The surface fractal dimension of the soil–pore interface as measured by image analysis
Identifieur interne : 001997 ( Istex/Corpus ); précédent : 001996; suivant : 001998The surface fractal dimension of the soil–pore interface as measured by image analysis
Auteurs : A. Dathe ; S. Eins ; J. Niemeyer ; G. GeroldSource :
- Geoderma [ 0016-7061 ] ; 2001.
Abstract
There is general interest in quantifying soil structure in order to obtain physically based parameters relevant to transport processes. To measure the surface fractal dimension of the pore–solid interface we use approaches known from fractal geometry. The characteristics of this interface, expressed by its fractal dimension, are descriptors of the heterogeneity and complexity of soil structure. Samples of the Bt horizon of a Luvisol in loess were taken near Göttingen, Germany. To prepare thin sections, the material was dehydrated and embedded in resin. We obtained digital images at different magnifications from a field emission scanning electron microscope. Automatic image analysis was used to determine the corresponding surface fractal dimension by using the box counting and dilation methods, respectively. As the fractal dimension of a line (DL) within a plain has been measured, the surface fractal dimension DS is obtained by DS=DL+1 assuming isotropy. We strongly focussed the calculation of the fractal dimension from the measured data files. The decision as to which data should be included between the lower and upper cutoffs is of fundamental significance to the final result. For the upper cutoff, we followed the convention that the scale range should not exceed 30% of the characteristic length (object or image size). Data derived from outside both cutoffs reflect structural properties, either of pixels (lower cutoff) or of structuring elements (upper cutoff). Different methods were used to derive a mean surface fractal dimension for one magnification for (i) single images and (ii) each measurement step. Within the same range of scale, differences between the two methods (box counting and dilation) were smaller than the standard deviation of DS. In contrast to our expectations for a mathematical fractal, we found decreasing values for DS with increasing magnification. The values drift from DS=2.91 for a resolution of 2.44 μm/pixel to DS=2.58 for a resolution of 0.05 μm/pixel. By fitting two straight lines to the log–log plot, we found a crossover-point at a scale of about 14 μm, forming the border between textural and structural fractality. In addition, we will discuss further results obtained as well as possible sources of error.
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DOI: 10.1016/S0016-7061(01)00077-5
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title>The surface fractal dimension of the soil–pore interface as measured by image analysis</title><author><name sortKey="Dathe, A" sort="Dathe, A" uniqKey="Dathe A" first="A" last="Dathe">A. Dathe</name><affiliation><mods:affiliation>E-mail: adathe@gwdg.de</mods:affiliation></affiliation><affiliation><mods:affiliation>Landscape Ecology Unit, Institute of Geography, University of Göttingen, Goldschmidtstrasse 5, 37077 Göttingen, Germany</mods:affiliation></affiliation></author><author><name sortKey="Eins, S" sort="Eins, S" uniqKey="Eins S" first="S" last="Eins">S. Eins</name><affiliation><mods:affiliation>Developmental Neurobiology Unit, Department of Anatomy, University of Göttingen, 37075 Göttingen, Germany</mods:affiliation></affiliation></author><author><name sortKey="Niemeyer, J" sort="Niemeyer, J" uniqKey="Niemeyer J" first="J" last="Niemeyer">J. Niemeyer</name><affiliation><mods:affiliation>Department of Soil Science, Faculty of Geography/Geoscience, University of Trier, 54286 Trier, Germany</mods:affiliation></affiliation></author><author><name sortKey="Gerold, G" sort="Gerold, G" uniqKey="Gerold G" first="G" last="Gerold">G. 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Dathe</name><affiliation><mods:affiliation>E-mail: adathe@gwdg.de</mods:affiliation></affiliation><affiliation><mods:affiliation>Landscape Ecology Unit, Institute of Geography, University of Göttingen, Goldschmidtstrasse 5, 37077 Göttingen, Germany</mods:affiliation></affiliation></author><author><name sortKey="Eins, S" sort="Eins, S" uniqKey="Eins S" first="S" last="Eins">S. Eins</name><affiliation><mods:affiliation>Developmental Neurobiology Unit, Department of Anatomy, University of Göttingen, 37075 Göttingen, Germany</mods:affiliation></affiliation></author><author><name sortKey="Niemeyer, J" sort="Niemeyer, J" uniqKey="Niemeyer J" first="J" last="Niemeyer">J. Niemeyer</name><affiliation><mods:affiliation>Department of Soil Science, Faculty of Geography/Geoscience, University of Trier, 54286 Trier, Germany</mods:affiliation></affiliation></author><author><name sortKey="Gerold, G" sort="Gerold, G" uniqKey="Gerold G" first="G" last="Gerold">G. Gerold</name><affiliation><mods:affiliation>Landscape Ecology Unit, Institute of Geography, University