Analysis and simulation of scale-dependent fractal surfaces with application to seafloor morphology
Identifieur interne : 001995 ( Istex/Corpus ); précédent : 001994; suivant : 001996Analysis and simulation of scale-dependent fractal surfaces with application to seafloor morphology
Auteurs : Ute Christina Herzfeld ; Christoph OverbeckSource :
- Computers and Geosciences [ 0098-3004 ] ; 1999.
Abstract
Common theories on fractal surfaces as observed in geology assume a universality law, in most instances the simplest type of universality, which is self-similarity or self-affinity; in the case of multifractals, another well-known special type of fractals, a more complex form of scale invariance is described using one generating process. The assumption of a scale-invariant universality law, however, implies that a geological object was created by a single underlying process, which is clearly in contradiction to geological knowledge and measurable observations. The processes of crust generation, seafloor spreading, sediment deposition, and erosion work at different specific homogeneity ranges of scale, and such scale dependency is observed in many data sets collected for topographic surfaces. This necessitates the design of methods and algorithms for analysis and simulation of fractal surfaces with scale-dependent spatial characteristica. A suite of algorithms and programs for this purpose is compiled and presented in this paper. Numerical algorithms build on geostatistics, Fourier theory, and some “fractal” methods. The approach presented here uses a dimension parameter for characterization of roughness and an anisotropy factor, given with respect to a principal direction, to capture anisotropic properties. Analytical methods are an isarithm method, a variogram method, a Fourier method, and an isarithm-type Fourier method for estimation of a dimension parameter. In applications to bathymetric data from the western flank of the mid-Atlantic ridge, the variogram method is found most accurate and produces results consistent with geological observations. Interpolation, unconditional simulation and conditional simulation algorithms based on Fourier methods and Fractional Brownian Surfaces localized in scale are combined to construct and merge grids of different scales with specific roughness and anisotropy characteristics, resultant in surfaces with scale-dependent properties which almost exactly reproduce those observed from geophysical data in the seafloor case studies. The scale-dependent simulation methods serve to (1) extrapolate in scale beyond the observed resolution, if roughness and anisotropic properties are known from another area with similar characteristics, and thus provide information on subscale properties for surveys with instrumentation of lower resolution, and (2) extrapolate and simulate in space, if an area has only partly been covered by a survey.
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DOI: 10.1016/S0098-3004(99)00062-X
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title>Analysis and simulation of scale-dependent fractal surfaces with application to seafloor morphology</title><author><name sortKey="Herzfeld, Ute Christina" sort="Herzfeld, Ute Christina" uniqKey="Herzfeld U" first="Ute Christina" last="Herzfeld">Ute Christina Herzfeld</name><affiliation><mods:affiliation>E-mail: uch@denali.uni-trier.de</mods:affiliation></affiliation><affiliation><mods:affiliation>Geomathematik, Fachbereich Geographie/Geowissenschaften, Universität Trier, D-54286, Trier, Germany</mods:affiliation></affiliation></author><author><name sortKey="Overbeck, Christoph" sort="Overbeck, Christoph" uniqKey="Overbeck C" first="Christoph" last="Overbeck">Christoph Overbeck</name><affiliation><mods:affiliation>Geomathematik, Fachbereich Geographie/Geowissenschaften, Universität Trier, D-54286, Trier, Germany</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:38D8D615E2B53196C0C5A9AE21A65C04551168D2</idno><date when="1999" year="1999">1999</date><idno type="doi">10.1016/S0098-3004(99)00062-X</idno><idno type="url">https://api.istex.fr/document/38D8D615E2B53196C0C5A9AE21A65C04551168D2/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">001995</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001995</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a">Analysis and simulation of scale-dependent fractal surfaces with application to seafloor morphology</title><author><name sortKey="Herzfeld, Ute Christina" sort="Herzfeld, Ute Christina" uniqKey="Herzfeld U" first="Ute Christina" last="Herzfeld">Ute Christina Herzfeld</name><affiliation><mods:affiliation>E-mail: uch@denali.uni-trier.de</mods:affiliation></affiliation><affiliation><mods:affiliation>Geomathematik, Fachbereich Geographie/Geowissenschaften, Universität Trier, D-54286, Trier, Germany</mods:affiliation></affiliation></author><author><name