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161           
162  <h1>Source code for disaggregation</h1><div class="highlight"><pre>
163<span></span><span class="ch">#!/usr/bin/env python3</span>
164<span class="c1"># -*- coding: utf-8 -*-</span>
165<span class="c1">#*******************************************************************************</span>
166<span class="c1"># @Author: Anne Philipp (University of Vienna)</span>
167<span class="c1">#</span>
168<span class="c1"># @Date: March 2018</span>
169<span class="c1">#</span>
170<span class="c1"># @Change History:</span>
171<span class="c1">#</span>
172<span class="c1">#    November 2015 - Leopold Haimberger (University of Vienna):</span>
173<span class="c1">#        - migration of the methods dapoly and darain from Fortran</span>
174<span class="c1">#          (flex_extract_v6 and earlier) to Python</span>
175<span class="c1">#</span>
176<span class="c1">#    April 2018 - Anne Philipp (University of Vienna):</span>
177<span class="c1">#        - applied PEP8 style guide</span>
178<span class="c1">#        - added structured documentation</span>
179<span class="c1">#        - outsourced the disaggregation functions dapoly and darain</span>
180<span class="c1">#          to a new module named disaggregation</span>
181<span class="c1">#        - added the new disaggregation method for precipitation</span>
182<span class="c1">#</span>
183<span class="c1">#    June 2020 - Anne Philipp (University of Vienna):</span>
184<span class="c1">#        - reformulated formular for dapoly</span>
185<span class="c1">#</span>
186<span class="c1"># @License:</span>
187<span class="c1">#    (C) Copyright 2014-2020.</span>
188<span class="c1">#    Anne Philipp, Leopold Haimberger</span>
189<span class="c1">#</span>
190<span class="c1">#    SPDX-License-Identifier: CC-BY-4.0</span>
191<span class="c1">#</span>
192<span class="c1">#    This work is licensed under the Creative Commons Attribution 4.0</span>
193<span class="c1">#    International License. To view a copy of this license, visit</span>
194<span class="c1">#    http://creativecommons.org/licenses/by/4.0/ or send a letter to</span>
195<span class="c1">#    Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.</span>
196<span class="c1">#</span>
197<span class="c1"># @Methods:</span>
198<span class="c1">#    - dapoly</span>
199<span class="c1">#    - darain</span>
200<span class="c1">#    - IA3</span>
201<span class="c1">#*******************************************************************************</span>
202<span class="sd">&#39;&#39;&#39;Disaggregation of deaccumulated flux data from an ECMWF model FG field.</span>
203
204<span class="sd">Initially the flux data to be concerned are:</span>
205<span class="sd">    - large-scale precipitation</span>
206<span class="sd">    - convective precipitation</span>
207<span class="sd">    - surface sensible heat flux</span>
208<span class="sd">    - surface solar radiation</span>
209<span class="sd">    - u stress</span>
210<span class="sd">    - v stress</span>
211
212<span class="sd">Different versions of disaggregation is provided for rainfall</span>
213<span class="sd">data (darain, modified linear) and the surface fluxes and</span>
214<span class="sd">stress data (dapoly, cubic polynomial).</span>
215<span class="sd">&#39;&#39;&#39;</span>
216
217<span class="c1"># ------------------------------------------------------------------------------</span>
218<span class="c1"># MODULES</span>
219<span class="c1"># ------------------------------------------------------------------------------</span>
220
221<span class="c1"># ------------------------------------------------------------------------------</span>
222<span class="c1"># FUNCTIONS</span>
223<span class="c1"># ------------------------------------------------------------------------------</span>
224<div class="viewcode-block" id="dapoly"><a class="viewcode-back" href="../Documentation/Api/api_python.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>
225    <span class="sd">&quot;&quot;&quot;Cubic polynomial interpolation of deaccumulated fluxes.</span>
226
227<span class="sd">    Interpolation of deaccumulated fluxes of an ECMWF model FG field</span>
228<span class="sd">    using a cubic polynomial solution which conserves the integrals</span>
229<span class="sd">    of the fluxes within each timespan.</span>
230<span class="sd">    Disaggregation is done for 4 accumluated timespans which</span>
231<span class="sd">    generates a new, disaggregated value which is output at the</span>
232<span class="sd">    central point of the 4 accumulation timespans.</span>
233<span class="sd">    This new point is used for linear interpolation of the complete</span>
234<span class="sd">    timeseries afterwards.</span>
235
236<span class="sd">    Parameters</span>
237<span class="sd">    ----------</span>
238<span class="sd">    alist : list of array of float</span>
239<span class="sd">        List of 4 timespans as 2-dimensional, horizontal fields.</span>
240<span class="sd">        E.g. [[array_t1], [array_t2], [array_t3], [array_t4]]</span>
241
242<span class="sd">    Return</span>
243<span class="sd">    ------</span>
244<span class="sd">    nfield : array of float</span>
245<span class="sd">        Interpolated flux at central point of accumulation timespan.</span>
246
247<span class="sd">    Note</span>
248<span class="sd">    ----</span>
249<span class="sd">    March 2000    : P. JAMES</span>
250<span class="sd">        Original author</span>
251
252<span class="sd">    June 2003     : A. BECK</span>
253<span class="sd">        Adaptations</span>
254
255<span class="sd">    November 2015 : Leopold Haimberger (University of Vienna)</span>
