[GRASS-SVN] r69653 - grass-addons/grass7/vector/v.kriging
svn_grass at osgeo.org
svn_grass at osgeo.org
Mon Oct 3 09:39:47 PDT 2016
Author: evas
Date: 2016-10-03 09:39:47 -0700 (Mon, 03 Oct 2016)
New Revision: 69653
Modified:
grass-addons/grass7/vector/v.kriging/v.kriging.html
Log:
v.kriging: html modified
Modified: grass-addons/grass7/vector/v.kriging/v.kriging.html
===================================================================
--- grass-addons/grass7/vector/v.kriging/v.kriging.html 2016-10-03 15:52:37 UTC (rev 69652)
+++ grass-addons/grass7/vector/v.kriging/v.kriging.html 2016-10-03 16:39:47 UTC (rev 69653)
@@ -1,952 +1,68 @@
<h2>DESCRIPTION</h2>
-<em>v.kriging</em> interpolates unknown values using method <i>ordinary
-kriging</i>. Output can be 2D or 3D.
+<em>v.kriging</em> constructs 2D / 3D raster from the values located on discrete points using interpolation method <i>ordinary kriging</i>. In order to let the user decide on the process and necessary parameters, the module performance is divided into three phases:
+<ul>
+ <li><b>initial phase</b> computes experimental variogram.
+ <ul>
+ <li>Please set up a name of the <b>report file</b>. The file will be created automatically in working directory to enable import of parameters from current to following phases. If the file has been deleted during the module performance, the user is asked to start interpolation again from the initial phase.</li>
+ <li>Warning about particular point and "<i>less than 2 neighbours in its closest surrounding. The perimeter of the surrounding will be increased...</i>" indicates that variogram range should be shortened.</li>
+ <li>There will appear some temporary files during variogram computation. They will be deleted automatically in following phase. If missing, the user is asked to repeat initial phase.</li>
+ <li>It is not necessary to save experimental variogram plots. They just help to estimate parameters of theoretical variogram that will be computed in following step (output contains experimental and theoretical variogram plotted together).</li>
+ </ul>
+ </li>
+ <li>in the <b>middle phase</b>, the user estimates theoretical variogram setting up the range (if necessary, the sill and the nugget effect as well) to fit the experimental variogram from previous phase.
+ <ul>
+ <li>Default <i>sill</i> is calculated from variogram values, more details in (<i>Stopkova, 2014</i>).</li>
+ <li>Save horizontal and vertical variogram plots using <i>file=extension</i>.</li>
+ <li>Experimental anisotropic / bivariate variogram is plotted as a base for final theoretical variogram parameters estimation in final phase.</li>
+ </ul>
+ </li>
+ <li><b>final phase</b> performs interpolation based on parameters of theoretical variogram.
+ <ul>
+ <li>Save anisotropic or bivariate variogram plot using <i>file=extension</i>.</li>
+ </ul>
+ </li>
+</ul>
+
<h2>EXAMPLES</h2>
-Two case studies have been prepared to outline that there is no universal rule how to use the module <i>v.kriging</i>.
-Bahaviour of the phenomena is quite variable and this influences how to interpolate the dataset properly. Every point
-dataset is special and requires trying different anisotropic ratios, variogram functions and careful analysis of
-the results to get relevant interpolated (2D / 3D) raster model.
+To get optimal results, it is necessary to test various initial settings, anisotropic ratios and variogram functions. Input (2D or 3D point layer) must contain values to be interpolated in the attribute table.
+
<h3>3D kriging</h3>
-Input layer should contain 3D coordinates (xyz) and
-values to be interpolated (in attribute table). The commands can look
-like this:
+<b>General commands</b>
<div class="code"><pre>
v.kriging phase=initial in=input_layer icol=name report=report_file.txt file=png
</pre></div>
<div class="code"><pre>
-v.kriging in=input_layer phase=middle hz_fun=exponential vert_fun=exponential ic=name file=png hz_range=double vert_range=double -b
+v.kriging in=input_layer phase=middle hz_fun=exponential vert_fun=exponential ic=name file=png \
+hz_range=double vert_range=double [hz_sill=double vert_sill=double hz_nugget=double vert_nugget=double] -u
</pre></div>
<div class="code"><pre>
-v.kriging in=input_layer phase=final final_fun=bivariate icol=name file=png out=name crossval=crossval_file.txt
+v.kriging in=input_layer phase=final final_fun=exponential final_range=double \
+[final_sill=double final_nugget=double] icol=name file=png out=name crossval=crossval_file.txt
</pre></div>
-<p>In the middle phase, there is possible also to modify nugget effect (default: 0.0) and sill (default: calculated from
-variogram values, more details in (<i>Stopkova, 2014</i>).
-
-<h4>Case study: Slovakia 3D precipitation</h4>
<p>
-The case study is based on the input points of annual precipitation
-<a href="http://grass.osgeo.org/download/sample-data/" target="_blank">dataset</a>.
-Although positions of the points are given in three-dimensional space, the points are concentrated at the terrain and thus
-interpolation above and below the terrain would become imprecise in deeper / higher areas of the dataset. That is the reason
-to test just cross-section from interpolated 3D raster comparing it with cross-section from the RST result obtained using
-<a href="v.vol.rst.html">v.vol.rst</a> according to (<i>Neteler and Mitasova, 2004</i>).
+Commands based on the <a href="http://grass.osgeo.org/download/sample-data/" target="_blank">dataset</a> of <b>Slovakia 3D precipitation</b> (<i>Mitasova and Hofierka, 2004</i>). Another examples of 3D interpolation are available in (<i>Stopkova, 2014</i>).