sortKey="Overbeck, Christoph" sort="Overbeck, Christoph" uniqKey="Overbeck C" first="Christoph" last="Overbeck">Christoph Overbeck</name><affiliation><mods:affiliation>Geomathematik, Fachbereich Geographie/Geowissenschaften, Universität Trier, D-54286, Trier, Germany</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">Computers and Geosciences</title><title level="j" type="abbrev">CAGEO</title><idno type="ISSN">0098-3004</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="1999">1999</date><biblScope unit="volume">25</biblScope><biblScope unit="issue">9</biblScope><biblScope unit="page" from="979">979</biblScope><biblScope unit="page" to="1007">1007</biblScope></imprint><idno type="ISSN">0098-3004</idno></series><idno type="istex">38D8D615E2B53196C0C5A9AE21A65C04551168D2</idno><idno type="DOI">10.1016/S0098-3004(99)00062-X</idno><idno type="PII">S0098-3004(99)00062-X</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0098-3004</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Common theories on fractal surfaces as observed in geology assume a universality law, in most instances the simplest type of universality, which is self-similarity or self-affinity; in the case of multifractals, another well-known special type of fractals, a more complex form of scale invariance is described using one generating process. The assumption of a scale-invariant universality law, however, implies that a geological object was created by a single underlying process, which is clearly in contradiction to geological knowledge and measurable observations. The processes of crust generation, seafloor spreading, sediment deposition, and erosion work at different specific homogeneity ranges of scale, and such scale dependency is observed in many data sets collected for topographic surfaces. This necessitates the design of methods and algorithms for analysis and simulation of fractal surfaces with scale-dependent spatial characteristica. A suite of algorithms and programs for this purpose is compiled and presented in this paper. Numerical algorithms build on geostatistics, Fourier theory, and some “fractal” methods. The approach presented here uses a dimension parameter for characterization of roughness and an anisotropy factor, given with respect to a principal direction, to capture anisotropic properties. Analytical methods are an isarithm method, a variogram method, a Fourier method, and an isarithm-type Fourier method for estimation of a dimension parameter. In applications to bathymetric data from the western flank of the mid-Atlantic ridge, the variogram method is found most accurate and produces results consistent with geological observations. Interpolation, unconditional simulation and conditional simulation algorithms based on Fourier methods and Fractional Brownian Surfaces localized in scale are combined to construct and merge grids of different scales with specific roughness and anisotropy characteristics, resultant in surfaces with scale-dependent properties which almost exactly reproduce those observed from geophysical data in the seafloor case studies. The scale-dependent simulation methods serve to (1) extrapolate in scale beyond the observed resolution, if roughness and anisotropic properties are known from another area with similar characteristics, and thus provide information on subscale properties for surveys with instrumentation of lower resolution, and (2) extrapolate and simulate in space, if an area has only partly been covered by a survey.</div></front></TEI><istex><corpusName>elsevier</corpusName><author><json:item><name>Ute Christina Herzfeld</name><affiliations><json:string>E-mail: uch@denali.uni-trier.de</json:string><json:string>Geomathematik, Fachbereich Geographie/Geowissenschaften, Universität Trier, D-54286, Trier, Germany</json:string></affiliations></json:item><json:item><name>Christoph Overbeck</name><affiliations><json:string>Geomathematik, Fachbereich Geographie/Geowissenschaften, Universität Trier, D-54286, Trier, Germany</json:string></affiliations></json:item></author><subject><json:item><lang><json:string>eng</json:string></lang><value>Surface roughness</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Anisotropy</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Subscale information</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Extrapolation</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Variogram method</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Isarithm method</value></json:item><json:item><lang><json:string>eng</json:string></lang><value>Fourier method</value></json:item></subject><language><json:string>eng</json:string></language><originalGenre><json:string>Full-length article</json:string></originalGenre><abstract>Common theories on fractal surfaces as observed in geology assume a universality law, in most instances the simplest type of universality, which is self-similarity or self-affinity; in