1 | #!/usr/bin/env python3 |
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2 | # -*- coding: utf-8 -*- |
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3 | #******************************************************************************* |
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4 | # @Author: Anne Philipp (University of Vienna) |
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5 | # |
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6 | # @Date: March 2018 |
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7 | # |
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8 | # @Change History: |
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9 | # |
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10 | # November 2015 - Leopold Haimberger (University of Vienna): |
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11 | # - migration of the methods dapoly and darain from Fortran |
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12 | # (flex_extract_v6 and earlier) to Python |
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13 | # |
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14 | # April 2018 - Anne Philipp (University of Vienna): |
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15 | # - applied PEP8 style guide |
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16 | # - added structured documentation |
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17 | # - outsourced the disaggregation functions dapoly and darain |
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18 | # to a new module named disaggregation |
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19 | # - added the new disaggregation method for precipitation |
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20 | # |
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21 | # June 2020 - Anne Philipp (University of Vienna): |
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22 | # - reformulated formular for dapoly |
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23 | # |
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24 | # @License: |
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25 | # (C) Copyright 2014-2020. |
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26 | # Anne Philipp, Leopold Haimberger |
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27 | # |
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28 | # SPDX-License-Identifier: CC-BY-4.0 |
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29 | # |
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30 | # This work is licensed under the Creative Commons Attribution 4.0 |
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31 | # International License. To view a copy of this license, visit |
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32 | # http://creativecommons.org/licenses/by/4.0/ or send a letter to |
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33 | # Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. |
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34 | # |
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35 | # @Methods: |
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36 | # - dapoly |
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37 | # - darain |
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38 | # - IA3 |
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39 | #******************************************************************************* |
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40 | '''Disaggregation of deaccumulated flux data from an ECMWF model FG field. |
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41 | |
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42 | Initially the flux data to be concerned are: |
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43 | - large-scale precipitation |
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44 | - convective precipitation |
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45 | - surface sensible heat flux |
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46 | - surface solar radiation |
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47 | - u stress |
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48 | - v stress |
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49 | |
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50 | Different versions of disaggregation is provided for rainfall |
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51 | data (darain, modified linear) and the surface fluxes and |
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52 | stress data (dapoly, cubic polynomial). |
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53 | ''' |
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54 | |
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55 | # ------------------------------------------------------------------------------ |
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56 | # MODULES |
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57 | # ------------------------------------------------------------------------------ |
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58 | |
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59 | # ------------------------------------------------------------------------------ |
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60 | # FUNCTIONS |
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61 | # ------------------------------------------------------------------------------ |
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62 | def dapoly(alist): |
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63 | """Cubic polynomial interpolation of deaccumulated fluxes. |
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64 | |
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65 | Interpolation of deaccumulated fluxes of an ECMWF model FG field |
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66 | using a cubic polynomial solution which conserves the integrals |
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67 | of the fluxes within each timespan. |
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68 | Disaggregation is done for 4 accumluated timespans which |
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69 | generates a new, disaggregated value which is output at the |
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70 | central point of the 4 accumulation timespans. |
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71 | This new point is used for linear interpolation of the complete |
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72 | timeseries afterwards. |
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73 | |
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74 | Parameters |
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75 | ---------- |
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76 | alist : list of array of float |
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77 | List of 4 timespans as 2-dimensional, horizontal fields. |
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78 | E.g. [[array_t1], [array_t2], [array_t3], [array_t4]] |
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79 | |
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80 | Return |
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81 | ------ |
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82 | nfield : array of float |
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83 | Interpolated flux at central point of accumulation timespan. |
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84 | |
