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# Advanced GIS EES 6513

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This 86 page Class Notes was uploaded by Viviane Runolfsdottir on Thursday October 29, 2015. The Class Notes belongs to EES 6513 at University of Texas at San Antonio taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/231365/ees-6513-university-of-texas-at-san-antonio in Earth Sciences at University of Texas at San Antonio.

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Date Created: 10/29/15

Spatial Analysis Raster data analysis Lecture 45 211182007 Recap raster data ESRI grid image format geodatabase raster dataset and raster catalog Column and row Cell Cell value Cell size Mixed pixels Raster data structure Cell by cell DEM and satellite image 39 BSQ bandinterleaved easy to retrieve spatial information 39 BIP pixelinterleaved easy to retrieve spectral information 39 BIL rowinterleaved fairly for both spatial and spectral Runlength encoding Quadtree Wavelet MrSID ECW JPEGZOOO Raster Data Encoding Mixed Fier Problems Splithallle auraslit Spatial Finalyst Tools Conditional Density Distance Extraction Generalization Groundwater Hydrology Interpolation Local Map algebra Math lquot391IIltiyariate Neighborhood Dyerlay Raster Creation Retlass SurFate Eonal Details quot Spatial Analyst extension l Spatial analyst quot Lakeii I M 031 Of the Qistanca Ir functions in the Dansitvu S al Analyst lntrpnlate c Raster P Surface FH39IE 3939sis Conmur H TOOIS are also ICell Statistics Slope lnteg rated Into aighbcnrhcxnd Statistics ESPECt the Spatial gm Statistics illshacle Ana Eeclassifw H Eiewshed eXte n S O n Raster Calculator gluFi H anvert p gpticins Types of spatial analysis 1 Mapping distance Straight line or Euclidean distance Cost Weighted Distance l Mapping density Interpolating to raster Inverse Distance Weighted Spline Kriging Performing surface analysis Contour Slop Aspect Hillshade Viewshed cutfill Spatial analyst quot39 l LBJJETII Qistance Density interpolate to Raster Surface analysis Cell Statistics eighborhood Statistics zonal Statistics eclassify Raster Calculator Qonvert thions Qontour Slope aspect Hillshade Eiewshecl cuthilI Types of spatial analysis 2 Statistics Cell statistics Neighborhood statistics Zonal statistics Reclassification v Raster Calculator map algebra Conversion of vector and raster 39 39Spatial analyst V Layer I Qistance Density interpolate to Raster Surface analysis Cell Statistics eighborhood Statistics Zonal Statistics Beclassify Raster Calculator gonvert thions gontour Slope aspect illshade wewshed gut Fill 1 Mapping distance The Straight Line Distance function measures the straight line distance from each cell to the closest source Allocation function allows you to identify which cells belong to which source based on straight line distance The Cost Weighted Distance function modifies the Straight Line Distance by some other factor which is a cost to travel through any given cell For example it may be shorter to climb over the mountain to the destination but it is faster to walk around it Cost can be money time or preference The Distance and Direction raster datasets are normally created from Cost Weighted Distance function to serve as inputs to the pathfinding function the shortest or leastcost path Global Operations 0 Global operations 0 Operations that compute an output grid where the value of each outpm cell is potentially a function of all the cells in the input grid on Correspond to Tomiizn s focal operations in extended vicinities considered to the extreme case of focal operation where the neighborhood is potentially entire input grid 41 Example of operations or Giohai statistics and Data slicing o Eucliidean distance analysis a FindiDistance Euc istance Assign Proximity Euchliocation e Weighted distance measurement Cost istance or Leastcost path determination 39 05thth Stella may or Fem cm Global Statistical Examples 1 2 3 4 5 Globa 1M aximum B T B 9 391 2 3 4L 5 E Globa m l ll u l T a g 1 2 GlobalRange 3 4 5 a 7 G 0b HWHj 01ith Input Grid Straight lineIEuclidean Distance we Straight iLinei39Euclidean Distance 0 Measure the straight line distance from every cell to the nearest cell containing an abject of interest source a The distance is measured from cell center in cell center using the P hegarean theorem 1 Snurce 4 Veemr features point line pelygen Valid grid cells Men null cells 1 Output grids in Centiniuous distance grid 4 Diremien and allocation wilds Optimal Mild ns is Er Fart E39lJ Euclidean Distance Based Analysis 5 Straight LinellEuclidean Distance Measure the straight line distance tram every cell to the nearest cell containing an abject uf interest lseuree The distance is mesured from cell center to cell center using the P hagerean theerem Source Vector features mint line pelygon 1llaliizl grid cells Nomnull cells Output grids Continuum distance gnu iv Directien and allocation grids Optional Source Grid 1 El ELEI D E 10 2 I 35 15 1C D LEI 2E MI 22 1 k 1 I 11 22 32 2 D 22 1 B 22 2 5 35 113 134 2 2 32 3 E 4 2 EM 1C 2 l 3C 4 E El Distance Guild The straight J39r39ne distance In the nearest town from every Inca Iran Proximity 1 Euclidean Allocation In Assign Proximity lEucAllncatium r Allocate