Page 0 of 0 SPATIAL INTERPOLATION METHODS

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1 Page 0 of 0 SPATIAL INTERPOLATION METHODS 2018

2 1. Introducton Spatal nterpolaton s the procedure to predct the value of attrbutes at unobserved ponts wthn a study regon usng exstng observatons (Waters, 1989). Lam (1983) defnton of spatal nterpolaton s gven a set of spatal data ether n the form of dscrete ponts or for subareas, fnd the functon that wll best represent the whole surface and that wll predct values at ponts or for other subareas. Predctng the values of a varable at ponts outsde the regon covered by exstng observatons s called extrapolaton (Burrough and McDonnell, 1998). All spatal nterpolaton methods can be used to generate an extrapolaton (L and Heap 2008). Spatal Interpolaton s the process of usng ponts wth known values to estmate values at other ponts. Through Spatal Interpolaton, We can estmate the precptaton value at a locaton wth no recorded data by usng known precptaton readngs at nearby weather statons. Ratonale behnd spatal nterpolaton s the observaton that ponts close together n space are more lkely to have smlar values than ponts far apart (Tobler s Law of Geography). Spatal Interpolaton covers a varety of method ncludng trend surface models, thessen polygons, kernel densty estmaton, nverse dstance weghted, splnes, and Krgng. Spatal Interpolaton requres two basc nputs: Sample Ponts Spatal Interpolaton Method Sample Ponts Sample Ponts are ponts wth known values. Sample ponts provde the data necessary for the development of nterpolator for spatal nterpolaton. The number and dstrbuton of sample ponts can greatly nfluence the accuracy of spatal nterpolaton. 2. Spatal Interpolaton Methods Pont nterpolaton methods may be categorzed n several ways. local or global exact or approxmate

3 Reference # Page 2 of 2 gradual or abrupt determnstc or stochastc Global Interpolaton uses every known pont avalable to estmate an unknown value. Local Interpolaton uses a sample of known ponts to estmate an unknown value. Ths method s desgned to capture the local or short range varaton. Exact nterpolaton predcts a value at the pont locaton that s the same as s known value. Approxmate nterpolaton (nexact) predcts a value at the pont locaton that dffers from ts known value. Determnstc nterpolaton method provdes no assessment of errors wth predcted values. Stochastc nterpolaton methods offer assessment of predcton errors wth estmated varance. Gradual or abrupt nterpolators are dstngushed by the contnuty of the surface they produce. Gradual nterpolaton technques wll produce a smooth surface wth gradual changes occurrng between observed data ponts. A wde varety of spatal nterpolaton methods exst n the lterature. The spatal nterpolaton methods covered n ths revew are only those commonly used n the studes. 2.1.Thessen polygons Thessen polygons are an exact method of nterpolaton that assumes the unknown values of the ponts on a surface to be equal to the value of the nearest known pont. Ths can be consdered as a local nterpolator because the characterstcs of the global data set have no nfluence on the nterpolaton process. The steps to follow are: 1. Construct the Thessen Polygons 2. Determne the area proportons of each polygon and assgn a weght 3. Calculate the average value by usng the weghts (Thessen, 1911) The method of Thessen polygons s a robust technque and wll always produce the same surface from the same set of data ponts. Ths s a dsadvantage that the technque s unntellgent and unable to respond to external knowledge about factors whch may nfluence the values recorded at the observed data ponts. One of the most common uses for ths technque s for the generaton of area terrtores from a set of ponts.