the case of multifractals, another well-known special type of fractals, a more complex form of scale invariance is described using one generating process. The assumption of a scale-invariant universality law, however, implies that a geological object was created by a single underlying process, which is clearly in contradiction to geological knowledge and measurable observations. The processes of crust generation, seafloor spreading, sediment deposition, and erosion work at different specific homogeneity ranges of scale, and such scale dependency is observed in many data sets collected for topographic surfaces. This necessitates the design of methods and algorithms for analysis and simulation of fractal surfaces with scale-dependent spatial characteristica. A suite of algorithms and programs for this purpose is compiled and presented in this paper. Numerical algorithms build on geostatistics, Fourier theory, and some “fractal” methods. The approach presented here uses a dimension parameter for characterization of roughness and an anisotropy factor, given with respect to a principal direction, to capture anisotropic properties. Analytical methods are an isarithm method, a variogram method, a Fourier method, and an isarithm-type Fourier method for estimation of a dimension parameter. In applications to bathymetric data from the western flank of the mid-Atlantic ridge, the variogram method is found most accurate and produces results consistent with geological observations. Interpolation, unconditional simulation and conditional simulation algorithms based on Fourier methods and Fractional Brownian Surfaces localized in scale are combined to construct and merge grids of different scales with specific roughness and anisotropy characteristics, resultant in surfaces with scale-dependent properties which almost exactly reproduce those observed from geophysical data in the seafloor case studies. The scale-dependent simulation methods serve to (1) extrapolate in scale beyond the observed resolution, if roughness and anisotropic properties are known from another area with similar characteristics, and thus provide information on subscale properties for surveys with instrumentation of lower resolution, and (2) extrapolate and simulate in space, if an area has only partly been covered by a survey.</abstract><qualityIndicators><score>8</score><pdfVersion>1.2</pdfVersion><pdfPageSize>595 x 792 pts</pdfPageSize><refBibsNative>true</refBibsNative><keywordCount>7</keywordCount><abstractCharCount>2508</abstractCharCount><pdfWordCount>12528</pdfWordCount><pdfCharCount>76439</pdfCharCount><pdfPageCount>29</pdfPageCount><abstractWordCount>350</abstractWordCount></qualityIndicators><title>Analysis and simulation of scale-dependent fractal surfaces with application to seafloor morphology</title><pii><json:string>S0098-3004(99)00062-X</json:string></pii><refBibs><json:item><author><json:item><name>F.P. 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Voss</name></json:item></author><host><pages><last>70</last><first>21</first></pages><author></author><title>The Science of Fractal Images</title></host><title>Fractals in nature: From characterization to simulation</title></json:item></refBibs><genre><json:string>research-article</json:string></genre><host><volume>25</volume><pii><json:string>S0098-3004(00)X0048-9</json:string></pii><pages><last>1007</last><first>979</first></pages><issn><json:string>0098-3004</json:string></issn><issue>9</issue><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><title>Computers and Geosciences</title><publicationDate>1999</publicationDate></host><publicationDate>1999</publicationDate><copyrightDate>1999</copyrightDate><doi><json:string>10.1016/S0098-3004(99)00062-X</json:string></doi><id>38D8D615E2B53196C0C5A9AE21A65C04551168D2</id><score>0.8267195</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/38D8D615E2B53196C0C5A9AE21A65C04551168D2/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/38D8D615E2B53196C0C5A9AE21A65C04551168D2/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/38D8D615E2B53196C0C5A9AE21A65C04551168D2/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a">Analysis and simulation of scale-dependent fractal surfaces with application to seafloor morphology</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>ELSEVIER</publisher><availability><p>©1999 Elsevier Science Ltd</p></availability><date>1999</date></publicationStmt><notesStmt><note type="content">Fig. 1: Shaded-relief map of 600 km×250 km survey area on western flank of Mid-Atlantic Ridge at 25°25′N to 27°10′N with study areas indicated. Eastmost track captures ridge. UTM coordinates with respect to middle meridian 45°W (Zone 23). Area A1 is 80 km×80 km, UTM 155,300-235,300E/2,908,000–2,988,000N/depth 3085–5331 m. Area B is 45 km E–W by 80 km N–S, UTM 111,800–156,800E/2,884,630–2,964,630N/depth 3150–5249 m. Area A2 is a 10 km×10 km subarea of A1, 160,000–170,000E/2,940,000–2,950,000N/depth 3451–4702 m.