-<p>As the algorithm still needs to be optimized for large datasets, the original region was used with lower resolution
-(hz: 5000 m, vert: 500 m). To test also original resolution, the points in smaller region (<b>Tab. 1</b>) were extracted.
-
-<p>
-<table border="1">
- <tr>
- <td colspan="2" align="center">N = 5 468 000 m</td>
- </tr>
- <tr>
- <td>W = 4 361 000 m</td>
- <td>E = 4 465 500 m</td>
- </tr>
- <tr>
- <td colspan="2" align="center">S = 5 374 500 m</td>
- </tr>
- <tr>
- <td>top: 2 250 m</td>
- <td>bottom: 200 m</td>
- </tr>
- <caption>
- <b>Tab. 1:</b> Smaller region extent (resolution hz: 500 m, vert: 50 m)
- </caption>
-</table>
-
-<p>
-In the initial phase, <b>experimental variograms</b> (horizontal and vertical) were computed:
<div class="code"><pre>
v.kriging phase=initial in=precip3d at PERMANENT ic=precip report=precip3d.txt file=png --o
</pre></div>
-
-In the middle phase, there were empirically estimated types of the function and coefficients of <b>theoretical variograms</b>:
<div class="code"><pre>
v.kriging in=precip3d at PERMANENT phase=middle hz_fun=exponential vert_fun=gaussian ic=precip file=png hz_range=100000. vert_range=1600 --o -u
</pre></div>
-
-<p>
-<table>
- <tr>
- <td>
- <b>Fig. 1:</b> Experimental and theoretical variograms in horizontal and vertical direction<br>
- </td>
- </tr>
- <tr>
- <td>
- a) whole region (horizontal direction):
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/v_kriging_variogram_hz_100000.png" alt="var_hz_initial" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- b) small region (horizontal direction):
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_hz_20000.png" alt="var_hz_small_initial" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- c) small region with modified horizontal range (horizontal direction):
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_hz_15000.png" alt="var_hz_small_add" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- d) whole region (vertical direction):
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/v_kriging_variogram_vertical_1600.png" alt="var_vert" style="float: left; width: 75%;
- margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- e) small region (vertical direction):
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_vertical_1200.png" alt="var_vert_add" style="float: left; width: 75%;
- margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
-</table>
-
-The final phase results into theoretical variogram and interpolated values of the 3D raster:
<div class="code"><pre>
v.kriging in=precip3d at PERMANENT phase=middle hz_fun=exponential vert_fun=gaussian ic=precip \
file=png hz_range=100000. vert_range=1600 --o -u
</pre></div>
-<b>Cross validation</b> results of all the interpolated 3D rasters were compared. RST interpolation was performed using
-modified settings according to (<i>Neteler and Mitasova, 2004</i>, page 173) - different tension and smoothing parameters were set up to obtain more accurate cross
-validation results:
+Note: 3D points in this example are concentrated on the Earth's surface. Thus the deeper / higher, the less accurate result of interpolation.
-<div class="code"><pre>
-v.vol.rst -c input="precip3d at PERMANENT" wcolumn="precip" tension=100. smooth=0. \
-cvdev="cxvalidation_rst_final" segmax=50 npmin=200 npmax=700 wscale=1.0 zscale=50
-</pre></div>
-
-<table>
- <tr>
- <td>
- a) linear
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/v_kriging_variogram_linear.png" alt="linear" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- b) exponential
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/v_kriging_variogram_exponential.png" alt="exponential" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- c) spherical
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/v_kriging_variogram_spherical.png" alt="spherical" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- d) Gaussian
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/v_kriging_variogram_gaussian.png" alt="gaussian" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <caption>
- <b>Fig. 2:</b> Variogram modelling of the whole region with horizontal range of 100 000 m and vertical range of 1 600 m
- </caption>
-</table>
-
-<table border="1" style="text-align: right;">
- <tr>
- <td></td>
- <td align="center"><a href="v.vol.rst.html" target="_blank">v.vol.rst</a></td>
- <td align="center">Linear</td>
- <td align="center">Exponential</td>
- <td align="center">Spherical</td>
- <td align="center">Gaussian</td>
- </tr>
- <tr>
- <td align="left">Minimum [mm]</td>
- <td>-816.805</td>
- <td>-684.258</td>
- <td>-686.063</td>
- <td>-683.580</td>
- <td>-741.081</td>
- </tr>
- <tr>
- <td align="left">Maximum [mm]</td>
- <td>175.831</td>
- <td>429.597</td>
- <td>424.419</td>
