
309<span class="sd">    the interpolation within the associated interval. Additionally, an advanced</span>
310<span class="sd">    monotonicity filter is applied to improve the monotonicity properties of</span>
311<span class="sd">    the series.</span>
312
313<span class="sd">    Note</span>
314<span class="sd">    ----</span>
315<span class="sd">    Copyright 2017</span>
316<span class="sd">    Sabine Hittmeir, Anne Philipp, Petra Seibert</span>
317
318<span class="sd">    This work is licensed under the Creative Commons Attribution 4.0</span>
319<span class="sd">    International License. To view a copy of this license, visit</span>
320<span class="sd">    http://creativecommons.org/licenses/by/4.0/ or send a letter to</span>
321<span class="sd">    Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.</span>
322
323<span class="sd">    Parameters</span>
324<span class="sd">    ----------</span>
325<span class="sd">    g : :obj:`list` of :obj:`float`</span>
326<span class="sd">        Complete data series that will be interpolated having</span>
327<span class="sd">        the dimension of the original raw series.</span>
328
329<span class="sd">    Return</span>
330<span class="sd">    ------</span>
331<span class="sd">    f : :obj:`list` of :obj:`float`</span>
332<span class="sd">        The interpolated data series with additional subgrid points.</span>
333<span class="sd">        Its dimension is equal to the length of the input data series</span>
334<span class="sd">        times three.</span>
335
336
337<span class="sd">    References</span>
338<span class="sd">    ----------</span>
339<span class="sd">    For more information see article:</span>
340<span class="sd">    Hittmeir, S.; Philipp, A.; Seibert, P. (2017): A conservative</span>
341<span class="sd">    interpolation scheme for extensive quantities with application to the</span>
342<span class="sd">    Lagrangian particle dispersion model FLEXPART.,</span>
343<span class="sd">    Geoscientific Model Development</span>
344<span class="sd">    &quot;&quot;&quot;</span>
345
346    <span class="c1">#######################  variable description #############################</span>
347    <span class="c1">#                                                                         #</span>
348    <span class="c1"># i      - index variable for looping over the data series                #</span>
349    <span class="c1"># g      - input data series                                              #</span>
350    <span class="c1"># f      - interpolated and filtered data series with additional          #</span>
351    <span class="c1">#          grid points                                                    #</span>
352    <span class="c1"># fi     - function value at position i, f_i                              #</span>
353    <span class="c1"># fi1    - first  sub-grid function value f_i^1                           #</span>
354    <span class="c1"># fi2    - second sub-grid function value f_i^2                           #</span>
355    <span class="c1"># fip1   - next function value at position i+1, f_(i+1)                   #</span>
356    <span class="c1"># dt     - time step                                                      #</span>
357    <span class="c1"># fmon   - monotonicity filter                                            #</span>
358    <span class="c1">#                                                                         #</span>
359    <span class="c1">###########################################################################</span>
360
361
362    <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
363
364    <span class="c1"># time step</span>
365    <span class="n">dt</span><span class="o">=</span><span class="mf">1.0</span>
366
367    <span class="c1">############### Non-negative Geometric Mean Based Algorithm ###############</span>
368
369    <span class="c1"># for the left boundary the following boundary condition is valid:</span>
370    <span class="c1"># the value at t=0 of the interpolation algorithm coincides with the</span>
371    <span class="c1"># first data value according to the persistence hypothesis</span>
372    <span class="n">f</span><span class="o">=</span><span class="p">[</span><span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
373
374    <span class="c1"># compute two first sub-grid intervals without monotonicity check</span>
375    <span class="c1"># go through the data series and extend each interval by two sub-grid</span>
376    <span class="c1"># points and interpolate the corresponding data values</span>
377    <span class="c1"># except for the last interval due to boundary conditions</span>
378    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">):</span>
379
380        <span class="c1"># as a requirement:</span>
381        <span class="c1"># if there is a zero data value such that g[i]=0, then the whole</span>
382        <span class="c1"># interval in f has to be zero to such that f[i+1]=f[i+2]=f[i+3]=0</span>
383        <span class="c1"># according to Eq. (6)</span>
384        <span class="k">if</span> <span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">==</span><span class="mf">0.</span><span class="p">:</span>
385            <span class="n">f</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span><span class="mf">0.</span><span class="p">,</span><span class="mf">0.</span><span class="p">])</span>
386
387        <span class="c1"># otherwise the sub-grid values are calculated and added to the list</span>
388        <span class="k">else</span><span class="p">:</span>
389            <span class="c1"># temporal save of last value in interpolated list</span>
390            <span class="c1"># since it is the left boundary and hence the new (fi) value</span>
391            <span class="n">fi</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
392
393            <span class="c1"># the value at the end of the interval (fip1) is prescribed by the</span>
394            <span class="c1"># geometric mean, restricted such that non-negativity is guaranteed</span>
395            <span class="c1"># according to Eq. (25)</span>