</note><note type="content">Fig. 2: Data distribution along ship’s tracks in study areas. All UTM coordinates with respect to middle meridian 45°W (Zone 23). (A) Area A1, 80 km×80 km, UTM 155,300–235,300E/2,908,000–2,988,000N/depth 3085–5331 m; (B) Area B, 45 km E–W by 80 km N–S, UTM 111,800–156,800E/2,884,630–2,964,630N/depth 3150–5249 m (adjacent to and slightly overlapping area A1); (C) Area A2, subarea of A1, 10 km×10 km, 160,000–170,000E/2,940,000–2,950,000N/depth 3451–4702 m.</note><note type="content">Fig. 3: Two unconditional simulations of 10 km×10 km area, run with the same parameters (characteristic of area A2): one scale break, dimension =2.3 below scale break, dimension =2.4 above scale break, anisotropy factor 1.6 with respect to principal direction 105°.</note><note type="content">Fig. 4: Conditional simulations of area A2, 10 km×10 km, UTM 160,000–170,000E/2,940,000–2,950,000N, resolution 200 m, anisotropy factor 1.6 with respect to principal direction 105°. (A) SIMA2-1 — interpolation using Shepard method for all scales (200 m–10000 m); (B) SIMA2-2 — interpolation using Shepard method above 1000 m resolution, simulation using four-point method and dimension 2.30 below 1000 m resolution; (C) SIMA2-3 — interpolation using Shepard method above 1000 m resolution, simulation using Shepard method and dimension 2.30 below 1000 m resolution; (D) SIMA2-4 — interpolation using Shepard method above 5000 m resolution, simulation using Shepard method and dimension 2.41 between 1000 m and 5000 m resolution, simulation using Shepard method and dimension 2.30 between 200 m and 1000 m resolution.</note><note type="content">Fig. 5: Conditional simulations of area A1, 80 km×80 km, UTM 155,300–235,300E/2,908,000–2,988,000N (surveyed with almost complete coverage), resolution 500 m, anisotropy factor 1.6 with respect to principal direction 105°. (A) SIMA1-1 — interpolation using Shepard method for all scales (500 m–80,000 m); (B) SIMA1-2 — interpolation using Shepard method above 2000 m resolution, simulation using four-point method and dimension 2.32 below 2000 m resolution; (C) SIMA1-3 — interpolation using Shepard method above 2000 m resolution, simulation using Shepard method and dimension 2.32 below 2000 m resolution; (D) SIMA1-4 — interpolation using Shepard method above 10000 m resolution, simulation using Shepard method and dimension 2.55 between 2000 m and 10,000 m resolution, simulation using Shepard method and dimension 2.32 between 500 m and 2000 m resolution.</note><note type="content">Fig. 6: Conditional simulations of area B, UTM 111,800–156,800E/2,884,630–2,964,630N (no data collected in northwestern part of map), resolution 500 m, anisotropy factor 1.6 with respect to principal direction 90°. Notice that area is 45 km E–W by 80 km N–S (Fig. 2(B)), but displayed grids appear square. (A) SIMB-1 — interpolation using Shepard method for all scales (500 m–80,000 m); (B) SIMB-2 — interpolation using Shepard method above 2000 m resolution, simulation using four-point method and dimension 2.29 below 2000 m resolution; (C) SIMB-3 — interpolation using Shepard method above 2000 m resolution, simulation using Shepard method and dimension 2.29 below 2000 m resolution; (D) SIMB-4 — interpolation using Shepard method above 10,000 m resolution, simulation using Shepard method and dimension 2.53 between 2000 m and 10,000 m resolution, simulation using Shepard method and dimension 2.29 between 500 m and 2000 m resolution.</note><note type="content">Table 1: Correspondence of mathematical and geological directions</note><note type="content">Table 2: Total box dimensions for area A1 (80 km×80 km)a</note><note type="content">Table 3: Directional box dimensions at 1000 m scale for area A1a</note><note type="content">Table 4: Total box dimensions for area A2 (10 km×10 km)a</note><note type="content">Table 5: Anisotropy numbers</note><note type="content">Table 6: Total box dimensions of area A2 (original) and two unconditional simulations (UCS-1, Fig. 3(A), and UCS-2, Fig. 3(B)) calculated with the residual variogram method V2 at different scales</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a">Analysis and simulation of scale-dependent fractal surfaces with application to seafloor morphology</title><author xml:id="author-1"><persName><forename type="first">Ute Christina</forename><surname>Herzfeld</surname></persName><email>uch@denali.uni-trier.de</email><note type="correspondence"><p>Corresponding author at: Geomathematik, Fachbereich Geopgraphie/Geowissenschaften, Universität Trier, D-54826 Trier, Germany</p></note><affiliation>Geomathematik, Fachbereich Geographie/Geowissenschaften, Universität Trier, D-54286, Trier, Germany</affiliation></author><author xml:id="author-2"><persName><forename type="first">Christoph</forename><surname>Overbeck</surname></persName><affiliation>Geomathematik, Fachbereich