each cell to the nearest suurce and assign it the lDlValue all the nearest snurce 1 The nearest smurce is determined by the Euclidean distanee d It is the GR equivalent of Thlessen polygon and a way of assigning service area Output Grid Euclidean Allocation swimmers Er Fan 14 WhyuCthe Allocation function locati 0 I1 Use the Allocation function to perform analyses such as Idcnljfying the customers scn39od by a series oiquot stems n i o 391 Finding out which llospilal is the closest Finding areas willh a shortage of fire hydrants Locating areas lhal are not served by a chain of superimrkcls Allows you to identify which cells belong to which source based on straight line distance function or cost weighted distance function line Cost weighted distance i 1 if 2 Allocating cells to sources Which areas a NoDala 1 2 are Shopping Centers are served by which town Cost weighted distance L c Find the least accumulative cost from each cell to the nearest cheapest source Cost can be money time or preference The lmmime that penfem met weighted ellis anee mapping are stutter tea the Stet ight Lime De ance neet39iema but in stead mtquot ealeuhit img the Etteil dismince nal line penimt tea annthew ther enmpute the Hemmullutilt e Wit f travelling frem meh eel tn the BENCH mum baseI39ll rm the ee l s d i39sttmee 1me etieh muree and the east tntn 39el thmugh it fnremtmp e irt iseasiler te 1tir39 llle thrmg h 1 mutellnaw than 1 Revamp lulu39thr use the eet Weighted Dietenee funetien C39mt weighted distmee mndeling i5 use Jl whenever mmrement is tin gengraphie thietn re each as animal mi gratinn etudiee er ennzmmer travel hehatrier Cmll weighted distance mar alert he used tn minimize ennetruetinn mete fer renting new Haida transmitsinn Iinee er pipelines The etmight line distanee between twn paints is net nm i ly the best In the graphie tn the left the ehmteet path WET the metmtain takes three heme The hunger path amend takes twn heme llftinie were a emit then the mute with the hanger distanee e hnnld he taken Hewever the aim maybe tn elimh WE l39tlE menntain Applyin g eeet weighted dietanee enables you tel epeeifyptefeteneee in your input data It ma fer eieimple tale lenge r tn travel mrer the quotmeuntain due tn eteep slepes an eteep shapes 5heuld he git en a higher east when nding a suitable pat h fmm A In El 39Eigh llg datath men djng tn petcult influenee The next step in predueing the east I39 StEl i5 tn add the reclassi ed damsets tegethe39r The simplest approach is to just add them mgether Hewever yen meyknew that some factors are more intpm39ttmt than nthem Fm instinee weiding 5th shapes may be twice 15 impelmm 35 the ht39nduse type 50 gnu mi g ht for emmplle give this dutuset an in uence nffi percent and the Ianduse detain an in uence Df 34 percent In nuke 1m percent The fellum39ng diagram Sherws the enneeptut pmeezss Ltmdm e DirEmitter The emitweighted dishinee meter tells yen the least timunmlatedl east elf getting from each eelll tn the ne39treat murem hurt it dmsn t tell which way tn gen ten get there The dirmtien raster pmvidee 1 mad n39um identifyin g the mute to take frrmt any ee l 1leung the leasteast path back to the nearest mume Goat Mbtghted Direth Sheena Coding The algnrithnt fur enntptlting the direction rusterussigns 1 etude tea eaeh eell that identi es which me of its neighboring cells is in the lammet path hack to the EGHBSI murm In the dirmrirm endin g diagram nhmre I represent everr eeil in the eastweighted dimmer raster Each eelll is teesigned n wahre representing the direetiean nf the nearest ehenpest enquot in the mute if the least ereHIr path If the neare muree Shortest path The shortest path function determines the path from a destination point to a source Once you have performed the cost weighted distance function creating distance and direction raster you can then computer the leastcost or shortest path from a chosen destination to your source point The purple line represents a cost distance where each input raster landuse and slope had the same influence The red line represents a cost distance where the slope input raster had a weight or influence of 66 percent 2 Mapping density o Spread point values over a surface The graph gives an example of density surface when added together the population values of all the cells equal to the sum of the population of the original point layer 3 spatial interpolation Will be covered in the geostatistic lectures 4 Surface analysis Contours Contours are polylines that connect points of equal value such as elevation temperature precipitation pollution or atmospheric pressure E Record NI 4 12 Shoch Septth Ft Input etevatton dataset Output contour dataset I 0 7 I 7 15 15 23 Slope I 2339 3149 39 47 47 55 55 63 63 70 70 73 Output slope daraser rise Degree of slope 9 Percent of slope e 100 run rise tan 6 run rise Degree of slope 30 45 76 Percent of slope 58 100 375 Aspect It is measured eleei mrise in degrees from iii due north m 3 60 again due nerth eeming full circle The value el eaeh eel i11311 aspect damsel indicates the directiml the cell s slope faces Flat slopes have 110 direction and are given a xalue of L Flat iii lllii 2 mm ITIZZ 2E Hi m Hillshade Setting a hypothetical light source and calculating the illumination values for