396            <span class="n">fip1</span><span class="o">=</span><span class="nb">min</span><span class="p">(</span> <span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">,</span> <span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="p">)</span>
397
398            <span class="c1"># the function value at the first sub-grid point (fi1) is determined</span>
399            <span class="c1"># according to the equal area condition with Eq. (19)</span>
400            <span class="n">fi1</span><span class="o">=</span><span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fip1</span><span class="o">-</span><span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fi</span>
401
402            <span class="c1"># the function value at the second sub-grid point (fi2) is determined</span>
403            <span class="c1"># according Eq. (18)</span>
404            <span class="n">fi2</span><span class="o">=</span><span class="n">fi1</span><span class="o">+</span><span class="mf">1.</span><span class="o">/</span><span class="mf">3.</span><span class="o">*</span><span class="p">(</span><span class="n">fip1</span><span class="o">-</span><span class="n">fi</span><span class="p">)</span>
405
406            <span class="c1"># add next interval of interpolated (sub-)grid values</span>
407            <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fi1</span><span class="p">)</span>
408            <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fi2</span><span class="p">)</span>
409            <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fip1</span><span class="p">)</span>
410
411    <span class="c1"># compute rest of the data series intervals</span>
412    <span class="c1"># go through the data series and extend each interval by two sub-grid</span>
413    <span class="c1"># points and interpolate the corresponding data values</span>
414    <span class="c1"># except for the last interval due to boundary conditions</span>
415    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
416
417        <span class="c1"># as a requirement:</span>
418        <span class="c1"># if there is a zero data value such that g[i]=0, then the whole</span>
419        <span class="c1"># interval in f has to be zero to such that f[i+1]=f[i+2]=f[i+3]=0</span>
420        <span class="c1"># according to Eq. (6)</span>
421        <span class="k">if</span> <span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">==</span><span class="mf">0.</span><span class="p">:</span>
422            <span class="c1"># apply monotonicity filter for interval before</span>
423            <span class="c1"># check if there is &quot;M&quot; or &quot;W&quot; shape</span>
424            <span class="k">if</span>     <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span> \
425               <span class="ow">and</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span> \
426               <span class="ow">and</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span><span class="p">:</span>
427
428                <span class="c1"># the monotonicity filter corrects the value at (fim1) by</span>
429                <span class="c1"># substituting (fim1) with (fmon), see Eq. (27), (28) and (29)</span>
430                <span class="n">fmon</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> \
431                           <span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> \
432                           <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,(</span><span class="mf">18.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="mf">5.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">])</span> <span class="o">*</span>
433                                         <span class="p">(</span><span class="mf">18.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mf">5.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]))))</span>
434
435                <span class="c1"># recomputation of the sub-grid interval values while the</span>
436                <span class="c1"># interval boundaries (fi) and (fip2) remains unchanged</span>
437                <span class="c1"># see Eq. (18) and (19)</span>
438                <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">=</span><span class="n">fmon</span>
439                <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">]</span><span class="o">=</span><span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fmon</span><span class="o">-</span><span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">]</span>
440                <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">]</span><span class="o">=</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="n">fmon</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">])</span><span class="o">/</span><span class="mf">3.</span>
441                <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">=</span><span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fmon</span>
442                <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">=</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">fmon</span><span class="p">)</span><span class="o">/</span><span class="mf">3.</span>
443
444            <span class="n">f</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span><span class="mf">0.</span><span class="p">,</span><span class="mf">0.</span><span class="p">])</span>
445
446        <span class="c1"># otherwise the sub-grid values are calculated and added to the list</span>
447        <span class="k">else</span><span class="p">:</span>
448            <span class="c1"># temporal save of last value in interpolated list</span>
449            <span class="c1"># since it is the left boundary and hence the new (fi) value</span>
450            <span class="n">fi</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
451
452            <span class="c1"># the value at the end of the interval (fip1) is prescribed by the</span>
453            <span class="c1"># geometric mean, restricted such that non-negativity is guaranteed</span>
454            <span class="c1"># according to Eq. (25)</span>
455            <span class="n">fip1</span><span class="o">=</span><span class="nb">min</span><span class="p">(</span> <span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">,</span> <span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="p">)</span>
456