- <td>424.228</td>
- <td>444.747</td>
- </tr>
- <tr>
- <td align="left">Mean [mm]</td>
- <td>0.171</td>
- <td>0.131</td>
- <td>0.699</td>
- <td>0.653</td>
- <td>0.721</td>
- </tr>
- <tr>
- <td align="left">Variance [mm<sup>2</sup>]</td>
- <td>9876.69</td>
- <td>5740.89</td>
- <td>5926.62</td>
- <td>5950.70</td>
- <td>5979.79</td>
- </tr>
- <tr>
- <td align="left">Std. deviation [mm]</td>
- <td>99.381</td>
- <td>75.769</td>
- <td>76.985</td>
- <td>77.141</td>
- <td>77.329</td>
- </tr>
- <caption>
- <b>Tab. 2:</b> Cross validation of the whole region with horizontal range of 100 000 m and vertical range of 1 600 m
- </caption>
-</table>
-
-<p>
-<table>
- <tr>
- <td>
- a) linear:
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_linear.png" alt="small_linear" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- b) exponential:
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_exponential.png" alt="small_exp" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- c) spherical:
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_spherical.png" alt="small_spher" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- d) Gaussian:
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_gauss.png" alt="small_gauss" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <caption>
- <b>Fig. 3:</b> Variogram modelling of small region with horizontal range of 20 000 m and vertical range of 1 200 m
- </caption>
-</table>
-
-<table border="1" style="text-align: right;">
- <tr>
- <td></td>
- <td align="center"><a href="v.vol.rst.html" target="_blank">v.vol.rst</a></td>
- <td align="center">Linear</td>
- <td align="center" style="color: greenyellow;">Exponential</td>
- <td align="center">Spherical</td>
- <td align="center">Gaussian</td>
- </tr>
- <tr>
- <td align="left">Minimum [mm]</td>
- <td>-908.303</td>
- <td>-709.02</td>
- <td>-755.531</td>
- <td>-678.237</td>
- <td>-786.613</td>
- </tr>
- <tr>
- <td align="left">Maximum [mm]</td>
- <td>257.69</td>
- <td>420.689</td>
- <td>340.874</td>
- <td>384.109</td>
- <td>315.248</td>
- </tr>
- <tr>
- <td align="left">Mean [mm]</td>
- <td>0.554</td>
- <td>0.908</td>
- <td style="color: greenyellow;">-0.310</td>
- <td>1.102</td>
- <td>-0.739</td>
- </tr>
- <tr>
- <td align="left">Variance [mm<sup>2</sup>]</td>
- <td>28891.6</td>
- <td>15638.7</td>
- <td>21259.5</td>
- <td>19754.8</td>
- <td>21243.2</td>
- </tr>
- <tr>
- <td align="left">Std. deviation [mm]</td>
- <td>169.975</td>
- <td>125.055</td>
- <td>145.806</td>
- <td>140.552</td>
- <td>145.751</td>
- </tr>
- <caption>
- <b>Tab. 3:</b> Cross validation of small region <br>with horizontal range of 20 000 m and vertical range of 1 200 m
- </caption>
-</table>
-
-<p>
-<table>
- <tr>
- <td>
- a) linear
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_linear_1.png" alt="small_linear_1" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- b) exponential
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_exponential_1.png" alt="small_exp_1" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- c) spherical
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_spherical_1.png" alt="small_spher_1" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- d) Gaussian
- </td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_gauss_1.png" alt="small_gauss_1" style="float: left; width: 75%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <caption>
- <b>Fig. 4:</b> Variogram modelling of small region with horizontal range of 15 000 m and vertical range of 1 200 m
- </caption>
-</table>
-
-<table border="1" style="text-align: right;">
- <tr>
- <td></td>
- <td align="center"><a href="v.vol.rst.html" target="_blank">v.vol.rst</a></td>
- <td align="center" style="color: greenyellow;">Linear</td>
- <td align="center">Exponential</td>
- <td align="center">Spherical</td>
- <td align="center" style="color: greenyellow;">Gaussian</td>
- </tr>
- <tr>
- <td align="left">Minimum [mm]</td>
- <td>-908.303</td>
- <td>-711.755</td>
- <td>-756.805</td>
- <td>-687.44</td>
- <td>-529.702</td>
- </tr>
- <tr>
- <td align="left">Maximum [mm]</td>
- <td>257.69</td>
- <td>397.666</td>
- <td>326.057</td>
- <td>357.990</td>
- <td>425.637</td>
- </tr>
- <tr>
- <td align="left">Mean [mm]</td>
- <td>0.554</td>
- <td style="color: greenyellow;">0.208</td>
- <td>-1.775</td>
- <td>1.823</td>
- <td style="color: greenyellow;">0.008</td>
- </tr>
- <tr>
- <td align="left">Variance [mm<sup>2</sup>]</td>
- <td>28891.6</td>
- <td>15873</td>
- <td>21438.5</td>
- <td>20930.9</td>
- <td>20473.1</td>
- </tr>
- <tr>
- <td align="left">Std. deviation [mm]</td>
- <td>169.975</td>
- <td>125.988</td>
- <td>146.419</td>
- <td>144.675</td>
- <td>143.084</td>
- </tr>
- <caption>
- <b>Tab. 4:</b> Cross validation of small region <br>with horizontal range of 15 000 m and vertical range of 1 200 m