each cell in relation to neighboring cells It can greatly enhance the visualization of a surface for analysis or graphical display 1 zimuth is the angular direction of the SL111 111eaSuredl 139om north in clockwise degrees E from t to 3 60 An azimuth of 90 is east The delault is 3 15 NW Altitude is the slope or angle of the illumination source above the horizon The units are in degrees from 0 011 the horizon to 90 degrees et erheadl The default is 45 degrees Viewshed Viewshed identifies the cells in an input raster that can be seen from one or more observation points or lines It is useful for finding the visibility For instance finding a well exposed places for communication towers The elevamn in the area ofth Green cells are visible from the hillshaded DEM as background obsawaHon paint observation point red cells are nor lemme 5 Statistics Cell statistics local function a statistic for each cell in an output raster is based on the values of each cell of multiple input rasters for instance to analyze the average crop yield over a 10year period Majority maximum mean median minimum minority range standard deviation sum variety If all the inputs are integer the output is integer If any of the inputs are floating point the output is floating point If any ofthe input is NODATA the output is NODATA DUTGHID VALUENODATA INGHID3 Expression MAHINGFIIDL IMGRID2 INGHIM DUTGFHID ll VALUENGDATA ING FWIIB Expm un HEANIIIMGFIIDL IHEHIIDE IMGHIDE Neighborhood statistics focal A statistic for each cell in an output raster is based on the values of cells within a specified neighborhood rectangle circle annulus and wedge Majority maximum mean median minimum A minority range standard Input processing raster 0WD raster deviation sum variety 0 Focal Minority the least frequent value in neighborhood Sum of 3 X 3 ce neighborhood 0 Foal Majority the most frequent value in neighborhood 0 Focal Minimum the minimum value in neighborhood 0 Focal Maximum the maximum value in neighborhood Focal Sum the total of all values in neighborhood 9 Focal Mean the average of all values in neighborhood Range max m In 0 Focal Variety the number of different values in neighborhood Moving Windows Useful for calculating local statistical functions or edge detection Kernel a set of constants applied with a function such as 19 being the mean of the center cell Other configurations may be used when dealing with diagonal or adjacent cells 39 Paul Bolstad GIS Fundamentals Examples of Focal Statistic Operations Use a 3 x 3 window Test l39linimum Nlaximum 0 l I 2 0 0 I U D D U I 2 1 2 D 0 0 0 D D U I 2 I I I D 0 II 0 0 D D 0 I D o o 1 o o n a n n u u n D 1 I II 0 0 U D D U E lV39Iean 1M HE 516 MS 216 0 118 18 5 9 4K 219 0 116 4J9 419 3 U D 1 5 219 215 11399 l1 0 1M 113 113 US ID 0 Zonal statistics computer statistics for each zone of a zone dataset based on the information in a value raster zone dataset can be feature or raster the value raster must be a raster j L39 L nilI W i mournl mm 0 Zone 9 Any unique areas I group of cells with the same value 9 In most GIS the value for zones must be integer o A zone may consist of noncontiguous cell or areas 9 Zones can not overlap Spatial Analysis Dire Fang Qiu Zonal Operations 0 Compute a new value for each location as a function of the existing values associated with a zone containing that location 0 Zone layer Define the zones their shape values and locations 9 Value layer contains the input values used in calculating output for each zone for some zonal operations 0 Output layer repeats the output value in each cell of the zone Zonal Statistics Example Zone Grid Zouallmean m 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 7 Value Grid 44 44 53 42 42 44 44 53 42 42 5 5 5 42 53 5 44 5 42 53 44 44 53 53 53 Output Grid Why use zonal statistics You might calculate the mean elevation for each forest zone or the number of accidents along each of the roads in a town Alternatively you might want to know how many di erent types of vegetation there are in each elevation zone variety The graphics below Show an example of the inputs and outputs from the Zonal Statistics function The variety of vegetation species per elevation zone is displayed in the output table and chart The most variety of species occurs at elevation levels of around 2500 meters mun nun Input zone dataset elevation zones amp fable Elevation range from 1547 to 3358 meters 3 Variety of quotVegetation lynesquotWilhin Zone 0 quotElevationquot 1999 2226 Zones 2452 2579 2905 3132 Output chart 6 Recalssification What is reclassification Reclassifying your data simply means replacing input cell values with new mitput cell Values The input data can be any supported raster format lfyou acid a multiband raster the rst band will be taken and usedl in the reclassification Why reclassify your data There are many reasons why you might want to reclassify your data Some of the most eonuuon reasons are To replace values based on new information To group certain values together a To reclassify values to a eonuuou scale for example for use in suitability analysis or for creating a cost raster for use in the Cost Weighted Distance function To set specific values to NoData or to set NoData cells to a value ecl Ty E inDJl raster WI Input raster