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85 | Note |
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86 | ---- |
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87 | March 2000 : P. JAMES |
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88 | Original author |
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89 | |
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90 | June 2003 : A. BECK |
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91 | Adaptations |
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92 | |
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93 | November 2015 : Leopold Haimberger (University of Vienna) |
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94 | Migration from Fortran to Python |
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95 | |
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96 | """ |
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97 | |
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98 | nfield = -1./12.*alist[0] + \ |
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99 | 7./12.*alist[1] + \ |
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100 | 7./12.*alist[2] - \ |
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101 | 1./12.*alist[3] |
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102 | |
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103 | return nfield |
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104 | |
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105 | |
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106 | def darain(alist): |
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107 | """Linear interpolation of deaccumulated fluxes. |
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108 | |
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109 | Interpolation of deaccumulated fluxes of an ECMWF model FG rainfall |
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110 | field using a modified linear solution which conserves the integrals |
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111 | of the fluxes within each timespan. |
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112 | Disaggregation is done for 4 accumluated timespans which generates |
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113 | a new, disaggregated value which is output at the central point |
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114 | of the 4 accumulation timespans. This new point is used for linear |
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115 | interpolation of the complete timeseries afterwards. |
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116 | |
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117 | Parameters |
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118 | ---------- |
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119 | alist : list of array of float |
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120 | List of 4 timespans as 2-dimensional, horizontal fields. |
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121 | E.g. [[array_t1], [array_t2], [array_t3], [array_t4]] |
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122 | |
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123 | Return |
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124 | ------ |
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125 | nfield : array of float |
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126 | Interpolated flux at central point of accumulation timespan. |
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127 | |
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128 | Note |
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129 | ---- |
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130 | March 2000 : P. JAMES |
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131 | Original author |
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132 | |
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133 | June 2003 : A. BECK |
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134 | Adaptations |
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135 | |
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136 | November 2015 : Leopold Haimberger (University of Vienna) |
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137 | Migration from Fortran to Python |
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138 | """ |
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139 | |
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140 | xa = alist[0] |
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141 | xb = alist[1] |
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142 | xc = alist[2] |
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143 | xd = alist[3] |
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144 | xa[xa < 0.] = 0. |
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145 | xb[xb < 0.] = 0. |
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146 | xc[xc < 0.] = 0. |
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147 | xd[xd < 0.] = 0. |
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148 | |
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149 | xac = 0.5 * xb |
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150 | mask = xa + xc > 0. |
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151 | xac[mask] = xb[mask] * xc[mask] / (xa[mask] + xc[mask]) |
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152 | xbd = 0.5 * xc |
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153 | mask = xb + xd > 0. |
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154 | xbd[mask] = xb[mask] * xc[mask] / (xb[mask] + xd[mask]) |
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155 | nfield = xac + xbd |
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156 | |
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157 | return nfield |
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158 | |
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159 | def IA3(g): |
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160 | """ Interpolation with a non-negative geometric mean based algorithm. |
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161 | |
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162 | The original grid is reconstructed by adding two sampling points in each |
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163 | data series interval. This subgrid is used to keep all information during |
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164 | the interpolation within the associated interval. Additionally, an advanced |
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165 | monotonicity filter is applied to improve the monotonicity properties of |
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166 | the series. |
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167 | |
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168 | Note |
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169 | ---- |
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170 | (C) Copyright 2017-2019 |
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171 | Sabine Hittmeir, Anne Philipp, Petra Seibert |
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172 | |
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173 | This work is licensed under the Creative Commons Attribution 4.0 |
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174 | International License. To view a copy of this license, visit |
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175 | http://creativecommons.org/licenses/by/4.0/ or send a letter to |