- </caption>
-</table>
-
-<p>
-<p>
-<table>
- <tr>
- <td>
- <img src="images/small/v_kriging_variogram_exponential_2.png" alt="small_exp_2" style="float: left; width: 32%; margin-right: 1%; margin-bottom: 0.5em;">
- <img src="images/small/v_kriging_variogram_spherical_2.png" alt="small_spher_2" style="float: left; width: 32%; margin-right: 1%; margin-bottom: 0.5em;">
- <img src="images/small/v_kriging_variogram_gauss_2.png" alt="small_gauss_2" style="float: left; width: 32%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <caption>
- <b>Fig. 5:</b> Variogram modelling of small region with horizontal range of 15 000 m and vertical range of 1 200 m
- </caption>
-</table>
-
-<table border="1" style="text-align: right;">
- <tr>
- <td></td>
- <td align="center"><a href="v.vol.rst.html" target="_blank">v.vol.rst</a></td>
- <td align="center">Exponential</td>
- <td align="center">Spherical</td>
- <td align="center">Gaussian</td>
- </tr>
- <tr>
- <td align="left">Minimum [mm]</td>
- <td>-908.303</td>
- <td>-812.347</td>
- <td>-792.391</td>
- <td>-783.897</td>
- </tr>
- <tr>
- <td align="left">Maximum [mm]</td>
- <td>257.69</td>
- <td>279.905</td>
- <td>284.76</td>
- <td>293.146</td>
- </tr>
- <tr>
- <td align="left">Mean [mm]</td>
- <td>0.554</td>
- <td>2.530</td>
- <td>2.229</td>
- <td>2.054</td>
- </tr>
- <tr>
- <td align="left">Variance [mm<sup>2</sup>]</td>
- <td>28891.6</td>
- <td>23983</td>
- <td>22586.9</td>
- <td>22209.2</td>
- </tr>
- <tr>
- <td align="left">Std. deviation [mm]</td>
- <td>169.975</td>
- <td>154.865</td>
- <td>150.289</td>
- <td>149.028</td>
- </tr>
- <caption>
- <b>Tab. 5:</b> Cross validation of small region <br>with horizontal range of 15000 m and vertical range of 1200 m<br>
- with different settings of theoretical variogram
- </caption>
-</table>
-
-<p>
-<b>The results</b> were compared just using cross sections because of the distribution of the points. The column <i>RST</i>
- in following tables summarize statistical characteristics of the cross section by <i><a href="v.vol.rst.html" target="_blank">v.vol.rst</a></i>. Other columns represent
- statistical characteristics of the cross section <i>values</i> and of the <i>differences</i> (between RST and kriging cross sections)
- produced by each type of variogram.
-
- <p>
-<table border="1" style="text-align: right;">
- <tr>
- <td rowspan="2"></td>
- <td rowspan="2" align="center"><a href="v.vol.rst.html" target="_blank">v.vol.rst</a></td>
- <td colspan="2" align="center">Linear</td>
- <td colspan="2" align="center">Exponential</td>
- <td colspan="2" align="center">Spherical</td>
- <td colspan="2" align="center">Gaussian</td>
- </tr>
- <tr>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- </tr>
- <tr>
- <td># of cells</td>
- <td>196105</td>
- <td colspan="2" align="center">196105</td>
- <td colspan="2" align="center">196105</td>
- <td colspan="2" align="center">196105</td>
- <td colspan="2" align="center">196105</td>
- </tr>
- <tr>
- <td align="left">Minimum [mm]</td>
- <td>491.985</td>
- <td>551.835</td>
- <td>-643.783</td>
- <td>543.503</td>
- <td>-603.59</td>
- <td>536.309</td>
- <td>-629.652</td>
- <td>550.935</td>
- <td>-622.887</td>
- </tr>
- <tr>
- <td align="left">Maximum [mm]</td>
- <td>1563.660</td>
- <td>1580.700</td>
- <td>317.011</td>
- <td>1572.000</td>
- <td>316.938</td>
- <td>1578.820</td>
- <td>320.219</td>
- <td>1548.51</td>
- <td>356.881</td>
- </tr>
- <tr>
- <td align="left">Mean [mm]</td>
- <td>722.036</td>
- <td>744.689</td>
- <td>-22.657</td>
- <td>740.308</td>
- <td>-18.271</td>
- <td>740.968</td>
- <td>-18.932</td>
- <td>747.655</td>
- <td>-25.6238</td>
- </tr>
- <tr>
- <td align="left">Variance mm<sup>2</sup></td>
- <td>14472.9</td>
- <td>29200.5</td>
- <td>7978.76</td>
- <td>28614.7</td>
- <td>6978.35</td>
- <td>29263.9</td>
- <td>7338.56</td>
- <td>31396.1</td>
- <td>9139.46</td>
- </tr>
- <tr>
- <td align="left">Std. deviation [mm]</td>
- <td>120.303</td>
- <td>170.882</td>
- <td>89.324</td>
- <td>169.159</td>
- <td>83.5365</td>
- <td>171.067</td>
- <td>85.665</td>
- <td>177.189</td>
- <td>95.601</td>
- </tr>
- <caption>
- <b>Tab. 6:</b> Statistical characteristics of the results for the whole region<br>with horizontal range of 100 000 m and vertical range of 1 600 m
- </caption>
-</table>
-
-<p>
-<table>
- <tr>
- <td>left: linear variogram; right: exponential variogram</td>
- </tr>
- <tr>
- <td>
- <img src="images/v_kriging_result_linear.png" alt="res_lin" style="float: left; width: 35%; margin-right: 1%; margin-bottom: 0.5em;">
- <img src="images/v_kriging_result_exponential.png" alt="res_exp" style="float: left; width: 35%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <caption>
- <b>Fig. 6:</b> Interpolated rasters in the whole region