I landusa V l Henlasx field I Landuse HEBlass he I Lmduse Set 39 39 39 Set values made if I Ellcl values I New values Clm iv I Uld values i N ew values I Clauiy I 39 Agriculuc Unique I 5 Um I HnDafa NnDaia AddEnlry MdEnny Delete Ertriesl Load I Save J 39 Changa missing value to Na aka Load I Save I Change missing values In N uDala Trath ran Ian I Elulput raster IltTemmrawgt Dulput rapter UKI I Cancel I Cancel I ekatialdataelevatinn Input rasier V Heclass eld IValue r i i i Uassifyj Unique I AddEntry I vI I b DeleteEntriesI Load I Save I 39 Change missing values to NuData Output raster F39IErnpmam v I U K I Cancel What can you do with the Raster Calculator 7 Raster calculator TllC Raster Calculator provides Vou39itlt a powerful tool for perforn ting multiple tasks You can tpc in t tt apA fgebra syntax to perform mathematical ealculations using operators and mctions set up selection queries or type in Spatial Analyst function syntax inputs can be grid datasets or raster layers sltape les coverages tables constants and numbers Mathematical operators and functions Operators and functions eralulate the expression only for input cells that are spatially coincident With the output eell Operators Functions Boolean operators There are four groups oftnatltmttatieal functions Logarithmic And or XOrI Not Aritltittetie Trigonontetrie and Powers Relational operators gt lt ltgt gt lt Exponential and logarithmic I I I I I I I Log Exp Sin Cos Sin Cos Tan Asin Acos Atan Sqrt Sqrt Pow Map Algebra 0 Map algebra oUses mathlike expressions to create new grid themes eSinglefaotor maps are treated as variables in the expression The expression can be made up of variables maps connected with operators and functions eProposed by C Dana Tomlin 19 Geographic Information System and Cartographic Modeling Englewood Cliffs Prentice Hall quotx Hist ost xquot J x fquot x 4quot an r 7 V I AverageCost HisCost MyCost I 2 sauna Maiysis or Fang on Operations on Raster Layers 0 Input variables in the form of rastervector layers 9 Operations are distinct and welidefined data processing activities including operators and functions 0 New raster layer output variable is created after applying an operations to one or more input rastervector layers 0 Complex anaiysis can be performed by a sequence of operations Spatial Analysis Dr Fang Qiu gt gt Classification on Operations o Tomlin 1983 1990 clefined and organized operations on raster data moclel as local focal and zonal according to the spatial scope of the operations Geographic Information System and Cartographic Modeling Englewood Cliffs Prentice Hall 1990 o Menton et al 1991 add global operations 0 Application operations are also defined to perform a specific application j Map Algebra Operations 9 Local operation 9 The value at each location in the output grid depends only on the value of corresponding cells at that location in the input grids 9 Focal and block operation 0 The value at each cell in the output grid depends the value of a specified neighborhood or block of that cell in the input grid 9 Zonal and regional operation 9 The value of each cell in the output grid depends on the value of all cells in the input grid that share the same zone or region A zone is a set of locations that have the same value and a region is a set of connected location having same value 9 Global operation 9 The value of each cell in the output grid potentially depends on all cells in the input grids Map Algebra Operations 0 Application Operations oDensity 9 Surface analysis oSurface generation oSiope oHydrological analysis OAspect oResolution altering oHiilshade Generalization oViewshed oGeometric transformation Cuwature oData Reclassification CDMOUF oDistance analysis 9 Straight line 0 Cost Weighted Allocation 9 Shortest Path NODATA in Map Algebra 4 NODATA 9 inadequate information or absence of data to characterize value in a cell ie value does not exist 9 NODAT A 0 Default is 9999 but user can set its value 9 Computation with NoData 9 Return NoData for the location no matter what 0 Ignore the NoData and computer with the available values General rule If any input cell is NODATA the output cell is NODATA EB 3 mmmm 2 7 3 7 1 1s1s13is 9 271u outgrd ingrd1 ingrdz Map Algebra Expression Components 9 Objects m ma 39 i I 9 o m u u 9a A o Grids shape files 39 1 0 TJ u m 0 Numbers Strings J JJ ILI quot quotquot T T l n t r M new I Auuueuonncn mummy d m L m o Operators 1 w w ll 9 add divid gt d w mm J greater than etc m I c Hz 0 FunctionsIRequests o Abs Algebraic Expression Focalmeanl DiffDem Abs deem SplineDem Zonalgeometryo 1 gtlt ff etc Ill 2lt x I f39f ff 39 Objects Operators Functions Requests Map Algebra Rules in Map Calculator O 9009 O 0 Evaluate from left to right no precedence for different operator Grid1Grid2Grid3Grid 1Grid2Grid3 l lee parenthesis to control the order of evaluation Grid1Grid2Grid3 Grids are in brackets Elevation Grids can have path names Eg c ldatagrcl1 if not listed in the View Functions must be followed by 0 with parameters inside Valid requests automatically create grid theme in the View 0 AspectElevation Parameters must be comma delimited and in parentheses o Hillshade Elevation 315 45 mil Function parameters may be other function nesting o llillshadeAspectElevationn35270245ni