Geographie/Geowissenschaften, Universität Trier, D-54286, Trier, Germany</affiliation></author></analytic><monogr><title level="j">Computers and Geosciences</title><title level="j" type="abbrev">CAGEO</title><idno type="pISSN">0098-3004</idno><idno type="PII">S0098-3004(00)X0048-9</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="1999"></date><biblScope unit="volume">25</biblScope><biblScope unit="issue">9</biblScope><biblScope unit="page" from="979">979</biblScope><biblScope unit="page" to="1007">1007</biblScope></imprint></monogr><idno type="istex">38D8D615E2B53196C0C5A9AE21A65C04551168D2</idno><idno type="DOI">10.1016/S0098-3004(99)00062-X</idno><idno type="PII">S0098-3004(99)00062-X</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1999</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>Common theories on fractal surfaces as observed in geology assume a universality law, in most instances the simplest type of universality, which is self-similarity or self-affinity; in the case of multifractals, another well-known special type of fractals, a more complex form of scale invariance is described using one generating process. The assumption of a scale-invariant universality law, however, implies that a geological object was created by a single underlying process, which is clearly in contradiction to geological knowledge and measurable observations. The processes of crust generation, seafloor spreading, sediment deposition, and erosion work at different specific homogeneity ranges of scale, and such scale dependency is observed in many data sets collected for topographic surfaces. This necessitates the design of methods and algorithms for analysis and simulation of fractal surfaces with scale-dependent spatial characteristica. A suite of algorithms and programs for this purpose is compiled and presented in this paper. Numerical algorithms build on geostatistics, Fourier theory, and some “fractal” methods. The approach presented here uses a dimension parameter for characterization of roughness and an anisotropy factor, given with respect to a principal direction, to capture anisotropic properties. Analytical methods are an isarithm method, a variogram method, a Fourier method, and an isarithm-type Fourier method for estimation of a dimension parameter. In applications to bathymetric data from the western flank of the mid-Atlantic ridge, the variogram method is found most accurate and produces results consistent with geological observations. Interpolation, unconditional simulation and conditional simulation algorithms based on Fourier methods and Fractional Brownian Surfaces localized in scale are combined to construct and merge grids of different scales with specific roughness and anisotropy characteristics, resultant in surfaces with scale-dependent properties which almost exactly reproduce those observed from geophysical data in the seafloor case studies. The scale-dependent simulation methods serve to (1) extrapolate in scale beyond the observed resolution, if roughness and anisotropic properties are known from another area with similar characteristics, and thus provide information on subscale properties for surveys with instrumentation of lower resolution, and (2) extrapolate and simulate in space, if an area has only partly been covered by a survey.</p></abstract><textClass><keywords scheme="keyword"><list><head>Keywords</head><item><term>Surface roughness</term></item><item><term>Anisotropy</term></item><item><term>Subscale information</term></item><item><term>Extrapolation</term></item><item><term>Variogram method</term></item><item><term>Isarithm method</term></item><item><term>Fourier method</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1999-02-12">Modified</change><change when="1999">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/38D8D615E2B53196C0C5A9AE21A65C04551168D2/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="gr3" NDATA="IMAGE" name="gr3"></istex:entity><istex:entity SYSTEM="gr4" NDATA="IMAGE" name="gr4"></istex:entity><istex:entity SYSTEM="gr6" NDATA="IMAGE" name="gr6"></istex:entity><istex:entity SYSTEM="fx1" NDATA="IMAGE" name="fx1"></istex:entity><istex:entity SYSTEM="fx2" NDATA="IMAGE" name="fx2"></istex:entity><istex:entity SYSTEM="fx3" NDATA="IMAGE" name="fx3"></istex:entity><istex:entity SYSTEM="fx4" NDATA="IMAGE" name="fx4"></istex:entity><istex:entity SYSTEM="fx5" NDATA="IMAGE" name="fx5"></istex:entity><istex:entity SYSTEM="fx6" NDATA="IMAGE" name="fx6"></istex:entity><istex:entity SYSTEM="gr2a" NDATA="IMAGE" name="gr2a"></istex:entity><istex:entity SYSTEM="gr2b" NDATA="IMAGE" name="gr2b"></istex:entity><istex:entity SYSTEM="gr2c" NDATA="IMAGE" name="gr2c"></istex:entity><istex:entity SYSTEM="gr5ab" NDATA="IMAGE" name="gr5ab"></istex:entity><istex:entity SYSTEM="gr5cd" NDATA="IMAGE" name="gr5cd"></istex:entity></istex:docType><istex:document><converted-article