457            <span class="c1"># the function value at the first sub-grid point (fi1) is determined</span>
458            <span class="c1"># according to the equal area condition with Eq. (19)</span>
459            <span class="n">fi1</span><span class="o">=</span><span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fip1</span><span class="o">-</span><span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fi</span>
460
461            <span class="c1"># the function value at the second sub-grid point (fi2) is determined</span>
462            <span class="c1"># according Eq. (18)</span>
463            <span class="n">fi2</span><span class="o">=</span><span class="n">fi1</span><span class="o">+</span><span class="mf">1.</span><span class="o">/</span><span class="mf">3.</span><span class="o">*</span><span class="p">(</span><span class="n">fip1</span><span class="o">-</span><span class="n">fi</span><span class="p">)</span>
464
465            <span class="c1"># apply monotonicity filter for interval before</span>
466            <span class="c1"># check if there is &quot;M&quot; or &quot;W&quot; shape</span>
467            <span class="k">if</span>     <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span> \
468               <span class="ow">and</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span> \
469               <span class="ow">and</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span><span class="p">:</span>
470
471                <span class="c1"># the monotonicity filter corrects the value at (fim1) by</span>
472                <span class="c1"># substituting (fim1) with fmon, see Eq. (27), (28) and (29)</span>
473                <span class="n">fmon</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> \
474                           <span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> \
475                           <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,(</span><span class="mf">18.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="mf">5.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">])</span> <span class="o">*</span>
476                                         <span class="p">(</span><span class="mf">18.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mf">5.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]))))</span>
477
478                <span class="c1"># recomputation of the sub-grid interval values while the</span>
479                <span class="c1"># interval boundaries (fi) and (fip2) remains unchanged</span>
480                <span class="c1"># see Eq. (18) and (19)</span>
481                <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">=</span><span class="n">fmon</span>
482                <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">]</span><span class="o">=</span><span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fmon</span><span class="o">-</span><span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">]</span>
483                <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">]</span><span class="o">=</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="n">fmon</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">])</span><span class="o">/</span><span class="mf">3.</span>
484                <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">=</span><span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fmon</span>
485                <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">=</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">fmon</span><span class="p">)</span><span class="o">/</span><span class="mf">3.</span>
486
487            <span class="c1"># add next interval of interpolated (sub-)grid values</span>
488            <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fi1</span><span class="p">)</span>
489            <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fi2</span><span class="p">)</span>
490            <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fip1</span><span class="p">)</span>
491
492    <span class="c1"># separate treatment of the final interval</span>
493
494    <span class="c1"># as a requirement:</span>
495    <span class="c1"># if there is a zero data value such that g[i]=0, then the whole</span>
496    <span class="c1"># interval in f has to be zero to such that f[i+1]=f[i+2]=f[i+3]=0</span>
497    <span class="c1"># according to Eq. (6)</span>
498    <span class="k">if</span> <span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="mf">0.</span><span class="p">:</span>
499        <span class="c1"># apply monotonicity filter for interval before</span>
500        <span class="c1"># check if there is &quot;M&quot; or &quot;W&quot; shape</span>
501        <span class="k">if</span>     <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span> \
502           <span class="ow">and</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span> \
503           <span class="ow">and</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span><span class="p">:</span>
504
505            <span class="c1"># the monotonicity filter corrects the value at (fim1) by</span>
506            <span class="c1"># substituting (fim1) with (fmon), see Eq. (27), (28) and (29)</span>
507            <span class="n">fmon</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">],</span> \
508                       <span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> \
509                       <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,(</span><span class="mf">18.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span> <span class="o">-</span> <span class="mf">5.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">])</span> <span class="o">*</span>
510                                     <span class="p">(</span><span class="mf">18.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="mf">5.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]))))</span>
511
512            <span class="c1"># recomputation of the sub-grid interval values while the</span>
513            <span class="c1"># interval boundaries (fi) and (fip2) remains unchanged</span>
514            <span class="c1"># see Eq. (18) and (19)</span>
515            <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">=</span><span class="n">fmon</span>