of Göttingen, Goldschmidtstrasse 5, 37077 Göttingen, Germany</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Geoderma</title><title level="j" type="abbrev">GEODER</title><idno type="ISSN">0016-7061</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="2001">2001</date><biblScope unit="volume">103</biblScope><biblScope unit="issue">1–2</biblScope><biblScope unit="page" from="203">203</biblScope><biblScope unit="page" to="229">229</biblScope></imprint><idno type="ISSN">0016-7061</idno></series><idno type="istex">D8BFD3DFD4D6144F166B69600C3A4EA058C10D0B</idno><idno type="DOI">10.1016/S0016-7061(01)00077-5</idno><idno type="PII">S0016-7061(01)00077-5</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0016-7061</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">There is general interest in quantifying soil structure in order to obtain physically based parameters relevant to transport processes. To measure the surface fractal dimension of the pore–solid interface we use approaches known from fractal geometry. The characteristics of this interface, expressed by its fractal dimension, are descriptors of the heterogeneity and complexity of soil structure. Samples of the Bt horizon of a Luvisol in loess were taken near Göttingen, Germany. To prepare thin sections, the material was dehydrated and embedded in resin. We obtained digital images at different magnifications from a field emission scanning electron microscope. Automatic image analysis was used to determine the corresponding surface fractal dimension by using the box counting and dilation methods, respectively. As the fractal dimension of a line (DL) within a plain has been measured, the surface fractal dimension DS is obtained by DS=DL+1 assuming isotropy. We strongly focussed the calculation of the fractal dimension from the measured data files. The decision as to which data should be included between the lower and upper cutoffs is of fundamental significance to the final result. For the upper cutoff, we followed the convention that the scale range should not exceed 30% of the characteristic length (object or image size). Data derived from outside both cutoffs reflect structural properties, either of pixels (lower cutoff) or of structuring elements (upper cutoff). Different methods were used to derive a mean surface fractal dimension for one magnification for (i) single images and (ii) each measurement step. Within the same range of scale, differences between the two methods (box counting and dilation) were smaller than the standard deviation of DS. In contrast to our expectations for a mathematical fractal, we found decreasing values for DS with increasing magnification. The values drift from DS=2.91 for a resolution of 2.44 μm/pixel to DS=2.58 for a resolution of 0.05 μm/pixel. By fitting two straight lines to the log–log plot, we found a crossover-point at a scale of about 14 μm, forming the border between textural and structural fractality. In addition, we will discuss further results obtained as well as possible sources of error.</div></front></TEI><istex><corpusName>elsevier</corpusName><author><json:item><name>A Dathe</name><affiliations><json:string>E-mail: adathe@gwdg.de</json:string><json:string>Landscape Ecology Unit, Institute of Geography, University of Göttingen, Goldschmidtstrasse 5, 37077 Göttingen, Germany</json:string></affiliations></json:item><json:item><name>S Eins</name><affiliations><json:string>Developmental Neurobiology Unit, Department of Anatomy, University of Göttingen, 37075 Göttingen, Germany</json:string></affiliations></json:item><json:item><name>J Niemeyer</name><affiliations><json:string>Department of Soil Science, Faculty of Geography/Geoscience, University of Trier, 54286 Trier, Germany</json:string></affiliations></json:item><json:item><name>G Gerold</name><affiliations><json:string>Landscape Ecology Unit, Institute of Geography, University of Göttingen, Goldschmidtstrasse 5, 37077 Göttingen, Germany</json:string></affiliations></json:item></author><subject><json:item><lang><json:string>eng</json:string></lang><value>Soil–pore interface</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Fractal dimension</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Image analysis</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Box counting</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Dilation</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Scanning electron microscopy (SEM)</value></json:item></subject><language><json:string>eng</json:string></language><originalGenre><json:string>Full-length article</json:string></originalGenre><abstract>There is general interest in quantifying soil structure in order to obtain physically based parameters relevant to transport processes. To measure the surface fractal dimension of the pore–solid interface we use approaches known from fractal geometry. The characteristics of this interface, expressed by its fractal dimension, are descriptors of the heterogeneity and complexity of soil structure. Samples