version="4.5.2" docsubtype="fla"><item-info><jid>CAGEO</jid><aid>759</aid><ce:pii>S0098-3004(99)00062-X</ce:pii><ce:doi>10.1016/S0098-3004(99)00062-X</ce:doi><ce:copyright type="full-transfer" year="1999">Elsevier Science Ltd</ce:copyright></item-info><head><ce:title>Analysis and simulation of scale-dependent fractal surfaces with application to seafloor morphology</ce:title><ce:author-group><ce:author><ce:given-name>Ute Christina</ce:given-name><ce:surname>Herzfeld</ce:surname><ce:cross-ref refid="AFF1"><ce:sup>a</ce:sup></ce:cross-ref><ce:cross-ref refid="AFF2"><ce:sup>b</ce:sup></ce:cross-ref><ce:cross-ref refid="CORR1">*</ce:cross-ref><ce:e-address>uch@denali.uni-trier.de</ce:e-address></ce:author><ce:author><ce:given-name>Christoph</ce:given-name><ce:surname>Overbeck</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>Geomathematik, Fachbereich Geographie/Geowissenschaften, Universität Trier, D-54286, Trier, Germany</ce:textfn></ce:affiliation><ce:affiliation id="AFF2"><ce:label>b</ce:label><ce:textfn>Institute of Arctic and Alpine Research, University of Colorado, Boulder, CO, 80309-0450, USA</ce:textfn></ce:affiliation><ce:correspondence id="CORR1"><ce:label>*</ce:label><ce:text>Corresponding author at: Geomathematik, Fachbereich Geopgraphie/Geowissenschaften, Universität Trier, D-54826 Trier, Germany</ce:text></ce:correspondence></ce:author-group><ce:date-received day="7" month="9" year="1998"></ce:date-received><ce:date-revised day="12" month="2" year="1999"></ce:date-revised><ce:date-accepted day="18" month="3" year="1999"></ce:date-accepted><ce:abstract><ce:section-title>Abstract</ce:section-title><ce:abstract-sec><ce:simple-para>Common theories on fractal surfaces as observed in geology assume a universality law, in most instances the simplest type of universality, which is self-similarity or self-affinity; in the case of multifractals, another well-known special type of fractals, a more complex form of scale invariance is described using one generating process. The assumption of a scale-invariant universality law, however, implies that a geological object was created by a single underlying process, which is clearly in contradiction to geological knowledge and measurable observations. The processes of crust generation, seafloor spreading, sediment deposition, and erosion work at different specific homogeneity ranges of scale, and such scale dependency is observed in many data sets collected for topographic surfaces. This necessitates the design of methods and algorithms for analysis and simulation of fractal surfaces with scale-dependent spatial characteristica. A suite of algorithms and programs for this purpose is compiled and presented in this paper. Numerical algorithms build on geostatistics, Fourier theory, and some “fractal” methods. The approach presented here uses a dimension parameter for characterization of roughness and an anisotropy factor, given with respect to a principal direction, to capture anisotropic properties. Analytical methods are an isarithm method, a variogram method, a Fourier method, and an isarithm-type Fourier method for estimation of a dimension parameter. In applications to bathymetric data from the western flank of the mid-Atlantic ridge, the variogram method is found most accurate and produces results consistent with geological observations. Interpolation, unconditional simulation and conditional simulation algorithms based on Fourier methods and Fractional Brownian Surfaces localized in scale are combined to construct and merge grids of different scales with specific roughness and anisotropy characteristics, resultant in surfaces with scale-dependent properties which almost exactly reproduce those observed from geophysical data in the seafloor case studies. The scale-dependent simulation methods serve to <ce:cross-ref refid="FD1">(1)</ce:cross-ref> extrapolate in scale beyond the observed resolution, if roughness and anisotropic properties are known from another area with similar characteristics, and thus provide information on subscale properties for surveys with instrumentation of lower resolution, and <ce:cross-ref refid="FD2">(2)</ce:cross-ref> extrapolate and simulate in space, if an area has only partly been covered by a survey.