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176 | Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. |
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177 | |
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178 | Parameters |
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179 | ---------- |
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180 | g : list of float |
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181 | Complete data series that will be interpolated having |
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182 | the dimension of the original raw series. |
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183 | |
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184 | Return |
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185 | ------ |
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186 | f : list of float |
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187 | The interpolated data series with additional subgrid points. |
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188 | Its dimension is equal to the length of the input data series |
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189 | times three. |
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190 | |
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191 | |
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192 | References |
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193 | ---------- |
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194 | For more information see article: |
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195 | Hittmeir, S.; Philipp, A.; Seibert, P. (2017): A conservative |
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196 | interpolation scheme for extensive quantities with application to the |
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197 | Lagrangian particle dispersion model FLEXPART., |
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198 | Geoscientific Model Development |
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199 | """ |
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200 | |
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201 | ####################### variable description ############################# |
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202 | # # |
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203 | # i - index variable for looping over the data series # |
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204 | # g - input data series # |
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205 | # f - interpolated and filtered data series with additional # |
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206 | # grid points # |
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207 | # fi - function value at position i, f_i # |
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208 | # fi1 - first sub-grid function value f_i^1 # |
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209 | # fi2 - second sub-grid function value f_i^2 # |
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210 | # fip1 - next function value at position i+1, f_(i+1) # |
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211 | # dt - time step # |
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212 | # fmon - monotonicity filter # |
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213 | # # |
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214 | ########################################################################### |
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215 | |
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216 | |
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217 | import numpy as np |
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218 | |
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219 | # time step |
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220 | dt = 1.0 |
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221 | |
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222 | ############### Non-negative Geometric Mean Based Algorithm ############### |
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223 | |
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224 | # for the left boundary the following boundary condition is valid: |
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225 | # the value at t=0 of the interpolation algorithm coincides with the |
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226 | # first data value according to the persistence hypothesis |
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227 | f = [g[0]] |
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228 | |
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229 | # compute two first sub-grid intervals without monotonicity check |
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230 | # go through the data series and extend each interval by two sub-grid |
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231 | # points and interpolate the corresponding data values |
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232 | # except for the last interval due to boundary conditions |
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233 | for i in range(0, 2): |
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234 | |
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235 | # as a requirement: |
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236 | # if there is a zero data value such that g[i]=0, then the whole |
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237 | # interval in f has to be zero to such that f[i+1]=f[i+2]=f[i+3]=0 |
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238 | # according to Eq. (6) |
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239 | if g[i] == 0.: |
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240 | f.extend([0., 0., 0.]) |
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241 | |
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242 | # otherwise the sub-grid values are calculated and added to the list |
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243 | else: |
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244 | # temporal save of last value in interpolated list |
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245 | # since it is the left boundary and hence the new (fi) value |
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246 | fi = f[-1] |
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247 | |
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248 | # the value at the end of the interval (fip1) is prescribed by the |
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249 | # geometric mean, restricted such that non-negativity is guaranteed |
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250 | # according to Eq. (25) |
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251 | fip1 = min(3. * g[i], 3. * g[i + 1], np.sqrt(g[i + 1] * g[i])) |
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252 | |
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253 | # the function value at the first sub-grid point (fi1) is determined |
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254 | # according to the equal area condition with Eq. (19) |
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255 | fi1 = 3./2.*g[i]-5./12.*fip1-1./12.*fi |
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256 | |
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257 | # the function value at the second sub-grid point (fi2) is determined |
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258 | # according Eq. (18) |