516            <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">]</span><span class="o">=</span><span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fmon</span><span class="o">-</span><span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">]</span>
517            <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">]</span><span class="o">=</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="n">fmon</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">])</span><span class="o">/</span><span class="mf">3.</span>
518            <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">=</span><span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fmon</span>
519            <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">=</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">fmon</span><span class="p">)</span><span class="o">/</span><span class="mf">3.</span>
520
521        <span class="n">f</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span><span class="mf">0.</span><span class="p">,</span><span class="mf">0.</span><span class="p">])</span>
522
523    <span class="c1"># otherwise the sub-grid values are calculated and added to the list</span>
524    <span class="c1"># using the persistence hypothesis as boundary condition</span>
525    <span class="k">else</span><span class="p">:</span>
526        <span class="c1"># temporal save of last value in interpolated list</span>
527        <span class="c1"># since it is the left boundary and hence the new (fi) value</span>
528        <span class="n">fi</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
529        <span class="c1"># since last interval in series, last value is also fip1</span>
530        <span class="n">fip1</span> <span class="o">=</span> <span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
531        <span class="c1"># the function value at the first sub-grid point (fi1) is determined</span>
532        <span class="c1"># according to the equal area condition with Eq. (19)</span>
533        <span class="n">fi1</span> <span class="o">=</span> <span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fip1</span><span class="o">-</span><span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fi</span>
534        <span class="c1"># the function value at the second sub-grid point (fi2) is determined</span>
535        <span class="c1"># according Eq. (18)</span>
536        <span class="n">fi2</span> <span class="o">=</span> <span class="n">fi1</span><span class="o">+</span><span class="n">dt</span><span class="o">/</span><span class="mf">3.</span><span class="o">*</span><span class="p">(</span><span class="n">fip1</span><span class="o">-</span><span class="n">fi</span><span class="p">)</span>
537
538        <span class="c1"># apply monotonicity filter for interval before</span>
539        <span class="c1"># check if there is &quot;M&quot; or &quot;W&quot; shape</span>
540        <span class="k">if</span>     <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span> \
541           <span class="ow">and</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span> \
542           <span class="ow">and</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">])</span><span class="o">==-</span><span class="mi">1</span><span class="p">:</span>
543
544            <span class="c1"># the monotonicity filter corrects the value at (fim1) by</span>
545            <span class="c1"># substituting (fim1) with (fmon), see Eq. (27), (28) and (29)</span>
546            <span class="n">fmon</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">],</span> \
547                       <span class="mf">3.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> \
548                       <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,(</span><span class="mf">18.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span> <span class="o">-</span> <span class="mf">5.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">])</span> <span class="o">*</span>
549                                     <span class="p">(</span><span class="mf">18.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="mf">5.</span><span class="o">/</span><span class="mf">13.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]))))</span>
550
551            <span class="c1"># recomputation of the sub-grid interval values while the</span>
552            <span class="c1"># interval boundaries (fi) and (fip2) remains unchanged</span>
553            <span class="c1"># see Eq. (18) and (19)</span>
554            <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span><span class="o">=</span><span class="n">fmon</span>
555            <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">]</span><span class="o">=</span><span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fmon</span><span class="o">-</span><span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">]</span>
556            <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">]</span><span class="o">=</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">6</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="n">fmon</span><span class="o">-</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">7</span><span class="p">])</span><span class="o">/</span><span class="mf">3.</span>
557            <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">=</span><span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">fmon</span>
558            <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">=</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span><span class="o">+</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">fmon</span><span class="p">)</span><span class="o">/</span><span class="mf">3.</span>
559
560        <span class="c1"># add next interval of interpolated (sub-)grid values</span>
561        <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fi1</span><span class="p">)</span>
562        <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fi2</span><span class="p">)</span>
563        <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fip1</span><span class="p">)</span>
564
565    <span class="k">return</span> <span class="n">f</span></div>
566</pre></div>
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