of the Bt horizon of a Luvisol in loess were taken near Göttingen, Germany. To prepare thin sections, the material was dehydrated and embedded in resin. We obtained digital images at different magnifications from a field emission scanning electron microscope. Automatic image analysis was used to determine the corresponding surface fractal dimension by using the box counting and dilation methods, respectively. As the fractal dimension of a line (DL) within a plain has been measured, the surface fractal dimension DS is obtained by DS=DL+1 assuming isotropy. We strongly focussed the calculation of the fractal dimension from the measured data files. The decision as to which data should be included between the lower and upper cutoffs is of fundamental significance to the final result. For the upper cutoff, we followed the convention that the scale range should not exceed 30% of the characteristic length (object or image size). Data derived from outside both cutoffs reflect structural properties, either of pixels (lower cutoff) or of structuring elements (upper cutoff). Different methods were used to derive a mean surface fractal dimension for one magnification for (i) single images and (ii) each measurement step. Within the same range of scale, differences between the two methods (box counting and dilation) were smaller than the standard deviation of DS. In contrast to our expectations for a mathematical fractal, we found decreasing values for DS with increasing magnification. The values drift from DS=2.91 for a resolution of 2.44 μm/pixel to DS=2.58 for a resolution of 0.05 μm/pixel. By fitting two straight lines to the log–log plot, we found a crossover-point at a scale of about 14 μm, forming the border between textural and structural fractality. In addition, we will discuss further results obtained as well as possible sources of error.</abstract><qualityIndicators><score>8</score><pdfVersion>1.2</pdfVersion><pdfPageSize>435 x 643 pts</pdfPageSize><refBibsNative>true</refBibsNative><keywordCount>6</keywordCount><abstractCharCount>2272</abstractCharCount><pdfWordCount>8396</pdfWordCount><pdfCharCount>48718</pdfCharCount><pdfPageCount>27</pdfPageCount><abstractWordCount>350</abstractWordCount></qualityIndicators><title>The surface fractal dimension of the soil–pore interface as measured by image analysis</title><pii><json:string>S0016-7061(01)00077-5</json:string></pii><refBibs><json:item><author><json:item><name>C Ahl</name></json:item><json:item><name>J Niemeyer</name></json:item></author><host><volume>152</volume><pages><last>458</last><first>457</first></pages><author></author><title>Z. Pflanzenernaehr. 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McBratney</name></json:item></editor><pages><last>229</last><first>203</first></pages><conference><json:item><name>Estimating uncertainty in soil models Pedometrics '99</name></json:item></conference><issn><json:string>0016-7061</json:string></issn><issue>1–2</issue><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><title>Geoderma</title><publicationDate>2001</publicationDate></host><categories><wos><json:string>science</json:string><json:string>soil science</json:string></wos><scienceMetrix><json:string>applied sciences</json:string><json:string>agriculture, fisheries & forestry</json:string><json:string>agronomy & agriculture</json:string></scienceMetrix></categories><publicationDate>2001</publicationDate><copyrightDate>2001</copyrightDate><doi><json:string>10.1016/S0016-7061(01)00077-5</json:string></doi><id>D8BFD3DFD4D6144F166B69600C3A4EA058C10D0B</id><score>0.81212854</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/D8BFD3DFD4D6144F166B69600C3A4EA058C10D0B/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/D8BFD3DFD4D6144F166B69600C3A4EA058C10D0B/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/D8BFD3DFD4D6144F166B69600C3A4EA058C10D0B/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a">The surface fractal dimension of the soil–pore interface as measured by image analysis</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>ELSEVIER</publisher><availability><p>©2001 Elsevier Science B.V.</p></availability><date>2001</date></publicationStmt><notesStmt><note type="content">Fig. 1: Original SEM image with overlaid pore contours. The contour line will be covered with boxes or will be dilated. The soil grains appear as the bright phase and the pore space appears dark.</note><note type="content">Fig. 2: (a) Original image obtained by bright field microscopy followed by RGB-to-grey transformation using the KS400®-Software. Section thickness is ca. 30 μm. (b) Grey-level histogram of image (a).</note><note type="content">Fig. 3: (a) Field emission SEM image composed of back-scattered and secondary electrons. (b) Grey-level histogram of image (a).</note><note type="content">Fig. 4: Lower cutoff for box counting. The correlation coefficient of the corresponding regression line is changed by rejecting a growing number of data points in the Richardson plot. Calculations for the combined data pool (method B, Table 2). ‘0’ on the abscissa means the full data set, ‘1’ means omitting data corresponding to 1 pixel resolution, and so on, by amounts of 2 pixels.