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259 | fi2 = fi1+1./3.*(fip1-fi) |
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260 | |
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261 | # add next interval of interpolated (sub-)grid values |
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262 | f.append(fi1) |
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263 | f.append(fi2) |
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264 | f.append(fip1) |
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265 | |
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266 | # compute rest of the data series intervals |
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267 | # go through the data series and extend each interval by two sub-grid |
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268 | # points and interpolate the corresponding data values |
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269 | # except for the last interval due to boundary conditions |
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270 | for i in range(2, len(g)-1): |
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271 | |
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272 | # as a requirement: |
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273 | # if there is a zero data value such that g[i]=0, then the whole |
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274 | # interval in f has to be zero to such that f[i+1]=f[i+2]=f[i+3]=0 |
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275 | # according to Eq. (6) |
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276 | if g[i] == 0.: |
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277 | # apply monotonicity filter for interval before |
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278 | # check if there is "M" or "W" shape |
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279 | if np.sign(f[-5]-f[-6]) * np.sign(f[-4]-f[-5]) == -1 \ |
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280 | and np.sign(f[-4]-f[-5]) * np.sign(f[-3]-f[-4]) == -1 \ |
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281 | and np.sign(f[-3]-f[-4]) * np.sign(f[-2]-f[-3]) == -1: |
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282 | |
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283 | # the monotonicity filter corrects the value at (fim1) by |
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284 | # substituting (fim1) with (fmon), see Eq. (27), (28) and (29) |
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285 | fmon = min(3. * g[i - 2], |
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286 | 3. * g[i - 1], |
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287 | np.sqrt(max(0, (18. / 13. * g[i - 2] - 5. / 13. * f[-7]) * |
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288 | (18. / 13. * g[i - 1] - 5. / 13. * f[-1])))) |
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289 | |
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290 | # recomputation of the sub-grid interval values while the |
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291 | # interval boundaries (fi) and (fip2) remains unchanged |
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292 | # see Eq. (18) and (19) |
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293 | f[-4] = fmon |
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294 | f[-6] = 3./2.*g[i-2]-5./12.*fmon-1./12.*f[-7] |
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295 | f[-5] = f[-6]+(fmon-f[-7])/3. |
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296 | f[-3] = 3./2.*g[i-1]-5./12.*f[-1]-1./12.*fmon |
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297 | f[-2] = f[-3]+(f[-1]-fmon)/3. |
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298 | |
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299 | f.extend([0., 0., 0.]) |
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300 | |
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301 | # otherwise the sub-grid values are calculated and added to the list |
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302 | else: |
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303 | # temporal save of last value in interpolated list |
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304 | # since it is the left boundary and hence the new (fi) value |
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305 | fi = f[-1] |
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306 | |
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307 | # the value at the end of the interval (fip1) is prescribed by the |
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308 | # geometric mean, restricted such that non-negativity is guaranteed |
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309 | # according to Eq. (25) |
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310 | fip1 = min(3. * g[i], 3. * g[i + 1], np.sqrt(g[i + 1] * g[i])) |
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311 | |
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312 | # the function value at the first sub-grid point (fi1) is determined |
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313 | # according to the equal area condition with Eq. (19) |
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314 | fi1 = 3./2.*g[i]-5./12.*fip1-1./12.*fi |
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315 | |
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316 | # the function value at the second sub-grid point (fi2) is determined |
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317 | # according Eq. (18) |
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318 | fi2 = fi1+1./3.*(fip1-fi) |
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319 | |
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320 | # apply monotonicity filter for interval before |
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321 | # check if there is "M" or "W" shape |
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322 | if np.sign(f[-5]-f[-6]) * np.sign(f[-4]-f[-5]) == -1 \ |
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323 | and np.sign(f[-4]-f[-5]) * np.sign(f[-3]-f[-4]) == -1 \ |
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324 | and np.sign(f[-3]-f[-4]) * np.sign(f[-2]-f[-3]) == -1: |
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325 | |
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326 | # the monotonicity filter corrects the value at (fim1) by |
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327 | # substituting (fim1) with fmon, see Eq. (27), (28) and (29) |
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328 | fmon = min(3. * g[i - 2], |
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329 | 3. * g[i - 1], |
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330 | np.sqrt(max(0, (18. / 13. * g[i - 2] - 5. / 13. * f[-7]) * |
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331 | (18. / 13. * g[i - 1] - 5. / 13. * f[-1])))) |
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332 | |
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333 | # recomputation of the sub-grid interval values while the |
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334 | # interval boundaries (fi) and (fip2) remains unchanged |
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335 | # see Eq. (18) and (19) |
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336 | f[-4] = fmon |
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337 | f[-6] = 3./2.*g[i-2]-5./12.*fmon-1./12.*f[-7] |
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338 | f[-5] = f[-6]+(fmon-f[-7])/3. |
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339 | f[-3] = 3./2.*g[i-1]-5./12.*f[-1]-1./12.*fmon |