Strings are delimited with the double quote character Only requests that create a grid theme can be used in the Raster Calculator Spatial Analysis Dre Fang Qiu Local Operators o Arithmetic o Relational Raster Calculator ltgt lt gt lt gt Layers newdem usdem l Elk o Boolean AND 0R XOR NOT 0 Parentnesis Spatial Analysis Dr Fang Qiu I Modulus or division remainder quot raise to power Local Operators o Arithmetic anotherGrid Grid anotherGrid Grid anotherGrid Grid IanotherGrid Grid anotherGrid Grid quot asnotherGrid Grid Negate Grid 0 Reiatian lt anotherGrid Grid lt anotherGrid Grid ltgt anotherGrid Grid anotherGrid Grid gt anotherGrid Grid gt anotherGrid Grid Spatial Analysis D1 Fang Qiu o Bitwise oamp anotherGrid Grid 9 anotherGrid Grid o Grid 9 anotherGrid Grid gtgt anotherGrid Grid oltlt anothierGrid Grid 9 Logicai oAnd anotherGrid Grid oOr anotn erGrid Grid oNot Grid oXOr anotherGrid Grid Logic Value in Map Algebra 4 Logic values 9 True Any none zero va ue EXCEPT NO DATA 9 False Zero va ue General rule Nonzero values are logical TRUE zero value are FALSE I I I gt i 1 D Relational operators and some functions return Ioglcal 039s and 139s 1 m o 6 gt 1 T i I 2 0 36 O T U 2 S o 1 0 1 l I and 1 T 1 1 1 a m a AA AE I Luglcal grids zero and nonzero may be used as tests in expressions Conditional Processing O The Con request QCheck a condition for true of false oParameters include a true grid and a false grid oExample Coningrdgt1 1 2 ing rd outgrd D 1 2 2 3 2 1 1 lsNull and SetNull o IsNull tests for No Data and returns true or false 10 0 SetNuM assigns No Data to nonzero cetl 0 Both are often used with the Con request 0 con lsNullElevation 43999 ElevatiDnD o SetNull1aGrid4 aGrid 4 ISNullQIngrdD SetNulllI11g1 d4 IngrdD Spatial Analysis Dr Fang Qiu Local Operators Ow 35 LocaIFunc ons 0 Trigonometric F quot ti quot5 o Exponential and Logarithmic Functions 9 ACos Grid 5 Grid 0 ACosH Grid Expu Grid 0 ASin Grid Expz Grid 9 ASinH Grid Log Grid 0 Man Grid Log10 Grid 9 ATanZCVG ridii Grid Logz Grid 0 ATaNH Grid oPowanotherGrid Grid o Cos Grid sqr Grid 0 CosH Grid sqn Grid o Sin Grid 9 SinH Grid o Tan Grid 0 TanH Grid LocalFunc ons 9 LOCEIStatiStiCS 0 Local Reclassification o EqualTo aGridList iGrid o LookupaFieldNameGrid o GridsGreaterThaMaGridListGr ReclassaVTabfrmField id toFieldoutFieldnoDataGrid o GridsLessThaMaGridListGrid ReclassBycgassLisu o LocaliStatsaGridStaTypeEnum aFie dMalmejacmaSSList aGridListGrid quotonata 39G d o SWiceaGridSliceTypeEnum 0 Local Armthmetlc Functions numZones basehneyemd AbsGrid 9 Local Condition Checking o CeilGrid 9 ConyesGrid noGridGrid o FloatGrid o IsNulerrid 9 FloorzGrid 0 Pick aGridListGrid o FmodanotherGridGrid o SetNulI anotherGridGrid t IntzGrid Spatial Analysis DI Fang Diu Cell Statistics o Majority oDetermines the value that occurs most often on a cellbycelil basis between inputs 0 Maximum oDetermines the maximum value on a cellbycell basis between inputs 9 Mean oComputes the mean of the values on a cellbycell basis between inputs 9 Median oComputes the median of the values on a cellby cell basis between inputs 9 Minimum oDetermines the minimum value on a cellbycell basis between inputs Cell Statistics 9 Range Maxmin O Determines the range of values on a cellbycell basis between inputs 0 Standard Deviation oComputes the standard deviation of the values on a cellbycell basis between inputs 9 Sum oComputes the sum of the values on a cellbycell basis between inputs 0 Variety QDetermines the number of unique values on a cellbycell basis between inputs 0 Minority oDetermines the value that occurs least often on a cellbycelll basis between inputs Spatial Analysis Dr Fang Qiu LayIn nvwdem mdnmtlk music Local analysis change detection DateOne Date WU BathDate 0 Water on Both dates a Raster Camculatur DateOne1 AND DateTwo1 v DateOne quot DateTwo Spa all may 5 Dr Fungal EitherDate DateOne DateI n o Water on Either dates Raster Calcu atnr DateoneF 0R DataTwoF l DateOne Date39rwo DnlyDate 39 Water on Only one date Raster Calculator DateOneF 0R DateTwole m AND NOT DateOne1l AND DateTwo1 DateOne1 XOR DateTn MFD Date ne DateTwa Ihange ate t Other Hand uses change to water From DateOne to DateTwn Raster Cal cu ator 39 DateOme2 AND DateTwoN spmalmsyssznrfan can DateOne a u a 11 o 1 a o u u 2 2 2 u u u 1 0 1 2 2 2 1 u o o o 2 2 2 o n o DatETWO a 1 a 11 o 1 a u FromTo Raster Calculator DaleOneDateTwo Sammnnsnsmrj anaom n a O n D n n a on no no on on on an no 1 n n a on on an on on no on on 0 U 1 1 I I H 0 Dateone a 1 1 1 i D a on M on 10 on 01 no on n a 1 1 1 D n n on no 11 11 11 no no on a o o 1 n u n a a n u I u u n a 00 10 11 11 11 10 10 00 10 on on 11 11 11 Do an an on 131 co m on m on cm on no on no on no at cm D BIETWO FromTo o Raster Cal cu ator o DateOne10DateTwo yLocaI Analysis Mean Annual Temperature Prediction Temperature Elevation 9 Map Calculator Temp C 129 0003364 Elevation Temp C 427 000237 Elevation meter 0766quot Lattitudedegrees meter j l Local Analysis Water Volume Calculation 9 Data Sets 9 Digital Elevation Model DEM 0 Lake Shape File 0 Elevation of the Lake Calculation Volume Length Width Height 9 Water Volume Cell E aize2 Depth 9 Depth LakeSurface DEM Spatial Analysis Dr Fang Ceiu Complex operations and functions Sometimes hundreds