</note><note type="content">Fig. 5: Lower cutoff for dilation (displayed for one image, highest magnification). The local slopes from neighbouring data points and their differences are shown. As a decision criterion, only those data for which differences of the slope are within the interval of ±0.025 (area marked with grey background) were taken into account.</note><note type="content">Fig. 6: (a) Richardson plot in respect of cutoffs. The data were obtained by the box counting procedure for the image shown in Fig. 3a. According to Eq. (2), the slope of the regression line yields the box counting fractal dimension of the pore–solid interface (DLd). R2—coefficient of determination. (b) Results of the box counting method for different resolutions. The data of the combined data pool (method B, Table 2) are shown.</note><note type="content">Fig. 7: Demonstration of fractal analysis for dilation of one single image, shown in Fig. 1. (a) Richardson plot with regard to cutoffs. According to Eq. (4), DLd is determined as 1.77. (b) Same data set as in (a), but with an exponential increase of dilation steps.</note><note type="content">Fig. 8: Adjusted combined data pool obtained by the box counting procedure. (a) One straight line has been fitted with a robust technique. (b) Two straight lines have been fitted and the crossover point has been determined by minimizing the absolute deviation. The regression equations are shown for both figures, and the results for DSb, DS1b and DS2b are shown in Table 3.</note><note type="content">Fig. 9: Adjusted combined data pool obtained by the dilation procedure. (a) One straight line has been fitted with a robust technique. (b) Two straight lines have been fitted and the crossover point has been determined by minimizing the absolute deviation. The regression equations are shown for both figures, and the results for DSd, DS1d and DS2d are shown in Table 4.</note><note type="content">Table 1: Selected soil physical data for the sample under investigation The porosity is calculated using the bulk and particle densities.</note><note type="content">Table 2: Surface fractal dimensions for box counting DSb and dilation DSd and related measurement parameters of the given soil sample The standard deviations and standard errors, respectively, are given in brackets. The first value is the standard deviation for method A, where DS is calculated as the average of single images. The second value is the standard error of the estimated regression coefficient for the slope (method B), where DS is calculated for the combined data pool of all images (cf. Section 2.5).</note><note type="content">Table 3: Fractal dimensions measured by box counting and estimated for the adjusted data pool of all images (method B) and magnifications As a robust technique, local M-Estimates were used. The absolute deviations are given in brackets. The absolute deviation printed after DS2b applies to both DS1b and DS2b. The scale range for all data is from 1.38 to 185.60 μm. A linear increasing step size means all data were taken into account. An exponential increasing step size means the distance for scaling elements is increasing exponentially.</note><note type="content">Table 4: Fractal dimensions obtained from dilation data (cf. Table 3 for box counting) The absolute deviations are given in brackets. The absolute deviation printed after DS2d applies to both DS1d and DS2d. The range of all data is from 1.38 to 185.60 μm.</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a">The surface fractal dimension of the soil–pore interface as measured by image analysis</title><author xml:id="author-1"><persName><forename type="first">A</forename><surname>Dathe</surname></persName><email>adathe@gwdg.de</email><note type="correspondence"><p>Corresponding author. Tel. +49-551-394570; fax: +49-551-398006</p></note><affiliation>Landscape Ecology Unit, Institute of Geography, University of Göttingen, Goldschmidtstrasse 5, 37077 Göttingen, Germany</affiliation></author><author xml:id="author-2"><persName><forename type="first">S</forename><surname>Eins</surname></persName><affiliation>Developmental Neurobiology Unit, Department of Anatomy, University of Göttingen, 37075 Göttingen, Germany</affiliation></author><author xml:id="author-3"><persName><forename type="first">J</forename><surname>Niemeyer</surname></persName><affiliation>Department of Soil Science, Faculty of Geography/Geoscience, University of Trier, 54286 Trier, Germany</affiliation></author><author xml:id="author-4"><persName><forename type="first">G</forename><surname>Gerold</surname></persName><affiliation>Landscape Ecology Unit, Institute of Geography, University of Göttingen, Goldschmidtstrasse 5, 37077 Göttingen, Germany</affiliation></author></analytic><monogr><title level="j">Geoderma</title><title level="j" type="abbrev">GEODER</title><idno