- </caption>
-</table>
-<table>
- <tr>
- <td>left: spherical variogram; right: Gaussian variogram</td>
- </tr>
- <tr>
- <td>
- <img src="images/v_kriging_result_spherical.png" alt="res_spher" style="float: left; width: 35%; margin-right: 1%; margin-bottom: 0.5em;">
- <img src="images/v_kriging_result_gauss.png" alt="res_gauss" style="float: left; width: 35%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
-</table>
-
- <p>
-<table border="1" style="text-align: right;">
- <tr>
- <td rowspan="2"></td>
- <td rowspan="2" align="center"><a href="v.vol.rst.html" target="_blank">v.vol.rst</a></td>
- <td colspan="2" align="center">Linear</td>
- <td colspan="2" align="center" style="color: greenyellow;">Exponential</td>
- <td colspan="2" align="center">Spherical</td>
- <td colspan="2" align="center">Gaussian</td>
- </tr>
- <tr>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- </tr>
- <tr>
- <td># of cells</td>
- <td>37639</td>
- <td colspan="2" align="center">37639</td>
- <td colspan="2" align="center">37639</td>
- <td colspan="2" align="center">37639</td>
- <td colspan="2" align="center">37639</td>
- </tr>
- <tr>
- <td align="left">Minimum [mm]</td>
- <td>600.922</td>
- <td>580.245</td>
- <td>-669.397</td>
- <td>595.692</td>
- <td>-513.497</td>
- <td>591.531</td>
- <td>-639.105</td>
- <td>608.913</td>
- <td>-378.109</td>
- </tr>
- <tr>
- <td align="left">Maximum [mm]</td>
- <td>1675.090</td>
- <td>1916.93</td>
- <td>203.666</td>
- <td>1837.18</td>
- <td>172.858</td>
- <td>1936.17</td>
- <td>217.189</td>
- <td>1545.81</td>
- <td>244.875</td>
- </tr>
- <tr>
- <td align="left">Mean [mm]</td>
- <td>883.909</td>
- <td>893.167</td>
- <td>-9.258</td>
- <td>869.083</td>
- <td>14.826</td>
- <td>863.901</td>
- <td>20.0083</td>
- <td>871.364</td>
- <td>12.5451</td>
- </tr>
- <tr>
- <td align="left">Variance mm<sup>2</sup></td>
- <td>11102</td>
- <td>39059.1</td>
- <td>11442.7</td>
- <td>22743.8</td>
- <td>3941.73</td>
- <td>26359.9</td>
- <td>5983.39</td>
- <td>20350.4</td>
- <td>3796.5</td>
- </tr>
- <tr>
- <td align="left">Std. deviation [mm]</td>
- <td>105.366</td>
- <td>197.634</td>
- <td>106.97</td>
- <td>150.811</td>
- <td>62.7832</td>
- <td>162.357</td>
- <td>77.352</td>
- <td>142.655</td>
- <td>61.616</td>
- </tr>
- <caption>
- <b>Tab. 7:</b> Statistical characteristics of the results for the small region<br>with horizontal range of 20 000 m and vertical range of 1 200 m
- </caption>
-</table>
-
-<p>
-<table border="1" style="text-align: right;">
- <tr>
- <td rowspan="2"></td>
- <td rowspan="2" align="center"><a href="v.vol.rst.html" target="_blank">v.vol.rst</a></td>
- <td colspan="2" align="center" style="color: greenyellow;">Linear</td>
- <td colspan="2" align="center">Exponential</td>
- <td colspan="2" align="center">Spherical</td>
- <td colspan="2" align="center" style="color: greenyellow;">Gaussian</td>
- </tr>
- <tr>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- </tr>
- <tr>
- <td># of cells</td>
- <td>37639</td>
- <td colspan="2" align="center">37639</td>
- <td colspan="2" align="center">37639</td>
- <td colspan="2" align="center">37639</td>
- <td colspan="2" align="center">37639</td>
- </tr>
- <tr>
- <td align="left">Minimum [mm]</td>
- <td>600.922</td>
- <td>576.625</td>
- <td>-709.196</td>
- <td>595.284</td>
- <td>-580.65</td>
- <td>594.467</td>
- <td>-665.563</td>
- <td>556.744</td>
- <td>-937.124</td>
- </tr>
- <tr>
- <td align="left">Maximum [mm]</td>
- <td>1675.090</td>
- <td>1920.29</td>
- <td>217.844</td>
- <td>1847.160</td>
- <td>199.350</td>
- <td>1925.91</td>
- <td>209.544</td>
- <td>2218.170</td>
- <td>249.345</td>
- </tr>
- <tr>
- <td align="left">Mean [mm]</td>
- <td>883.909</td>
- <td>886.734</td>
- <td>-2.82485</td>
- <td>865.272</td>
- <td>18.6376</td>
- <td>860.772</td>
- <td>23.137</td>
- <td>860.294</td>
- <td>23.615</td>
- </tr>
- <tr>
- <td align="left">Variance mm<sup>2</sup></td>
- <td>11102</td>
- <td>37500.3</td>
- <td>10985.4</td>
- <td>23111.1</td>
- <td>4697.95</td>
- <td>25530.4</td>
- <td>6226.34</td>
- <td>32575</td>
- <td>10525.7</td>
- </tr>
- <tr>
- <td align="left">Std. deviation [mm]</td>
- <td>105.366</td>
- <td>193.65</td>
- <td>104.811</td>
- <td>152.023</td>
- <td>68.5416</td>
- <td>159.782</td>
- <td>78.907</td>
- <td>180.485</td>
- <td>102.595</td>
- </tr>
- <caption>
- <b>Tab. 8:</b> Statistical characteristics of the results for the small region<br>with horizontal range of 15 000 m and vertical range of 1 200 m
- </caption>
-</table>
-
-<p>
-<table border="1" style="text-align: right;">
- <tr>
- <td rowspan="2"></td>
- <td rowspan="2" align="center"><a href="v.vol.rst.html" target="_blank">v.vol.rst</a></td>
- <td colspan="2" align="center">Exponential</td>
- <td colspan="2" align="center">Spherical</td>
- <td colspan="2" align="center">Gaussian</td>