of operations and functions may be necessary DEMO 8 Conversion iii 1quot 11 1 Output raster tnput poiy nes L V 53 l 122 f 39 Vr 1 39 t 4 9 Input raster Output potyttnes Feature polygon polyline points to raster Raster to feature polygon polyline points KRIGING CampPE 940 19 October 2005 Geoff Bohling Assistant Scientist Kansas Geological Survey geoffkgskuedu 8642093 Overheads and other resources available at http peoplekuedugbohlingcpe940 What is Kriging Optimal interpolation based on regression against observed 2 values of surrounding data points weighted according to spatial covariance values Pronunciation Hard g as in Danie Krige or soft g a la Georges Matheron take your pick What is interpolation Estimation of a variable at an unmeasured location from observed values at surrounding locations For example estimating porosity at u 2000 m 4700 m based on porosity values at nearest six data points in our Zone A data 5700 e 0 4 1253 5200 7 l s 1441 03 1287 b 1 o o 1 Northing m 4200 7 12 1215 3700 039 011384 1 1 1 1 1 1 1 500 1000 1500 2000 2500 3000 3500 Easting m 1 4000 It would seem reasonable to estimate 1 by a weighted average 2 lad with weights 2 given by some decreasing function of the distance d from u to data point 0 All interpolation algorithms inverse distance squared splines radial basis functions triangulation etc estimate the value at a given location as a weighted sum of data values at surrounding locations Almost all assign weights according to functions that give a decreasing weight with increasing separation distance Kriging assigns weights according to a moderately datadriven weighting function rather than an arbitrary function but it is still just an interpolation algorithm and will give very similar results to others in many cases lsaaks and Srivastava 1989 In particular If the data locations are fairly dense and uniformly distributed throughout the study area you will get fairly good estimates regardless of interpolation algorithm If the data locations fall in a few clusters with large gaps in between you will get unreliable estimates regardless of interpolation algorithm Almost all interpolation algorithms will underestimate the highs and overestimate the lows this is inherent to averaging and if an interpolation algorithm didn t average we wouldn t consider it reasonable Some advantages of kriging Helps to compensate for the effects of data clustering assigning individual points within a cluster less weight than isolated data points or treating clusters more like single points Gives estimate of estimation error kriging variance along with estimate of the variable Z itself but error map is basically a scaled version of a map of distance to nearest data point so not that unique Availability of estimation error provides basis for stochastic simulation of possible realizations of Z 11 Kriging approach and terminology Goovaerts 1997 All kriging estimators are but variants of the basic linear regression estimator Z u de ned as 2u mu ma moan with uua location vectors for estimation point and one of the neighboring data points indexed by 05 nu number of data points in local neighborhood used for estimation of Z u mumua expected values means of Z 11 and Z no la 11 kriging weight assigned to datum 2ua for estimation location 11 same datum will receive different weight for different estimation location Z 11 is treated as a random field with a trend component mu and a residual component Ru Z u Kriging estimates residual at u as weighted sum of residuals at surrounding data points Kriging weights la are derived from covariance function or semivariogram which should characterize residual component Distinction between trend and residual somewhat arbitrary varies with scale Development here will follow that of Pierre Goovaerts 1997 Geoslalislz39cs for Natural Resources Evaluation Oxford University Press We Will continue working with our example porosity data including looking in detail at results in the sixdatapoint region shown earlier quot1 mg Nurth Porosity lSODD 39 M250quot 7500 375039 1 15000 20000 10000 Easting m Basics of Kriging Again the basic form of the kriging estimator is nu Z 11 mu Ella Zua mua The goal is to determine weights la that minimize the variance of the estimator a u VarZu Zu under the unbiasedness constraint E Z u Z u 0 The random field RF Zu is decomposed into residual and trend components Z 11 Ru mu with the residual component treated as an RF with a stationary mean of 0 and a stationary covariance a function of lag h but not of position 11 E ROD 0 CovRuRu h ERu Ru h CR The residual covariance function is generally derived from the input semivariogram model CR h 2 CR 0 7h Sill yh Thus the semivariogram we feed to a kriging program should represent the residual component of the variable The three main kriging variants simple ordinary and kriging with a trend differ in their treatments of the trend component mu Simple Kriging For simple kriging we assume that the trend component is a constant and known mean mu m so that zKugtm zKuzuagt mi This estimate is automatically unbiased since EZua m 0 so that ElZK uJ m The estimation error ZK u Zu is a linear combination of random variables representing residuals at the data points ua and the estimation