type="pISSN">0016-7061</idno><idno type="PII">S0016-7061(00)X0079-1</idno><meeting><addName>Estimating uncertainty in soil models</addName><addName>Pedometrics '99</addName></meeting><editor><persName>I.O.A. Odeh</persName></editor><editor><persName>A.B. McBratney</persName></editor><imprint><publisher>ELSEVIER</publisher><date type="published" when="2001"></date><biblScope unit="volume">103</biblScope><biblScope unit="issue">1–2</biblScope><biblScope unit="page" from="203">203</biblScope><biblScope unit="page" to="229">229</biblScope></imprint></monogr><idno type="istex">D8BFD3DFD4D6144F166B69600C3A4EA058C10D0B</idno><idno type="DOI">10.1016/S0016-7061(01)00077-5</idno><idno type="PII">S0016-7061(01)00077-5</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2001</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>There is general interest in quantifying soil structure in order to obtain physically based parameters relevant to transport processes. To measure the surface fractal dimension of the pore–solid interface we use approaches known from fractal geometry. The characteristics of this interface, expressed by its fractal dimension, are descriptors of the heterogeneity and complexity of soil structure. Samples of the Bt horizon of a Luvisol in loess were taken near Göttingen, Germany. To prepare thin sections, the material was dehydrated and embedded in resin. We obtained digital images at different magnifications from a field emission scanning electron microscope. Automatic image analysis was used to determine the corresponding surface fractal dimension by using the box counting and dilation methods, respectively. As the fractal dimension of a line (DL) within a plain has been measured, the surface fractal dimension DS is obtained by DS=DL+1 assuming isotropy. We strongly focussed the calculation of the fractal dimension from the measured data files. The decision as to which data should be included between the lower and upper cutoffs is of fundamental significance to the final result. For the upper cutoff, we followed the convention that the scale range should not exceed 30% of the characteristic length (object or image size). Data derived from outside both cutoffs reflect structural properties, either of pixels (lower cutoff) or of structuring elements (upper cutoff). Different methods were used to derive a mean surface fractal dimension for one magnification for (i) single images and (ii) each measurement step. Within the same range of scale, differences between the two methods (box counting and dilation) were smaller than the standard deviation of DS. In contrast to our expectations for a mathematical fractal, we found decreasing values for DS with increasing magnification. The values drift from DS=2.91 for a resolution of 2.44 μm/pixel to DS=2.58 for a resolution of 0.05 μm/pixel. By fitting two straight lines to the log–log plot, we found a crossover-point at a scale of about 14 μm, forming the border between textural and structural fractality. In addition, we will discuss further results obtained as well as possible sources of error.</p></abstract><textClass><keywords scheme="keyword"><list><head>Keywords</head><item><term>Soil–pore interface</term></item><item><term>Fractal dimension</term></item><item><term>Image analysis</term></item><item><term>Box counting</term></item><item><term>Dilation</term></item><item><term>Scanning electron microscopy (SEM)</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="2001-01-09">Modified</change><change when="2001">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/D8BFD3DFD4D6144F166B69600C3A4EA058C10D0B/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Elsevier, elements deleted: ce:floats; body; tail"><istex:xmlDeclaration>version="1.0" encoding="utf-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//ES//DTD journal article DTD version 4.5.2//EN//XML" URI="art452.dtd" name="istex:docType"><istex:entity SYSTEM="gr1" NDATA="IMAGE" name="gr1"></istex:entity><istex:entity SYSTEM="gr2" NDATA="IMAGE" name="gr2"></istex:entity><istex:entity SYSTEM="gr3" NDATA="IMAGE" name="gr3"></istex:entity><istex:entity SYSTEM="gr4" NDATA="IMAGE" name="gr4"></istex:entity><istex:entity SYSTEM="gr5" NDATA="IMAGE" name="gr5"></istex:entity><istex:entity SYSTEM="gr6" NDATA="IMAGE" name="gr6"></istex:entity><istex:entity SYSTEM="gr7" NDATA="IMAGE" name="gr7"></istex:entity><istex:entity SYSTEM="gr8" NDATA="IMAGE" name="gr8"></istex:entity><istex:entity SYSTEM="gr9" NDATA="IMAGE" name="gr9"></istex:entity></istex:docType><istex:document><converted-article version="4.5.2" docsubtype="fla"><item-info><jid>GEODER</jid><aid>1727</aid><ce:pii>S0016-7061(01)00077-5</ce:pii><ce:doi>10.1016/S0016-7061(01)00077-5</ce:doi><ce:copyright type="full-transfer" year="2001">Elsevier Science B.V.