256<span class="sd">        Migration from Fortran to Python</span>
257
258<span class="sd">    &quot;&quot;&quot;</span>
259   
260    <span class="n">nfield</span> <span class="o">=</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">alist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> \
261              <span class="mf">7.</span><span class="o">/</span><span class="mf">12.</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> \
262              <span class="mf">7.</span><span class="o">/</span><span class="mf">12.</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> \
263              <span class="mf">1.</span><span class="o">/</span><span class="mf">12.</span><span class="o">*</span><span class="n">alist</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
264
265    <span class="k">return</span> <span class="n">nfield</span></div>
266
267
268<div class="viewcode-block" id="darain"><a class="viewcode-back" href="../Documentation/Api/api_python.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>
269    <span class="sd">&quot;&quot;&quot;Linear interpolation of deaccumulated fluxes.</span>
270
271<span class="sd">    Interpolation of deaccumulated fluxes of an ECMWF model FG rainfall</span>
272<span class="sd">    field using a modified linear solution which conserves the integrals</span>
273<span class="sd">    of the fluxes within each timespan.</span>
274<span class="sd">    Disaggregation is done for 4 accumluated timespans which generates</span>
275<span class="sd">    a new, disaggregated value which is output at the central point</span>
276<span class="sd">    of the 4 accumulation timespans. This new point is used for linear</span>
277<span class="sd">    interpolation of the complete timeseries afterwards.</span>
278
279<span class="sd">    Parameters</span>
280<span class="sd">    ----------</span>
281<span class="sd">    alist : list of array of float</span>
282<span class="sd">        List of 4 timespans as 2-dimensional, horizontal fields.</span>
283<span class="sd">        E.g. [[array_t1], [array_t2], [array_t3], [array_t4]]</span>
284
285<span class="sd">    Return</span>
286<span class="sd">    ------</span>
287<span class="sd">    nfield : array of float</span>
288<span class="sd">        Interpolated flux at central point of accumulation timespan.</span>
289
290<span class="sd">    Note</span>
291<span class="sd">    ----</span>
292<span class="sd">    March 2000    : P. JAMES</span>
293<span class="sd">        Original author</span>
294
295<span class="sd">    June 2003     : A. BECK</span>
296<span class="sd">        Adaptations</span>
297
298<span class="sd">    November 2015 : Leopold Haimberger (University of Vienna)</span>
299<span class="sd">        Migration from Fortran to Python</span>
300<span class="sd">    &quot;&quot;&quot;</span>
301
302    <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>
303    <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>
304    <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>
305    <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>
306    <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>
307    <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>
308    <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>
309    <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>
310
311    <span class="n">xac</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">xb</span>
312    <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>
313    <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>
314    <span class="n">xbd</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">xc</span>
315    <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>
316    <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>
317    <span class="n">nfield</span> <span class="o">=</span> <span class="n">xac</span> <span class="o">+</span> <span class="n">xbd</span>
318
319    <span class="k">return</span> <span class="n">nfield</span></div>
320
321<div class="viewcode-block" id="IA3"><a class="viewcode-back" href="../Documentation/Api/api_python.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>
322    <span class="sd">&quot;&quot;&quot; Interpolation with a non-negative geometric mean based algorithm.</span>
323
324<span class="sd">    The original grid is reconstructed by adding two sampling points in each</span>
325<span class="sd">    data series interval. This subgrid is used to keep all information during</span>
326<span class="sd">    the interpolation within the associated interval. Additionally, an advanced</span>
327<span class="sd">    monotonicity filter is applied to improve the monotonicity properties of</span>
328<span class="sd">    the series.</span>
329
330<span class="sd">    Note</span>
331<span class="sd">    ----</span>
332<span class="sd">    (C) Copyright 2017-2019</span>
333<span class="sd">    Sabine Hittmeir, Anne Philipp, Petra Seibert</span>
334
335<span class="sd">    This work is licensed under the Creative Commons Attribution 4.0</span>
336<span class="sd">    International License. To view a copy of this license, visit</span>
337<span class="sd">    http://creativecommons.org/licenses/by/4.0/ or send a letter to</span>
338<span class="sd">    Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.</span>
339
340<span class="sd">    Parameters</span>
341<span class="sd">    ----------</span>
342<span class="sd">    g : list of float</span>
343<span class="sd">        Complete data series that will be interpolated having</span>
344<span class="sd">        the dimension of the original raw series.</span>
345
346<span class="sd">    Return</span>
347<span class="sd">    ------</span>
348<span class="sd">    f : list of float</span>