- </tr>
- <tr>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- <td align="center">Value</td>
- <td align="center">Diff.</td>
- </tr>
- <tr>
- <td># of cells</td>
- <td>37639</td>
- <td colspan="2" align="center">37639</td>
- <td colspan="2" align="center">37639</td>
- <td colspan="2" align="center">37639</td>
- </tr>
- <tr>
- <td align="left">Minimum [mm]</td>
- <td>600.922</td>
- <td>618.417</td>
- <td>-453.562</td>
- <td>600.733</td>
- <td>-665.563</td>
- <td>608.602</td>
- <td>-415.753</td>
- </tr>
- <tr>
- <td align="left">Maximum [mm]</td>
- <td>1675.090</td>
- <td>1687.83</td>
- <td>177.512</td>
- <td>1586.71</td>
- <td>209.544</td>
- <td>1571.55</td>
- <td>238.521</td>
- </tr>
- <tr>
- <td align="left">Mean [mm]</td>
- <td>883.909</td>
- <td>867.071</td>
- <td>16.8376</td>
- <td>867.047</td>
- <td>23.137</td>
- <td>867.075</td>
- <td>16.834</td>
- </tr>
- <tr>
- <td align="left">Variance mm<sup>2</sup></td>
- <td>11102</td>
- <td>18570.3</td>
- <td>3191.89</td>
- <td>19377.5</td>
- <td>6226.34</td>
- <td>19744.4</td>
- <td>4155.31</td>
- </tr>
- <tr>
- <td align="left">Std. deviation [mm]</td>
- <td>105.366</td>
- <td>136.273</td>
- <td>41.737</td>
- <td>139.203</td>
- <td>78.907</td>
- <td>140.515</td>
- <td>64.462</td>
- </tr>
- <caption>
- <b>Tab. 9:</b> Statistical characteristics of the results for the small region<br>with horizontal range of 15 000 m and vertical range of 1 200 m<br>with different settings of the theoretical variogram
- </caption>
-</table>
-
-<p>
-<table>
- <tr>
- <td> Gaussian variogram from <b>Tab. 4</b> vs. Exponential variogram from <b>Tab. 5</b></td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_result_gauss_1.png" alt="res_gauss_1" style="float: left; width: 35%; margin-right: 1%; margin-bottom: 0.5em;">
- <img src="images/small/v_kriging_result_exponential_2.png" alt="res_exponential_2" style="float: left; width: 35%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <caption>
- <b>Fig. 7:</b> Interpolated rasters with the best (on the left) and the worse (on the right) cross validation results.
- </caption>
-</table>
-<table>
- <tr>
- <td> Linear variogram from <b>Tab. 4</b> vs. Spherical variogram from <b>Tab. 5</b></td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_result_linear_1.png" alt="res_linear_1" style="float: left; width: 35%; margin-right: 1%; margin-bottom: 0.5em;">
- <img src="images/small/v_kriging_result_spherical_2.png" alt="res_spherical_2" style="float: left; width: 35%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
-</table>
-<table>
- <tr>
- <td> Exponential variogram from <b>Tab. 3</b> vs. Gaussian variogram from <b>Tab. 5</b></td>
- </tr>
- <tr>
- <td>
- <img src="images/small/v_kriging_result_exponential.png" alt="res_exponential" style="float: left; width: 35%; margin-right: 1%; margin-bottom: 0.5em;">
- <img src="images/small/v_kriging_result_gauss_2.png" alt="res_gauss_2" style="float: left; width: 35%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
-</table>
-
-<p>
-More case studies on 3D data are described in (<i>Stopkova, 2014</i>).
-
<h3>2D kriging</h3>
-Input layer should contain 2D coordinates (xy) and values to be interpolated (in attribute table). The commands can in general look like this:
+<p> <b>General commands</b>
+
<div class="code"><pre>
v.kriging phase=initial in=input_layer icol=name report=report_file.txt file=png -2
</pre></div>
@@ -955,276 +71,36 @@
out=name crossval=crossval_file.txt -2
</pre></div>
-<h4>Case study: elev_lid792_randpts</h4>
-<p>
-The case study is based on 500 random points that were extracted from input points of Digital Elevation Model (DEM)
-<i>elev_lid792_randpts</i> from the North Carolina
-<a href="http://grass.osgeo.org/download/sample-data/" target="_blank">dataset</a>.
-
-<p>In initial phase, temporary <b>experimental variogram</b> was computed:
+<p> Commands based on 500 random points extracted from input points of Digital Elevation Model (DEM) <i>elev_lid792_randpts</i> from the <b>North Carolina <a href="http://grass.osgeo.org/download/sample-data/" target="_blank">dataset</a></b> (<i>Neteler and Mitasova, 2004</i>).
+
<div class="code"><pre>
v.kriging phase=initial in=elev_lid792_selected ic=value azimuth=45. td=45. \
report=lid792_500_linear.txt -2 --o
</pre></div>
-
-Then, in final phase (middle phase is skipped in 2D kriging) the <b>theoretical variogram</b> was computed and
-interpolation of unknown values was performed:
-
<div class="code"><pre>
v.kriging in=elev_lid792_selected phase=final final_function=linear ic=value \
file=png out=lid792_500_linear crossval=lid792_500_xval_linear.txt -2 --o
</pre></div>
-Variogram modelling was compared with the result of variogram analysis in <i>Surfer </i> (<i>Golden Software, Inc.</i>).