</ce:copyright></item-info><head><ce:title>The surface fractal dimension of the soil–pore interface as measured by image analysis</ce:title><ce:author-group><ce:author><ce:given-name>A</ce:given-name><ce:surname>Dathe</ce:surname><ce:cross-ref refid="COR1">*</ce:cross-ref><ce:cross-ref refid="AFF1"><ce:sup>a</ce:sup></ce:cross-ref><ce:e-address>adathe@gwdg.de</ce:e-address></ce:author><ce:author><ce:given-name>S</ce:given-name><ce:surname>Eins</ce:surname><ce:cross-ref refid="AFF2"><ce:sup>b</ce:sup></ce:cross-ref></ce:author><ce:author><ce:given-name>J</ce:given-name><ce:surname>Niemeyer</ce:surname><ce:cross-ref refid="AFF3"><ce:sup>c</ce:sup></ce:cross-ref></ce:author><ce:author><ce:given-name>G</ce:given-name><ce:surname>Gerold</ce:surname><ce:cross-ref refid="AFF1"><ce:sup>a</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="AFF1"><ce:label>a</ce:label><ce:textfn>Landscape Ecology Unit, Institute of Geography, University of Göttingen, Goldschmidtstrasse 5, 37077 Göttingen, Germany</ce:textfn></ce:affiliation><ce:affiliation id="AFF2"><ce:label>b</ce:label><ce:textfn>Developmental Neurobiology Unit, Department of Anatomy, University of Göttingen, 37075 Göttingen, Germany</ce:textfn></ce:affiliation><ce:affiliation id="AFF3"><ce:label>c</ce:label><ce:textfn>Department of Soil Science, Faculty of Geography/Geoscience, University of Trier, 54286 Trier, Germany</ce:textfn></ce:affiliation><ce:correspondence id="COR1"><ce:label>*</ce:label><ce:text>Corresponding author. Tel. +49-551-394570; fax: +49-551-398006</ce:text></ce:correspondence></ce:author-group><ce:date-received day="15" month="2" year="2000"></ce:date-received><ce:date-revised day="9" month="1" year="2001"></ce:date-revised><ce:date-accepted day="9" month="2" year="2001"></ce:date-accepted><ce:abstract><ce:section-title>Abstract</ce:section-title><ce:abstract-sec><ce:simple-para>There is general interest in quantifying soil structure in order to obtain physically based parameters relevant to transport processes. To measure the surface fractal dimension of the pore–solid interface we use approaches known from fractal geometry. The characteristics of this interface, expressed by its fractal dimension, are descriptors of the heterogeneity and complexity of soil structure. Samples of the Bt horizon of a Luvisol in loess were taken near Göttingen, Germany. To prepare thin sections, the material was dehydrated and embedded in resin. We obtained digital images at different magnifications from a field emission scanning electron microscope. Automatic image analysis was used to determine the corresponding surface fractal dimension by using the box counting and dilation methods, respectively. As the fractal dimension of a line (<ce:italic>D</ce:italic><ce:inf>L</ce:inf>) within a plain has been measured, the surface fractal dimension <ce:italic>D</ce:italic><ce:inf>S</ce:inf> is obtained by <ce:italic>D</ce:italic><ce:inf>S</ce:inf>=<ce:italic>D</ce:italic><ce:inf>L</ce:inf>+1 assuming isotropy. We strongly focussed the calculation of the fractal dimension from the measured data files. The decision as to which data should be included between the lower and upper cutoffs is of fundamental significance to the final result. For the upper cutoff, we followed the convention that the scale range should not exceed 30% of the characteristic length (object or image size). Data derived from outside both cutoffs reflect structural properties, either of pixels (lower cutoff) or of structuring elements (upper cutoff). Different methods were used to derive a mean surface fractal dimension for one magnification for (i) single images and (ii) each measurement step. Within the same range of scale, differences between the two methods (box counting and dilation) were smaller than the standard deviation of <ce:italic>D</ce:italic><ce:inf>S</ce:inf>. In contrast to our expectations for a mathematical fractal, we found decreasing values for <ce:italic>D</ce:italic><ce:inf>S</ce:inf> with increasing magnification. The values drift from <ce:italic>D</ce:italic><ce:inf>S</ce:inf>=2.91 for a resolution of 2.44 μm/pixel to <ce:italic>D</ce:italic><ce:inf>S</ce:inf>=2.58 for a resolution of 0.05 μm/pixel. By fitting two straight lines to the log–log plot, we found a crossover-point at a scale of about 14 μm, forming the border between textural and structural fractality. In addition, we will discuss further results obtained as well as possible sources of error.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:keywords class="keyword"><ce:section-title>Keywords</ce:section-title><ce:keyword><ce:text>Soil–pore interface</ce:text></ce:keyword><ce:keyword><ce:text>Fractal dimension</ce:text></ce:keyword><ce:keyword><ce:text>Image analysis</ce:text></ce:keyword><ce:keyword><ce:text>Box counting</ce:text></ce:keyword><ce:keyword><ce:text>Dilation</ce:text></ce:keyword><ce:keyword><ce:text>Scanning electron microscopy (SEM)</ce:text></ce:keyword></ce:keywords></head></converted-article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo><title>The surface fractal dimension of the soil–pore interface as measured by image analysis</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>The surface fractal dimension of the soil–pore interface