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340 | f[-2] = f[-3]+(f[-1]-fmon)/3. |
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341 | |
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342 | # add next interval of interpolated (sub-)grid values |
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343 | f.append(fi1) |
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344 | f.append(fi2) |
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345 | f.append(fip1) |
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346 | |
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347 | # separate treatment of the final interval |
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348 | |
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349 | # as a requirement: |
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350 | # if there is a zero data value such that g[i]=0, then the whole |
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351 | # interval in f has to be zero to such that f[i+1]=f[i+2]=f[i+3]=0 |
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352 | # according to Eq. (6) |
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353 | if g[-1] == 0.: |
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354 | # apply monotonicity filter for interval before |
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355 | # check if there is "M" or "W" shape |
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356 | if np.sign(f[-5]-f[-6]) * np.sign(f[-4]-f[-5]) == -1 \ |
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357 | and np.sign(f[-4]-f[-5]) * np.sign(f[-3]-f[-4]) == -1 \ |
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358 | and np.sign(f[-3]-f[-4]) * np.sign(f[-2]-f[-3]) == -1: |
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359 | |
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360 | # the monotonicity filter corrects the value at (fim1) by |
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361 | # substituting (fim1) with (fmon), see Eq. (27), (28) and (29) |
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362 | fmon = min(3. * g[-3], |
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363 | 3. * g[-2], |
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364 | np.sqrt(max(0, (18. / 13. * g[-3] - 5. / 13. * f[-7]) * |
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365 | (18. / 13. * g[-2] - 5. / 13. * f[-1])))) |
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366 | |
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367 | # recomputation of the sub-grid interval values while the |
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368 | # interval boundaries (fi) and (fip2) remains unchanged |
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369 | # see Eq. (18) and (19) |
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370 | f[-4] = fmon |
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371 | f[-6] = 3./2.*g[-3]-5./12.*fmon-1./12.*f[-7] |
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372 | f[-5] = f[-6]+(fmon-f[-7])/3. |
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373 | f[-3] = 3./2.*g[-2]-5./12.*f[-1]-1./12.*fmon |
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374 | f[-2] = f[-3]+(f[-1]-fmon)/3. |
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375 | |
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376 | f.extend([0., 0., 0.]) |
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377 | |
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378 | # otherwise the sub-grid values are calculated and added to the list |
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379 | # using the persistence hypothesis as boundary condition |
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380 | else: |
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381 | # temporal save of last value in interpolated list |
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382 | # since it is the left boundary and hence the new (fi) value |
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383 | fi = f[-1] |
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384 | # since last interval in series, last value is also fip1 |
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385 | fip1 = g[-1] |
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386 | # the function value at the first sub-grid point (fi1) is determined |
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387 | # according to the equal area condition with Eq. (19) |
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388 | fi1 = 3./2.*g[-1]-5./12.*fip1-1./12.*fi |
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389 | # the function value at the second sub-grid point (fi2) is determined |
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390 | # according Eq. (18) |
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391 | fi2 = fi1+dt/3.*(fip1-fi) |
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392 | |
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393 | # apply monotonicity filter for interval before |
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394 | # check if there is "M" or "W" shape |
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395 | if np.sign(f[-5]-f[-6]) * np.sign(f[-4]-f[-5]) == -1 \ |
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396 | and np.sign(f[-4]-f[-5]) * np.sign(f[-3]-f[-4]) == -1 \ |
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397 | and np.sign(f[-3]-f[-4]) * np.sign(f[-2]-f[-3]) == -1: |
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398 | |
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399 | # the monotonicity filter corrects the value at (fim1) by |
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400 | # substituting (fim1) with (fmon), see Eq. (27), (28) and (29) |
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401 | fmon = min(3. * g[-3], |
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402 | 3. * g[-2], |
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403 | np.sqrt(max(0, (18. / 13. * g[-3] - 5. / 13. * f[-7]) * |
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404 | (18. / 13. * g[-2] - 5. / 13. * f[-1])))) |
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405 | |
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406 | # recomputation of the sub-grid interval values while the |
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407 | # interval boundaries (fi) and (fip2) remains unchanged |
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408 | # see Eq. (18) and (19) |
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409 | f[-4] = fmon |
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410 | f[-6] = 3./2.*g[-3]-5./12.*fmon-1./12.*f[-7] |
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411 | f[-5] = f[-6]+(fmon-f[-7])/3. |
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412 | f[-3] = 3./2.*g[-2]-5./12.*f[-1]-1./12.*fmon |
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413 | f[-2] = f[-3]+(f[-1]-fmon)/3. |
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414 | |
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415 | # add next interval of interpolated (sub-)grid values |
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416 | f.append(fi1) |
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417 | f.append(fi2) |
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418 | f.append(fip1) |
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419 | |
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420 | return f |
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