SIZE="-1"><STRONG>Purpose</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">2-D spatial filter for krigeddata.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT><FONTFACE="Helvetica" SIZE="-1"><STRONG>Synopsis</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">F = barnes (xi, yi, zi, c,g)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT><FONTFACE="Helvetica" SIZE="-1"><STRONG>Description</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">The Barnes' filter is a low-pass2-D filter whose mathematical description is:</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><img src="engli0{image10}.gif"width="162" height="28" align=bottom ></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">where <imgsrc="engli0{image11}.gif" width="163" height="102" align=bottom > ,<img src="engli0{image12}.gif" width="232" height="102" align=bottom> , </FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><img src="engli0{image13}.gif"width="178" height="29" align=bottom > and <imgsrc="engli0{image14}.gif" width="186" height="29" align=bottom>.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><U>Inputvariables:</U></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">xi:  column gridcoordinates</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">yi:  row gridcoordinates</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">zi:  grid data</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">c, g:  filterparameters</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><U>Outputvariable:</U></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">F:  filtered data </FONT></P><P><FONT FACE="Helvetica"SIZE="-1"><STRONG>Example</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">The tintore function provides agood example of spatial filters in physical oceanography.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT><FONTFACE="Helvetica" SIZE="-1"><STRONG>References</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Maddox, R.A., 1980.  AnObjective Technique for Separating Macroscale and Mesoscale Featuresin Meteorological Data.  Monthly Weather Rev., 108:1108-1121.<STRONG></STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Tintor&eacute; et al., 1991.Mesoscale Dynamics and Vertical Motion in the Alboran Sea.  J.Phys.  Oceanogr., 21:  811-823.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT><FONTFACE="Helvetica" SIZE="-1"><STRONG>See also</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">tintore, filresp</FONT></P><P><FONT FACE="Helvetica"><STRONG>cokri / cokri2</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1"></FONT><FONT FACE="Helvetica"SIZE="-1"><STRONG>Purpose</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">cokri:  Point or block cokrigingin D dimensions.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">cokri2:  Cokriging functioncalled from cokri.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT><FONTFACE="Helvetica" SIZE="-1"><STRONG>Synopsis</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">[x0s, s, sv, id, l] = cokri (x,x0, model, c, itype, avg, block, nd, ival, nk, rad, ntok,d)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">[x0s, s, id, l, k0] = cokri2 (x,x0, id, model, c, sv, itype, avg, ng, d)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT><FONTFACE="Helvetica" SIZE="-1"><STRONG>Description</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Cokriging means kriging withmore than one variable.  When cokri is called with only one variableat a time, the results will be those of simple kriging, ordinarykriging, universal kriging, point kriging or block kriging.  Moredetails can be found in the paper of Marcotte (1991).</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Available kriging options are:</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> simple cokriging </FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> ordinary cokriging with onenonbias condition (Isaaks and Srivastava)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> ordinary cokriging with pnonbias condition</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> universal cokriging with driftof order 1</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> universal cokriging with driftof order 2</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">99.cokriging is not performed,only sv is computed</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Available semivariogram modelsare:</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> nugget effect</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> exponential model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> gaussian model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> spherical model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> linear model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> quadratic model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> power (h<SUP>d</SUP>)model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> logarithmic</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> sinc(h)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> Bessel [ Jo (h) ]</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> exp(-h) * cos(dh)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> exp(-h) * Jo(dh)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> exp(-h<SUP>2</SUP>) *cos(dh)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> exp(-h<SUP>2</SUP>) *Jo(dh)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> exp(-h<SUP>2</SUP>) * (1 -dh<SUP>2</SUP>)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> 1 - 3*min(h,1) +2*min(h,1)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> h * log(max(h,eps))</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">New models can be added quiteeasily since models are calculated using the evalfunction.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><U>Inputvariables:</U></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">x:  data matrix [x y z var1 var2...]</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">x0:  grid coordinates [xi yizi]</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">model:  [models, a (h=r/a),rotation angles].</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">No rotation angle is requiredfor an isotropic distribution.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Example:  model = [1 10; 4 30]means that the distribution is isotropic and that it is representedby a nugget effect of range 10 plus a spherical model of range30.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">c:  amplitudes of themodels</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">itype:  krigingoption</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">block:  vector (1 x D), givingthe size of the block to estimate; for point cokriging:  block =ones(1,D)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">nd:  Vector (1 x D), giving thediscretization grid for block cokriging; for point cokriging:  nd =ones(1,D)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">ival:  Code forcross-validation.