349<span class="sd">        The interpolated data series with additional subgrid points.</span>
350<span class="sd">        Its dimension is equal to the length of the input data series</span>
351<span class="sd">        times three.</span>
352
353
354<span class="sd">    References</span>
355<span class="sd">    ----------</span>
356<span class="sd">    For more information see article:</span>
357<span class="sd">    Hittmeir, S.; Philipp, A.; Seibert, P. (2017): A conservative</span>
358<span class="sd">    interpolation scheme for extensive quantities with application to the</span>
359<span class="sd">    Lagrangian particle dispersion model FLEXPART.,</span>
360<span class="sd">    Geoscientific Model Development</span>
361<span class="sd">    &quot;&quot;&quot;</span>
362
363    <span class="c1">#######################  variable description #############################</span>
364    <span class="c1">#                                                                         #</span>
365    <span class="c1"># i      - index variable for looping over the data series                #</span>
366    <span class="c1"># g      - input data series                                              #</span>
367    <span class="c1"># f      - interpolated and filtered data series with additional          #</span>
368    <span class="c1">#          grid points                                                    #</span>
369    <span class="c1"># fi     - function value at position i, f_i                              #</span>
370    <span class="c1"># fi1    - first  sub-grid function value f_i^1                           #</span>
371    <span class="c1"># fi2    - second sub-grid function value f_i^2                           #</span>
372    <span class="c1"># fip1   - next function value at position i+1, f_(i+1)                   #</span>
373    <span class="c1"># dt     - time step                                                      #</span>
374    <span class="c1"># fmon   - monotonicity filter                                            #</span>
375    <span class="c1">#                                                                         #</span>
376    <span class="c1">###########################################################################</span>
377
378
379    <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
380
381    <span class="c1"># time step</span>
382    <span class="n">dt</span> <span class="o">=</span> <span class="mf">1.0</span>
383
384    <span class="c1">############### Non-negative Geometric Mean Based Algorithm ###############</span>
385
386    <span class="c1"># for the left boundary the following boundary condition is valid:</span>
387    <span class="c1"># the value at t=0 of the interpolation algorithm coincides with the</span>
388    <span class="c1"># first data value according to the persistence hypothesis</span>
389    <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>
390
391    <span class="c1"># compute two first sub-grid intervals without monotonicity check</span>
392    <span class="c1"># go through the data series and extend each interval by two sub-grid</span>
393    <span class="c1"># points and interpolate the corresponding data values</span>
394    <span class="c1"># except for the last interval due to boundary conditions</span>
395    <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>
396
397        <span class="c1"># as a requirement:</span>
398        <span class="c1"># if there is a zero data value such that g[i]=0, then the whole</span>
399        <span class="c1"># interval in f has to be zero to such that f[i+1]=f[i+2]=f[i+3]=0</span>
400        <span class="c1"># according to Eq. (6)</span>
401        <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>
402            <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>
403
404        <span class="c1"># otherwise the sub-grid values are calculated and added to the list</span>
405        <span class="k">else</span><span class="p">:</span>
406            <span class="c1"># temporal save of last value in interpolated list</span>
407            <span class="c1"># since it is the left boundary and hence the new (fi) value</span>
408            <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>
409
410            <span class="c1"># the value at the end of the interval (fip1) is prescribed by the</span>
411            <span class="c1"># geometric mean, restricted such that non-negativity is guaranteed</span>
412            <span class="c1"># according to Eq. (25)</span>
413            <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="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="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>
414
415            <span class="c1"># the function value at the first sub-grid point (fi1) is determined</span>
416            <span class="c1"># according to the equal area condition with Eq. (19)</span>
417            <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>
418
419            <span class="c1"># the function value at the second sub-grid point (fi2) is determined</span>
420            <span class="c1"># according Eq. (18)</span>
421            <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>
422
423            <span class="c1"># add next interval of interpolated (sub-)grid values</span>
424            <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>
425            <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>
426            <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>
427
428    <span class="c1"># compute rest of the data series intervals</span>
429    <span class="c1"># go through the data series and extend each interval by two sub-grid</span>