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:keywords class="keyword"><ce:section-title>Keywords</ce:section-title><ce:keyword><ce:text>Surface roughness</ce:text></ce:keyword><ce:keyword><ce:text>Anisotropy</ce:text></ce:keyword><ce:keyword><ce:text>Subscale information</ce:text></ce:keyword><ce:keyword><ce:text>Extrapolation</ce:text></ce:keyword><ce:keyword><ce:text>Variogram method</ce:text></ce:keyword><ce:keyword><ce:text>Isarithm method</ce:text></ce:keyword><ce:keyword><ce:text>Fourier method</ce:text></ce:keyword></ce:keywords></head></converted-article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo><title>Analysis and simulation of scale-dependent fractal surfaces with application to seafloor morphology</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Analysis and simulation of scale-dependent fractal surfaces with application to seafloor morphology</title></titleInfo><name type="personal"><namePart type="given">Ute Christina</namePart><namePart type="family">Herzfeld</namePart><affiliation>E-mail: uch@denali.uni-trier.de</affiliation><affiliation>Geomathematik, Fachbereich Geographie/Geowissenschaften, Universität Trier, D-54286, Trier, Germany</affiliation><description>Corresponding author at: Geomathematik, Fachbereich Geopgraphie/Geowissenschaften, Universität Trier, D-54826 Trier, Germany</description><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Christoph</namePart><namePart type="family">Overbeck</namePart><affiliation>Geomathematik, Fachbereich Geographie/Geowissenschaften, Universität Trier, D-54286, Trier, 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">1999</dateIssued><dateModified encoding="w3cdtf">1999-02-12</dateModified><copyrightDate encoding="w3cdtf">1999</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">Common theories on fractal surfaces as observed in geology assume a universality law, in most instances the simplest type of universality, which is self-similarity or self-affinity; in the case of multifractals, another well-known special type of fractals, a more complex form of scale invariance is described using one generating process. The assumption of a scale-invariant universality law, however, implies that a geological object was created by a single underlying process, which is clearly in contradiction to geological knowledge and measurable observations. The processes of crust generation, seafloor spreading, sediment deposition, and erosion work at different specific homogeneity ranges of scale, and such scale dependency is observed in many data sets collected for topographic surfaces. This necessitates the design of methods and algorithms for analysis and simulation of fractal surfaces with scale-dependent spatial characteristica. A suite of algorithms and programs for this purpose is compiled and presented in this paper. Numerical algorithms build on geostatistics, Fourier theory, and some “fractal” methods. The approach presented here uses a dimension parameter for characterization of roughness and an anisotropy factor, given with respect to a principal direction, to capture anisotropic properties. Analytical methods are an isarithm method, a variogram method, a Fourier method, and an isarithm-type Fourier method for estimation of a dimension parameter. In applications to bathymetric data from the western flank of the mid-Atlantic ridge, the variogram method is found most accurate and produces results consistent with geological observations. Interpolation, unconditional simulation and conditional simulation algorithms based on Fourier methods and Fractional Brownian Surfaces localized in scale are combined to construct and merge grids of different scales with specific roughness and anisotropy characteristics, resultant in surfaces with scale-dependent properties which almost exactly reproduce those observed from geophysical data in the seafloor case studies. The scale-dependent simulation methods serve to (1) extrapolate in scale beyond the observed resolution, if roughness and anisotropic properties are known from another area with similar characteristics, and thus provide information on subscale properties for surveys with instrumentation of lower resolution, and (2) extrapolate and simulate in space, if an area has only partly been covered by a survey.</abstract><note type="content">Fig. 1: Shaded-relief map of 600 km×250 km survey area on western flank of Mid-Atlantic Ridge at 25°25′N to 27°10′N with study areas indicated. Eastmost track captures ridge. UTM coordinates with respect to middle meridian 45°W (Zone 23). Area A1 is 80 km×80 km, UTM 155,300-235,300E/2,908,000–2,988,000N/depth 3085–5331 m. Area B is 45 km E–W by 80 km N–S, UTM 111,800–156,800E/2,884,630–2,964,630N/depth 3150–5249 m. Area A2 is a 10 km×10 km subarea of A1, 160,000–170,000E/2,940,000–2,950,000N/depth 3451–4702 m.</note><note type="content">Fig. 2: Data distribution along ship’s tracks in study areas. All UTM coordinates with respect to middle meridian 45°W (Zone 23). (A) Area A1, 80 km×80 km, UTM 155,300–235,300E/2,908,000–2,988,000N/depth 3085–5331 m; (B) Area B, 45 km E–W by 80 km N–S, UTM 111,800–156,800E/2,884,630–2,964,630N/depth 3150–5249 m (adjacent to and slightly overlapping area A1); (C) Area A2, subarea of A1, 10 km×10 km, 160,000–170,000E/2,940,000–2,950,000N/depth 3451–4702 m.</note><note type="content">Fig. 3: Two unconditional simulations of 10 km×10 km area, run with the same parameters (characteristic of area A2): one scale break, dimension =2.3 below scale break, dimension =2.4 above scale break, anisotropy factor 1.6 with respect to principal direction 105°.