-<p>
-<table>
- <tr>
- <td>
- <img src="images/v_kriging_variogram_lid792_500_linear.png" alt="variogram_500_linear" style="float: left; width: 50%; margin-right: 1%; margin-bottom: 0.5em;">
- <img src="images/v_kriging_variogram_lid792_500_surfer.png" alt="variogram_500_surfer" style="float: left; width: 35%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <caption>
- <b>Fig. 8:</b> Experimental and theoretical variogram (left: by <i>v.kriging</i>; right: by <i>Surfer</i> (<i>Golden Software, Inc.</i>))
- </caption>
-</table>
-The difference in coefficient of the linear function might be caused by using slightly different lag size or by different approach to the computation (spatial index in <i>v.kriging</i>, optimization algorithms in <i>Surfer</i> (<i>Golden Software, Inc.</i>) etc.).
+<h2>TODO</h2>
+<ul>
+<li>final phase still needs <b>optimization</b></li>
+<li><b>anisotropy</b> in horizontal direction missing
+<li>current version is suitable just for <b>metric coordinate systems</b>
+<li>enable <b>mask usage</b>
+<li><b>bivariate variogram</b> needs to be rebuilt (theory)
+<li><b>2D interpolation from 3D input layer</b> needs to be rebuilt (especially in case that there are too many points located on identical horizontal coordinates with different elevation)
+</ul>
-<p><b>The results</b> were compared with the values interpolated using the module <i><a href="v.surf.rst.html">v.surf.rst</a></i> and kriging tool of <i>Surfer</i> (<i>Golden Software, Inc.</i>). <b>Fig. 9a</b> - <b>9c</b> show interpolated DEM and the comparisons with the results of another interpolation tools. Statistical characteristics of the results are summarized in <b>Tab. 10</b>.
-<p>
-<table>
- <tr>
- <td>
- <img src="images/v_kriging_result_lid792_500.png" alt="elev_kriging" style="float: left; width: 30%; margin-right: 1%; margin-bottom: 0.5em;">
- <img src="images/v_kriging_diff_lid792_500_rst.png" alt="elev_diff_rst" style="float: left; width: 30%; margin-right: 1%; margin-bottom: 0.5em;">
- <img src="images/v_kriging_diff_lid792_500_surfer.png" alt="elev_diff_surfer" style="float: left; width: 30%; margin-right: 1%; margin-bottom: 0.5em;">
- </td>
- </tr>
- <tr>
- <td>
- <b>Fig. 9:</b> <b>a</b> - the DEM interpolated using <i>v.kriging</i>, <b>b</b> - the difference between <i>v.kriging</i> and <i><a href="v.surf.rst.html">v.surf.rst</a></i>, <b>c</b> - the difference between <i>v.kriging</i> and <i>Surfer</i> (<i>Golden Software, Inc.</i>))
- </td>
- </tr>
-</table>
+<h2>REFERENCES</h2>
-<br>
-
-<table border="1" style="text-align: right; margin-left: 70px;">
- <tr>
- <td rowspan="2" style="text-align: center;"><b>Results</b></td>
- <td rowspan="2" style="text-align: center;">v.kriging</td>
- <td colspan="2" style="text-align: center;"><a href="v.surf.rst.html">v.surf.rst</a></td>
- <td colspan="2" style="text-align: center;">Surfer (<i>Golden Software, Inc.</i>)</td>
- </tr>
- <tr>
- <td>Values</td>
- <td>Differences</td>
- <td>Values</td>
- <td>Differences</td>
- </tr>
- <tr>
- <td style="text-align: left;">Minimum [m]</td>
- <td>105.114</td>
- <td>105.061</td>
- <td>-1.065</td>
- <td>105.090</td>
- <td>-1.181</td>
- </tr>
- <tr>
- <td style="text-align: left;">Maximum [m]</td>
- <td>131.510</td>
- <td>131.570</td>
- <td>2.072</td>
- <td>131.510</td>
- <td>0.522</td>
- </tr>
- <tr>
- <td style="text-align: left;">Mean [m]</td>
- <td>120.763</td>
- <td>120.781</td>
- <td>0.018</td>
- <td>120.584</td>
- <td>-0.178</td>
- </tr>
-<tr>
- <td style="text-align: left;">Variance [m<sup>2</sup>]</td>
- <td>43.7367</td>
- <td>43.2701</td>
- <td>0.045581</td>
- <td>44.2389</td>
- <td>0.027244</td>
- </tr>
- <tr>
- <td style="text-align: left;">Standard deviation [m]</td>
- <td>6.613</td>
- <td>6.578</td>
- <td>0.165</td>
- <td>6.651</td>
- <td>0.213</td>
- </tr>
- <tr>
- <td style="text-align: left;">95% quantile [m]</td>
- <td>130.115</td>
- <td>130.109</td>
- <td>0.225</td>
- <td>130.088</td>
- <td>0.213</td>
- </tr>
- <tr>
- <td style="text-align: left;">75% quantile [m]</td>
- <td>126.580</td>
- <td>126.587</td>
- <td>0.046</td>
- <td>126.434</td>
- <td>-0.047</td>
- </tr>
- <tr>
- <td style="text-align: left;">50% quantile [m]</td>
- <td>121.315</td>
- <td>121.325</td>
- <td>0.000</td>
- <td>121.080</td>
- <td>-0.190</td>
- </tr>
- <tr>
- <td style="text-align: left;">25% quantile [m]</td>
- <td>115.749</td>
- <td>115.786</td>
- <td>-0.047</td>
- <td>115.489</td>
- <td>-0.328</td>
- </tr>
- <tr>
- <td style="text-align: left;">5% quantile [m]</td>
- <td>109.004</td>
- <td>109.115</td>
- <td>-0.139</td>
- <td>108.800</td>
- <td>-0.487</td>
- </tr>
- <caption>
- <b>Tab. 10:</b> Comparison of statistical characteristics of interpolated rasters
- </caption>
-</table>
-
-<br>
-
-Also <b>cross validation</b> results of all the methods mentioned above and their statistical
-characteristics have been compared. Cross validation using <i><a href="v.surf.rst.html">v.surf.rst</a></i>
-was performed with these settings:
-<div class="code"><pre>
-v.surf.rst -c input="elev_lid792_selected at test_kriging" layer="1" zcolumn="value" \
-cvdev="lid792_500_rst_xval" tension=40 segmax=30 npmin=120 dmin=5.000000 dmax=25.000000 zscale=1.0
-</pre></div>
-
-Cross validation points are shown in <b>Fig. 10</b> (bigger circles represent kriging results,
-smaller circles represent the results of RST). Statistical characteristics of the cross validation
-are summarized in <b>Tab. 11</b>.