point u ZKu Zu 23K u m Zu m 12KuRua Ru RKu Ru Using rules for the variance of a linear combination of random variables the error variance is then given by 0 u Var RK Var RSK u 2 Cov RK uRSK quotl0 quot 0 quot 0 Z Z AKu KuCR ua u CR 0 2 Z liKuCR ua u 1 31 061 To minimize the error variance we take the derivative of the above expression with respect to each of the kriging weights and set each derivative to zero This leads to the following system of equations mu E112KuCRua u CRua u 0lnu Because of the constant mean the covariance function for Zu is the same as that for the residual component C h C R h so that we can write the simple kriging system directly in terms of C h n ls KuCua u Cua u alnu 31 This can be written in matriX form as K XSK u k where K SK is the matriX of covariances between data points with elements KL 2 C u u j k is the vector of covariances between the data points and the estimation point with elements given by k1 C u u and ASK u is the vector of simple kriging weights for the surrounding data points If the covariance model is licit meaning the underlying semivariogram model is licit and no two data points are colocated then the data covariance matriX is positive definite and we can solve for the kriging weights using ASK K lk Once we have the kriging weights we can compute both the kriging estimate and the kriging variance which is given by Gina clto mm clto 39zquotjzKucua u after substituting the kriging weights into the error variance expression above o what 035 all nus math do7 It nds a set ofwexghts for csa atmg mc van his value atthe locaaon ufrom values at a set fnex hbon gdata 1 Thewe h hd gc 1y decrea s thh mcr stance to thatpomt m neral s casmg accordance thh mc decreasmg datartoresnmanon covananccs spen edm mc n htchand vector k However the set of Weights 1 es nedto account forredundancy among h matnXK Mumplymg pomts fallmgm clusters cclaavc to isolated pomts at mc Sam stance We Wm apply simple kngmg to ourporosxty data usmg the sphcnca1 semxvanogram thatwe tbefore wnh zero nugget a sxll ofU 78 and arange of4141m Semivanance 2000 Alum auuu amm mmm Lagmeters Since we are using a spherical semivariogram the covariance function is given by ChC0 yh 0781 15h4141 05h41413 for separation distances h up to 4141 m and 0 beyond that range The plot below shows the elements of the righthand vector k 038 056032 049 046 037 T obtained from plugging the datatoestimationpoint distances into this covariance function 5700 0 k4 C1044m 049 5200 k5 C1170m 046 O O k6 C1746m 032 E4700 7 k6 C1513m 031 E t e z 4200 0 k2 C806m 056 3700 0 k1 C1487m 030 0 500 1000 1500 2000 2500 3000 3500 4000 Easting m The matrix of distances between the pairs of data points rounded to the nearest meter is given by This translates into a data covariance matrix of 078 028 006 017 040 043 028 078 043 039 027 020 006 043 078 037 011 006 017 039 037 078 037 027 040 027 011 037 078 065 043 020 006 027 065 078 rounded to two decimal places Note in particular the relatively high correlation between points 5 and 6 separated by 447 m The resulting vector of kriging weights is 11 01475 12 04564 13 1 00205 14 Z 2 02709 15 02534 16 00266 Notice that data point 6 is assigned a very small weight relative to data point 1 even though they are both about the same distance from the estimation point and have about the same datapointto estimationpoint covariance k1 038 kg 037 This is because data point 6 is effectively screened by the nearby data point 5 Data points 5 and 6 are fairly strongly correlated with each other and 5 has a stronger correlation with the estimation point so data point 6 is effectively ignored Note that the covariances and thus the kriging weights are determined entirely by the data configuration and the covariance model not the actual data values The porosities at points 5 and 6 could in fact be very different and this would have no in uence on the kriging weights The mean porosity value for the 85 wells is 1470 and the porosity values at the siX example wells are 1384 1215 1287 1268 1441 and 1459 The estimated residual from the mean at u is given by the dot product of the kriging weights and the vector of residuals at the data points Ru 39R 086 255 183 015 046 002 027 025 003 201 187 028 010 Adding the mean back into this estimated residual gives an estimated porosity of Zu Ru m 187 1470 1283 Similarly plugging the kriging weights and the vector k into the eXpression for the estimation variance gives a variance of 0238 squared Given these two pieces of information we can represent the porosity at u 2000 m 4700 m as a normal distribution with a mean of 1283 and a standard deviation of 049 Note that like the kriging weights the variance estimate depends entirely on the data configuration and the covariance function not on the data values themselves The estimated kriging variance would be the same regardless of whether the actual porosity values in the neighborhood were very similar or highly variable The in uence of the data values through the fitting of the semivariogram model is quite indirect Here are the simple kriging estimates and standard deviation on a 100X80 g1id with lOOmeter spacing using the spheiical semivaiiogram model and estimating each g1id value from the 16 nearest