</note><note type="content">Fig. 4: Conditional simulations of area A2, 10 km×10 km, UTM 160,000–170,000E/2,940,000–2,950,000N, resolution 200 m, anisotropy factor 1.6 with respect to principal direction 105°. (A) SIMA2-1 — interpolation using Shepard method for all scales (200 m–10000 m); (B) SIMA2-2 — interpolation using Shepard method above 1000 m resolution, simulation using four-point method and dimension 2.30 below 1000 m resolution; (C) SIMA2-3 — interpolation using Shepard method above 1000 m resolution, simulation using Shepard method and dimension 2.30 below 1000 m resolution; (D) SIMA2-4 — interpolation using Shepard method above 5000 m resolution, simulation using Shepard method and dimension 2.41 between 1000 m and 5000 m resolution, simulation using Shepard method and dimension 2.30 between 200 m and 1000 m resolution.</note><note type="content">Fig. 5: Conditional simulations of area A1, 80 km×80 km, UTM 155,300–235,300E/2,908,000–2,988,000N (surveyed with almost complete coverage), resolution 500 m, anisotropy factor 1.6 with respect to principal direction 105°. (A) SIMA1-1 — interpolation using Shepard method for all scales (500 m–80,000 m); (B) SIMA1-2 — interpolation using Shepard method above 2000 m resolution, simulation using four-point method and dimension 2.32 below 2000 m resolution; (C) SIMA1-3 — interpolation using Shepard method above 2000 m resolution, simulation using Shepard method and dimension 2.32 below 2000 m resolution; (D) SIMA1-4 — interpolation using Shepard method above 10000 m resolution, simulation using Shepard method and dimension 2.55 between 2000 m and 10,000 m resolution, simulation using Shepard method and dimension 2.32 between 500 m and 2000 m resolution.</note><note type="content">Fig. 6: Conditional simulations of area B, UTM 111,800–156,800E/2,884,630–2,964,630N (no data collected in northwestern part of map), resolution 500 m, anisotropy factor 1.6 with respect to principal direction 90°. Notice that area is 45 km E–W by 80 km N–S (Fig. 2(B)), but displayed grids appear square. (A) SIMB-1 — interpolation using Shepard method for all scales (500 m–80,000 m); (B) SIMB-2 — interpolation using Shepard method above 2000 m resolution, simulation using four-point method and dimension 2.29 below 2000 m resolution; (C) SIMB-3 — interpolation using Shepard method above 2000 m resolution, simulation using Shepard method and dimension 2.29 below 2000 m resolution; (D) SIMB-4 — interpolation using Shepard method above 10,000 m resolution, simulation using Shepard method and dimension 2.53 between 2000 m and 10,000 m resolution, simulation using Shepard method and dimension 2.29 between 500 m and 2000 m resolution.</note><note type="content">Table 1: Correspondence of mathematical and geological directions</note><note type="content">Table 2: Total box dimensions for area A1 (80 km×80 km)a</note><note type="content">Table 3: Directional box dimensions at 1000 m scale for area A1a</note><note type="content">Table 4: Total box dimensions for area A2 (10 km×10 km)a</note><note type="content">Table 5: Anisotropy numbers</note><note type="content">Table 6: Total box dimensions of area A2 (original) and two unconditional simulations (UCS-1, Fig. 3(A), and UCS-2, Fig. 3(B)) calculated with the residual variogram method V2 at different scales</note><subject><genre>Keywords</genre><topic>Surface roughness</topic><topic>Anisotropy</topic><topic>Subscale information</topic><topic>Extrapolation</topic><topic>Variogram method</topic><topic>Isarithm method</topic><topic>Fourier method</topic></subject><relatedItem type="host"><titleInfo><title>Computers and Geosciences</title></titleInfo><titleInfo type="abbreviated"><title>CAGEO</title></titleInfo><genre type="journal">journal</genre><originInfo><dateIssued encoding="w3cdtf">19991115</dateIssued></originInfo><identifier type="ISSN">0098-3004</identifier><identifier type="PII">S0098-3004(00)X0048-9</identifier><part><date>19991115</date><detail type="volume"><number>25</number><caption>vol.</caption></detail><detail type="issue"><number>9</number><caption>no.</caption></detail><extent unit="issue pages"><start>947</start><end>1100</end></extent><extent unit="pages"><start>979</start><end>1007</end></extent></part></relatedItem><identifier type="istex">38D8D615E2B53196C0C5A9AE21A65C04551168D2</identifier><identifier type="DOI">10.1016/S0098-3004(99)00062-X</identifier><identifier type="PII">S0098-3004(99)00062-X</identifier><accessCondition type="use and reproduction" contentType="copyright">©1999 Elsevier Science Ltd</accessCondition><recordInfo><recordContentSource>ELSEVIER</recordContentSource><recordOrigin>Elsevier Science Ltd, ©1999</recordOrigin></recordInfo></mods></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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