-
-<br>
-
-<table style="float: left; align: left;">
- <tr>
- <td rowspan="11">
- <img src="images/v_kriging_xvalid_rst_krig.png" alt="xval" style="margin-right: 10%; margin-left: 10%;">
- </td>
- </tr>
- <tr>
- <td>
- <b>Fig. 10:</b> Cross validation by <i>v.kriging</i> and <i><a href="v.surf.rst.html">v.surf.rst</a></i>
- </td>
- </tr>
-</table>
-<table border="1" style="text-align: right; float: left; align: left; vertical-align: center;">
- <tr>
- <td style="text-align: center;"><b>Cross validation</b></td>
- <td>v.kriging</td>
- <td><a href="v.surf.rst.html">v.surf.rst</a></td>
- <td>Surfer (<i>Golden Software, Inc.</i>)</td>
- </tr>
- <tr>
- <td style="text-align: left;">Minimum [m]</td>
- <td>-2.990</td>
- <td>-1.593</td>
- <td>-2.344</td>
- </tr>
- <tr>
- <td style="text-align: left;">Maximum [m]</td>
- <td>2.399</td>
- <td>3.362</td>
- <td>3.011</td>
- </tr>
- <tr>
- <td style="text-align: left;">Mean [m]</td>
- <td>-0.004</td>
- <td>0.004</td>
- <td>0.003</td>
- </tr>
- <tr>
- <td style="text-align: left;">Variance [m<sup>2</sup>]</td>
- <td>0.132490</td>
- <td>0.143742</td>
- <td>0.133809</td>
- </tr>
- <tr>
- <td style="text-align: left;">Standard deviation [m]</td>
- <td>0.364</td>
- <td>0.379</td>
- <td>0.366</td>
- </tr>
- <tr>
- <td style="text-align: left;">95% quantile [m]</td>
- <td>0.434</td>
- <td>0.557</td>
- <td>0.584</td>
- </tr>
- <tr>
- <td style="text-align: left;">50% quantile [m]</td>
- <td>0.018</td>
- <td>-0.013</td>
- <td>-0.020</td>
- </tr>
- <tr>
- <td style="text-align: left;">25% quantile [m]</td>
- <td>-0.113</td>
- <td>-0.144</td>
- <td>-0.126</td>
- </tr>
- <tr>
- <td style="text-align: left;">5% quantile [m]</td>
- <td>-0.587</td>
- <td>-0.499</td>
- <td>-0.445</td>
- </tr>
- <caption>
- <b>Tab. 11:</b> Statistical characteristics<br>of the cross validation results
- </caption>
-</table>
- <p>
-
- <br style="clear:both">
- <p>
- <b>Recommendations</b>
- <table>
+<table>
<tr>
<td>
- In case of too much <b>warnings</b> about input points that have
- "<b>less than 2 neighbours in its closest surrounding</b>. The perimeter of the surrounding will be increased...", please
- consider shorter variogram range.
+ Mitasova, H. and Hofierka, J. (2004). <i>Slovakia Precipitation data</i>. Available at <a href="https://grass.osgeo.org/download/sample-data/">https://grass.osgeo.org/download/sample-data/</a>.
</td>
</tr>
<tr>
- <td>
- Save just figures with theoretical variogram (using <i>file=extension</i> in the middle and final phase).
- Experimental variograms are included in the theoretical variogram plot and separate "experimental" plots
- can be just temporal.
- </td>
- </tr>
- </table>
-
- <p>
- <b>References</b>
- <table>
<tr>
<td>
Neteler, M. and Mitasova, H. (2004). <i>Open Source GIS: A GRASS GIS Approach</i>.
@@ -1240,21 +116,6 @@
</tr>
</table>
-<h2>TODO</h2>
-<ul>
-<li>still need <b>optimization</b> for large datasets</li>
-<li><b>anisotropy</b> in horizontal direction missing
-<li>current version is suitable just for <b>metric coordinate systems</b>
-<li>enable <b>mask usage</b>
-<li><b>bivariate variogram</b> needs to be rebuilt (theory)
-<li><b>2D interpolation from 3D input layer</b> needs to be rebuilt (especially in case that there are too many points located on identical horizontal coordinates with different elevation)
-</ul>
-
-<h2>BUGS</h2>
-<ul>
-<li><b>bivariate variogram</b> - too inaccurate results
-</ul>
-
<h2>SEE ALSO</h2>
<em>
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