as measured by image analysis</title></titleInfo><name type="personal"><namePart type="given">A</namePart><namePart type="family">Dathe</namePart><affiliation>E-mail: adathe@gwdg.de</affiliation><affiliation>Landscape Ecology Unit, Institute of Geography, University of Göttingen, Goldschmidtstrasse 5, 37077 Göttingen, Germany</affiliation><description>Corresponding author. Tel. +49-551-394570; fax: +49-551-398006</description><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">S</namePart><namePart type="family">Eins</namePart><affiliation>Developmental Neurobiology Unit, Department of Anatomy, University of Göttingen, 37075 Göttingen, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">J</namePart><namePart type="family">Niemeyer</namePart><affiliation>Department of Soil Science, Faculty of Geography/Geoscience, University of Trier, 54286 Trier, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">G</namePart><namePart type="family">Gerold</namePart><affiliation>Landscape Ecology Unit, Institute of Geography, University of Göttingen, Goldschmidtstrasse 5, 37077 Göttingen, Germany</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="Full-length article"></genre><originInfo><publisher>ELSEVIER</publisher><dateIssued encoding="w3cdtf">2001</dateIssued><dateModified encoding="w3cdtf">2001-01-09</dateModified><copyrightDate encoding="w3cdtf">2001</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">There is general interest in quantifying soil structure in order to obtain physically based parameters relevant to transport processes. To measure the surface fractal dimension of the pore–solid interface we use approaches known from fractal geometry. The characteristics of this interface, expressed by its fractal dimension, are descriptors of the heterogeneity and complexity of soil structure. Samples of the Bt horizon of a Luvisol in loess were taken near Göttingen, Germany. To prepare thin sections, the material was dehydrated and embedded in resin. We obtained digital images at different magnifications from a field emission scanning electron microscope. Automatic image analysis was used to determine the corresponding surface fractal dimension by using the box counting and dilation methods, respectively. As the fractal dimension of a line (DL) within a plain has been measured, the surface fractal dimension DS is obtained by DS=DL+1 assuming isotropy. We strongly focussed the calculation of the fractal dimension from the measured data files. The decision as to which data should be included between the lower and upper cutoffs is of fundamental significance to the final result. For the upper cutoff, we followed the convention that the scale range should not exceed 30% of the characteristic length (object or image size). Data derived from outside both cutoffs reflect structural properties, either of pixels (lower cutoff) or of structuring elements (upper cutoff). Different methods were used to derive a mean surface fractal dimension for one magnification for (i) single images and (ii) each measurement step. Within the same range of scale, differences between the two methods (box counting and dilation) were smaller than the standard deviation of DS. In contrast to our expectations for a mathematical fractal, we found decreasing values for DS with increasing magnification. The values drift from DS=2.91 for a resolution of 2.44 μm/pixel to DS=2.58 for a resolution of 0.05 μm/pixel. By fitting two straight lines to the log–log plot, we found a crossover-point at a scale of about 14 μm, forming the border between textural and structural fractality. In addition, we will discuss further results obtained as well as possible sources of error.</abstract><note type="content">Fig. 1: Original SEM image with overlaid pore contours. The contour line will be covered with boxes or will be dilated. The soil grains appear as the bright phase and the pore space appears dark.</note><note type="content">Fig. 2: (a) Original image obtained by bright field microscopy followed by RGB-to-grey transformation using the KS400®-Software. Section thickness is ca. 30 μm. (b) Grey-level histogram of image (a).</note><note type="content">Fig. 3: (a) Field emission SEM image composed of back-scattered and secondary electrons. (b) Grey-level histogram of image (a).</note><note type="content">Fig. 4: Lower cutoff for box counting. The correlation coefficient of the corresponding regression line is changed by rejecting a growing number of data points in the Richardson plot. Calculations for the combined data pool (method B, Table 2). ‘0’ on the abscissa means the full data set, ‘1’ means omitting data corresponding to 1 pixel resolution, and so on, by amounts of 2 pixels.</note><note type="content">Fig. 5: Lower cutoff for dilation (displayed for one image, highest magnification). The local slopes from neighbouring data points and their differences are shown. As a decision criterion, only those data for which differences of the slope are within the interval of ±0.025 (area marked with grey background) were taken into account.