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> 0:  nocross-validation</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> 1:  cross-validation isperformed by removing one variable at a</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> time at a givenlocation.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> 2:  cross-validation isperformed by removing all variables at a</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> given location.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">nk:  number of nearest neighborsin x matrix to use in the cokriging.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">rad:  search radius.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">ntok:  points in x0 will bekriged by groups of ntok grid points.  </FONT></P><P><FONT FACE="Helvetica" SIZE="-1">d:  model coefficients.  Thiscoefficient has been added to the original Marcotte's function.Warning:  In cokri, models are defined in terms of h = r/a.  Invariogr, the dependent variable is r and hence d = b*a (seevariogr).</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><U>Outputvariables</U></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">x0s:  kriged data matrix at x0locations.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">s:  kriged data variance matrixat x0 locations.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">sv:  variance of eachvariable.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">id, l see Marcotte(1991)</FONT></P><P><FONT FACE="Helvetica"SIZE="-1"><STRONG>Reference</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Marcotte, D.  1991.  Cokrigeagewith matlab.  Computers &amp; Geosciences.  17(9):1265-1280.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT><FONTFACE="Helvetica" SIZE="-1"><STRONG>See also</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">cokri2, variogr, trans,means<STRONG></STRONG></FONT></P><P><FONT FACE="Helvetica"><STRONG>confint</STRONG></FONT><FONTFACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT></P><P><FONT FACE="Helvetica"SIZE="-1"><STRONG>Purpose</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Confidence intervals.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT><FONTFACE="Helvetica" SIZE="-1"><STRONG>Synopsis</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">[k2, k1] = confint (g, m,S2)</FONT></P><P><FONT FACE="Helvetica"SIZE="-1"><STRONG>Description</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Confidence intervals for thestructure function</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">CONF {k2 variance k1}(1)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">The structure function is ameasure of the variance of a given variable as a function ofdistance.  The estimation of the confidence intervals in such a caseis given by (1).</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">k1 = (n-1) * S<SUP>2</SUP> /c1</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">k2 = (n-1) * S<SUP>2</SUP> / c2(2)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> </FONT></P><P><FONT FACE="Helvetica" SIZE="-1">where n = sample size =m+1</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">m = number of degrees offreedom</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">S<SUP>2</SUP> = variance of thesample</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">c1 and c2 are determined by thesolution to the equations</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">F(c1) = (1-g) /2</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">F(c2) = (1+g) /2 (3)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">where g = confidence level (95%,99% or the like)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">F = <imgsrc="engli0{image15}.gif" width="15" height="20" align=bottom >distribution</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Solutions are obtained byfunction chitable (Chi Toolbox).</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT><FONTFACE="Helvetica" SIZE="-1"><STRONG>References</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Denman, K.  L.  and H.  J.Freeland (1985).  Correlation Scales, Objective Mapping and aStatistical Test of Geostrophy over the Continental Shelf.  J.  ofMar.  Res., 43:  517-539.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Kreyszig, E., 1988.<EM>Advanced Engineering Mathematics</EM>, sixth ed., John Wiley&amp; Sons, New York, p.1252</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG>Seealso</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">chitable</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT><FONTFACE="Helvetica"><STRONG>davis</STRONG></FONT><FONT FACE="Helvetica"SIZE="-1"><STRONG></STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG>Purpose</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Point kriging using Davis' setof equations A W = B</FONT></P><P><FONT FACE="Times New Roman" SIZE="-1"></FONT><FONT FACE="TimesNew Roman" SIZE="-1"> </FONT><FONT FACE="Helvetica"SIZE="-1">where</FONT><FONT FACE="Times New Roman" SIZE="-1"> <imgsrc="engli0{image16}.gif" width="245" height="108" align=bottom >,<img src="engli0{image17}.gif" width="67" height="102" align=bottom> </FONT><FONT FACE="Helvetica" SIZE="-1">and</FONT><FONTFACE="Times New Roman" SIZE="-1"> <img src="engli0{image18}.gif"width="83" height="105" align=bottom >.</FONT></P><P><FONT FACE="Helvetica" SIZE="-1">The g(h<SUB>ik</SUB>) is thesemivariance of sample pairs separated by distance h<SUB>ik</SUB>.Non bias conditions require the sum of W<SUB>i</SUB> to be equal to1.  In that case, one more degree of freedom must be introduced withthe use of a Lagrange multiplier<img src="engli0{image19}.gif"width="14" height="21" align=bottom > in order to minimize theestimation error.</FONT></P><P><FONT FACE="Helvetica"SIZE="-1"><STRONG>Synopsis</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">[Zp, Sp] = davis (data, x0,model, a, d, c, A)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"><STRONG></STRONG></FONT><FONTFACE="Helvetica" SIZE="-1"><STRONG>Description</STRONG></FONT></P><P><FONT FACE="Helvetica" SIZE="-1">Available models are:</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> <MULTICOL COLS="2" WIDTH="507"GUTTER="46"> nugget effect</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> exponential model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> gaussian model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> spherical model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> linear model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> quadratic model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> power model(h<SUP>d</SUP>)</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> logarithmic model</FONT></P><P><FONT FACE="Helvetica" SIZE="-1"> sinc(h)</FONT></P>