neighbor data points well locations Estimated Porosity Using Simple Kriging 15000 E 10000 V 15 07 5 o 14 Z 5000 13 12 10000 Easting m Simple Kriging Standard Deviation 15000 0 8 E 10000 0 6 07 E g 0 4 D Z 5000 0 2 0 0 10000 Easting m Some characteristics to note Smoothness Kriged surface will basically be as smooth as possible given the constraints of the data in many cases probably smoother than the true surface Bullseyes Because kriging averages between data points local extremes will usually be at well locations bullseyes are inevitable This is true of almost all interpolation algorithms Extreme form of this is artifact discontinuities at well locations when semivariogram model includes significant nugget Error map re ects data locations not data values Map of kriging standard deviation depends entirely on data configuration and covariance function essentially a map of distance to nearest well location scaled by covariance function Ordinary Kriging For ordinary kriging rather than assuming that the mean is constant over the entire domain we assume that it is constant in the local neighborhood of each estimation point that is that mua mu for each nearby data value Z um that we are using to estimate Z In this case the kriging estimator can be written 2umu jzauzua mu jith Mama lmon and we filter the unknown local mean by requiring that the kriging weights sum to 1 leading to an ordinary kriging estimator of quotIO nu ZOKU 213KUZUa with 213 1 061 06l In order to minimize the error variance subject to the unitsum constraint on the weights we actually set up the system minimize the error variance plus an additional term involving a Lagrange parameter uOK u L o u2u0Ku1 zquotjzau so that minimization with respect to the Lagrange parameter forces the constraint to be obeyed 16L nltugt 1 A 0 26y El 0611 In this case the system of equations for the kriging weights turns out to be nu 3ng u CR um u uOK u CRua u 06 lnu nu 212K u 1 31 where C R h is once again the covariance function for the residual component of the variable In simple kriging we could equate C R h and C h the covariance function for the variable itself due to the assumption of a constant mean That equality does not hold here but in practice the substitution is often made anyway on the assumption that the semivariogram from which C h is derived effectively filters the in uence of largescale trends in the mean In fact the unitsum constraint on the weights allows the ordinary kriging system to be stated directly in terms of the semivariogram in place of the C R h values above In a sense ordinary kriging is the interpolation approach that follows naturally from a semivariogram analysis since both tools tend to filter trends in the mean Once the kriging weights and Lagrange parameter are obtained the ordinary kriging error variance is given by can C0 39 amp3Kucua u u0K u In matrix terms the ordinary kriging system is an augmented version of the simple kriging system For our siXpoint example it would be 078 028 006 017 040 043 100 11 038 028 078 043 039 027 020 100 22 056 006 043 078 037 011 006 100 23 032 017 039 037 078 037 027 100 2L4 049 040 027 011 037 078 065 100 25 046 043 020 006 027 065 078 100 26 037 100 100 100 100 100 100 000 11 100 to which the solution is 21 01274 12 04515 13 00463 14 K 1k 02595 15 02528 16 00448 11 00288 The ordinary kriging estimate at u 2000 m 4700 m turns out to be 1293 with a standard deviation of 0490 only slightly different from the simple kriging values of 1283 and 0488 Again using 16 nearest neighbors for each estimation point the ordinary kriging porosity estimate and standard deViation look very much like those from simple kriging Estimated Porosity Using Ordinary Kriging Northing m 5000 10000 15000 Easting m Ordinary Kriging Standard Deviation 15000 0 8 E 10000 0 6 07 E g 0 4 D Z 5000 1 0 2 0 0 5000 10000 15000 Easting m Kriging with a Trend Kriging with a trend the method formerly known as universal kriging is much like ordinary kriging except that instead of fitting just a local mean in the neighborhood of the estimation point we fit a linear or higherorder trend in the xy coordinates of the data points A local linear aka firstorder trend model would be given by 17111 mxy Clo alxa2y Including such a model in the kriging system involves the same kind of extension as we used for ordinary kriging with the addition of two more Lagrange parameters and two extra columns and rows in the K matrix whose nonzero elements are the x and y coordinates of the data points Higherorder trends quadratic cubic could be handled in the same way but in practice it is rare to use anything higher than a firstorder trend Ordinary kriging is kriging with a zerothorder trend model If the variable of interest does exhibit a significant trend a typical approach would be to attempt to estimate a de trended semivariogram using one of the methods described in the semivariogram lecture and then feed this into kriging with a first order trend However Goovaerts 1997 warns against this approach and instead recommends performing simple kriging of the residuals from a global trend with a constant mean of 0 and then adding the kriged residuals back into the global trend

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