</note><note type="content">Fig. 6: (a) Richardson plot in respect of cutoffs. The data were obtained by the box counting procedure for the image shown in Fig. 3a. According to Eq. (2), the slope of the regression line yields the box counting fractal dimension of the pore–solid interface (DLd). R2—coefficient of determination. (b) Results of the box counting method for different resolutions. The data of the combined data pool (method B, Table 2) are shown.</note><note type="content">Fig. 7: Demonstration of fractal analysis for dilation of one single image, shown in Fig. 1. (a) Richardson plot with regard to cutoffs. According to Eq. (4), DLd is determined as 1.77. (b) Same data set as in (a), but with an exponential increase of dilation steps.</note><note type="content">Fig. 8: Adjusted combined data pool obtained by the box counting procedure. (a) One straight line has been fitted with a robust technique. (b) Two straight lines have been fitted and the crossover point has been determined by minimizing the absolute deviation. The regression equations are shown for both figures, and the results for DSb, DS1b and DS2b are shown in Table 3.</note><note type="content">Fig. 9: Adjusted combined data pool obtained by the dilation procedure. (a) One straight line has been fitted with a robust technique. (b) Two straight lines have been fitted and the crossover point has been determined by minimizing the absolute deviation. The regression equations are shown for both figures, and the results for DSd, DS1d and DS2d are shown in Table 4.</note><note type="content">Table 1: Selected soil physical data for the sample under investigation The porosity is calculated using the bulk and particle densities.</note><note type="content">Table 2: Surface fractal dimensions for box counting DSb and dilation DSd and related measurement parameters of the given soil sample The standard deviations and standard errors, respectively, are given in brackets. The first value is the standard deviation for method A, where DS is calculated as the average of single images. The second value is the standard error of the estimated regression coefficient for the slope (method B), where DS is calculated for the combined data pool of all images (cf. Section 2.5).</note><note type="content">Table 3: Fractal dimensions measured by box counting and estimated for the adjusted data pool of all images (method B) and magnifications As a robust technique, local M-Estimates were used. The absolute deviations are given in brackets. The absolute deviation printed after DS2b applies to both DS1b and DS2b. The scale range for all data is from 1.38 to 185.60 μm. A linear increasing step size means all data were taken into account. An exponential increasing step size means the distance for scaling elements is increasing exponentially.</note><note type="content">Table 4: Fractal dimensions obtained from dilation data (cf. Table 3 for box counting) The absolute deviations are given in brackets. The absolute deviation printed after DS2d applies to both DS1d and DS2d. The range of all data is from 1.38 to 185.60 μm.</note><subject><genre>Keywords</genre><topic>Soil–pore interface</topic><topic>Fractal dimension</topic><topic>Image analysis</topic><topic>Box counting</topic><topic>Dilation</topic><topic>Scanning electron microscopy (SEM)</topic></subject><relatedItem type="host"><titleInfo><title>Geoderma</title></titleInfo><titleInfo type="abbreviated"><title>GEODER</title></titleInfo><name type="conference"><namePart>Estimating uncertainty in soil models</namePart><namePart>Pedometrics '99</namePart></name><name type="personal"><namePart>I.O.A. Odeh</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart>A.B. McBratney</namePart><role><roleTerm type="text">editor</roleTerm></role></name><genre type="journal">journal</genre><originInfo><dateIssued encoding="w3cdtf">200109</dateIssued></originInfo><identifier type="ISSN">0016-7061</identifier><identifier type="PII">S0016-7061(00)X0079-1</identifier><part><date>200109</date><detail type="issue"><title>Estimating uncertainty in soil models</title></detail><detail type="volume"><number>103</number><caption>vol.</caption></detail><detail type="issue"><number>1–2</number><caption>no.</caption></detail><extent unit="issue pages"><start>1</start><end>230</end></extent><extent unit="pages"><start>203</start><end>229</end></extent></part></relatedItem><identifier type="istex">D8BFD3DFD4D6144F166B69600C3A4EA058C10D0B</identifier><identifier type="DOI">10.1016/S0016-7061(01)00077-5</identifier><identifier type="PII">S0016-7061(01)00077-5</identifier><accessCondition type="use and reproduction" contentType="copyright">©2001 Elsevier Science B.V.</accessCondition><recordInfo><recordContentSource>ELSEVIER</recordContentSource><recordOrigin>Elsevier Science B.V., ©2001</recordOrigin></recordInfo></mods></metadata></istex></record>Pour manipuler ce document sous Unix (Dilib)
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