430    <span class="c1"># points and interpolate the corresponding data values</span>
431    <span class="c1"># except for the last interval due to boundary conditions</span>
432    <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>
433
434        <span class="c1"># as a requirement:</span>
435        <span class="c1"># if there is a zero data value such that g[i]=0, then the whole</span>
436        <span class="c1"># interval in f has to be zero to such that f[i+1]=f[i+2]=f[i+3]=0</span>
437        <span class="c1"># according to Eq. (6)</span>
438        <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>
439            <span class="c1"># apply monotonicity filter for interval before</span>
440            <span class="c1"># check if there is &quot;M&quot; or &quot;W&quot; shape</span>
441            <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="o">-</span><span class="mi">1</span> \
442               <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="o">-</span><span class="mi">1</span> \
443               <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="o">-</span><span class="mi">1</span><span class="p">:</span>
444
445                <span class="c1"># the monotonicity filter corrects the value at (fim1) by</span>
446                <span class="c1"># substituting (fim1) with (fmon), see Eq. (27), (28) and (29)</span>
447                <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>
448                           <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>
449                           <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="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>
450                                       <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>
451
452                <span class="c1"># recomputation of the sub-grid interval values while the</span>
453                <span class="c1"># interval boundaries (fi) and (fip2) remains unchanged</span>
454                <span class="c1"># see Eq. (18) and (19)</span>
455                <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>
456                <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>
457                <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>
458                <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>
459                <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>
460
461            <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>
462
463        <span class="c1"># otherwise the sub-grid values are calculated and added to the list</span>
464        <span class="k">else</span><span class="p">:</span>
465            <span class="c1"># temporal save of last value in interpolated list</span>
466            <span class="c1"># since it is the left boundary and hence the new (fi) value</span>
467            <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>
468
469            <span class="c1"># the value at the end of the interval (fip1) is prescribed by the</span>
470            <span class="c1"># geometric mean, restricted such that non-negativity is guaranteed</span>
471            <span class="c1"># according to Eq. (25)</span>
472            <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="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="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>
473
474            <span class="c1"># the function value at the first sub-grid point (fi1) is determined</span>
475            <span class="c1"># according to the equal area condition with Eq. (19)</span>
476            <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>
477
478            <span class="c1"># the function value at the second sub-grid point (fi2) is determined</span>
479            <span class="c1"># according Eq. (18)</span>
480            <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>
481
482            <span class="c1"># apply monotonicity filter for interval before</span>
483            <span class="c1"># check if there is &quot;M&quot; or &quot;W&quot; shape</span>
484            <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="o">-</span><span class="mi">1</span> \
485               <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="o">-</span><span class="mi">1</span> \
486               <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="o">-</span><span class="mi">1</span><span class="p">:</span>
487
488                <span class="c1"># the monotonicity filter corrects the value at (fim1) by</span>
489                <span class="c1"># substituting (fim1) with fmon, see Eq. (27), (28) and (29)</span>
490                <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>
491                           <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>
492                           <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="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>
493                                       <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>
494
495                <span class="c1"># recomputation of the sub-grid interval values while the</span>
496                <span class="c1"># interval boundaries (fi) and (fip2) remains unchanged</span>
497                <span class="c1"># see Eq. (18) and (19)</span>
498                <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>
499                <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>
500                <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>
501                <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>
502                <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>
503
504            <span class="c1"># add next interval of interpolated (sub-)grid values</span>
505            <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>
506            <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>
507            <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>
508
509    <span class="c1"># separate treatment of the final interval</span>
510
511    <span class="c1"># as a requirement:</span>
512    <span class="c1"># if there is a zero data value such that g[i]=0, then the whole</span>
513    <span class="c1"># interval in f has to be zero to such that f[i+1]=f[i+2]=f[i+3]=0</span>
514    <span class="c1"># according to Eq. (6)</span>
515    <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>
516        <span class="c1"># apply monotonicity filter for interval before</span>
517        <span class="c1"># check if there is &quot;M&quot; or &quot;W&quot; shape</span>
518        <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="o">-</span><span class="mi">1</span> \
519           <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="o">-</span><span class="mi">1</span> \
520           <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="o">-</span><span class="mi">1</span><span class="p">:</span>
521
522            <span class="c1"># the monotonicity filter corrects the value at (fim1) by</span>
523            <span class="c1"># substituting (fim1) with (fmon), see Eq. (27), (28) and (29)</span>
524            <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>
525                       <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>
526                       <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="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>
527                                   <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>
528
529            <span class="c1"># recomputation of the sub-grid interval values while the</span>
530            <span class="c1"># interval boundaries (fi) and (fip2) remains unchanged</span>
531            <span class="c1"># see Eq. (18) and (19)</span>
532            <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>
533            <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>
534            <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>
535            <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>
536            <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>
537
538        <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>
539
540    <span class="c1"># otherwise the sub-grid values are calculated and added to the list</span>
541    <span class="c1"># using the persistence hypothesis as boundary condition</span>
542    <span class="k">else</span><span class="p">:</span>
543        <span class="c1"># temporal save of last value in interpolated list</span>
544        <span class="c1"># since it is the left boundary and hence the new (fi) value</span>
545        <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>
546        <span class="c1"># since last interval in series, last value is also fip1</span>
547        <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>
548        <span class="c1"># the function value at the first sub-grid point (fi1) is determined</span>
549        <span class="c1"># according to the equal area condition with Eq. (19)</span>
550        <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>
551        <span class="c1"># the function value at the second sub-grid point (fi2) is determined</span>
552        <span class="c1"># according Eq. (18)</span>
553        <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>
554
555        <span class="c1"># apply monotonicity filter for interval before</span>
556        <span class="c1"># check if there is &quot;M&quot; or &quot;W&quot; shape</span>
557        <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="o">-</span><span class="mi">1</span> \
558           <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="o">-</span><span class="mi">1</span> \
559           <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="o">-</span><span class="mi">1</span><span class="p">:</span>
560
561            <span class="c1"># the monotonicity filter corrects the value at (fim1) by</span>
562            <span class="c1"># substituting (fim1) with (fmon), see Eq. (27), (28) and (29)</span>
563            <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>
564                       <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>
565                       <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="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>
566                                   <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>
567
568            <span class="c1"># recomputation of the sub-grid interval values while the</span>
569            <span class="c1"># interval boundaries (fi) and (fip2) remains unchanged</span>
570            <span class="c1"># see Eq. (18) and (19)</span>
571            <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>
572            <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>
573            <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>
574            <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>
575            <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>
576
577        <span class="c1"># add next interval of interpolated (sub-)grid values</span>
578        <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>
579        <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>
580        <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>
581
582    <span class="k">return</span> <span class="n">f</span></div>
583</pre></div>
584
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