4 Reference # Page 3 of The Trangulated Irregular Network (TIN) The Trangulated Irregular Network (TIN) developed n the early 1970 s s a smple way to construct a surface from a set of known ponts. It s a partcularly useful technque for rregularly spaced ponts. The TIN approach s an exact nterpolaton method. In the TIN model, the known data ponts are connected by lnes to form a seres of trangles. A TIN s a terran model type that creates a set of contnuous, non-overlappng, connected trangles (faces), based on a so-called Delaunay trangulaton of rregularly spaced observatons. The corners of the trangles are dentcal to the observatons and wthn each trangle the surface s usually represented by a plane. A TIN model wll allow for dfferent densty n the data model n dfferent areas. The sze of the trangles can therefore be adjusted to reflect the degree of relef n the surface to be modelled, provded more data has been gathered n areas of varable elevaton characterstcs. The use of trangles ensures that each pece of the surface fts the neghbourng peces, and t s thus possble to create the desred contnuous seamless transton between the trangles, wthn the model Trend Surface Models An approxmate method, trend surface analyss approxmates ponts wth known values wth a polynomal equaton. A surface nterpolaton method that fts a polynomal surface by least-squares regresson through the sample data ponts. Ths method results n a surface that mnmzes the varance of the surface n relaton to the nput values. The resultng surface rarely goes through the sample data ponts. Ths s the smplest method for descrbng large varatons, but the trend surface s susceptble to outlers n the data. Trend surface analyss s used to fnd general tendences of the sample data, rather than to model a surface precsely (ESRI). The lnear equaton or the nterpolator can then be used to estmate values at other ponts. Z(x, y) = b 0+ b 1 x + b 2 y Where the attrbute value z s a functon of x and y coordnates. The b coeffcents are estmated from the known ponts. The quadratc nterpolator can be wrtten as gven below. Z(x, y) = b 0+ b 1 x 2 + b 2 y 2 + b 3 x + b 4 y + b 5 xy

5 Reference # Page 4 of 4 The two general classes of technques for estmatng a regular grd of ponts on a surface from scattered observatons are methods called "global ft" and "local ft." As the name suggests, globalft procedures calculate a sngle functon descrbng a surface that covers the entre map area. The functon s evaluated to obtan values at the grd nodes. In contrast, local-ft procedures estmate the surface at successve nodes n the grd usng only a selecton of the nearest data ponts. Trend surface analyss s the most wdely used global surface-fttng procedure. The mapped data are approxmated by a polynomal expanson of the geographc coordnates of the control ponts, and the coeffcents of the polynomal functon are found by the method of least squares, nsurng that the sum of the squared devatons from the trend surface s a mnmum. Each orgnal observaton s consdered to be the sum of a determnstc polynomal functon of the geographc coordnates plus a random error. The polynomal can be expanded to any desred degree, although there are computatonal lmts because of roundng error. The unknown coeffcents are found by solvng a set of smultaneous lnear equatons whch nclude the sums of powers and cross products of the X, Y, and Z values. Once the coeffcents have been estmated, the polynomal functon can be evaluated at any pont wthn the map area. It s a smple matter to create a grd matrx of values by substtutng the coordnates of the grd nodes nto the polynomal and calculatng an estmate of the surface for each node. Because of the least-squares fttng procedure, no other polynomal equaton of the same degree can provde a better approxmaton of the data. The trend surface method fts a surface to the known observaton ponts by a method known as least squares. The shape of the surface s determned by the mathematcal equaton, or polynomal, whch s used to descrbe t. A polynomal surface has some very mportant characterstcs; most mportantly t changes smoothly and predctably from one map edge to the other. The polynomal equaton may be lnear, quadratc or even cubc. A lnear equaton would be used to descrbe a tlted plane. A quadratc equaton would defne a smple hll or hollow and a cubc equaton would defne a more complex surface. The trend surface method fts a surface to the known observaton ponts by a method known as least squares. The shape of the surface s determned by the mathematcal equaton, or polynomal, whch s used to descrbe t. A polynomal surface has some very mportant characterstcs; most mportantly t changes smoothly and predctably from one map edge to the other. The polynomal equaton may be lnear, quadratc or even cubc. A lnear equaton would be used to descrbe a tlted plane. A quadratc equaton would defne a smple hll or hollow and a cubc equaton would defne a more complex surface.

6 Reference # Page 5 of Nearest neghbor nterpolaton Nearest-neghbor nterpolaton s a smple method of multvarate nterpolaton n one or more dmensons. Interpolaton s the problem of approxmatng the value of a functon for a non-gven pont n some space when gven the value of that functon n ponts around (neghborng) that pont. The nearest neghbor algorthm selects the value of the nearest pont and does not consder the values of neghborng ponts at all, yeldng a pecewse-constant nterpolant. The algorthm s very smple to mplement and s commonly used (usually along wth mpmappng) n real-tme 3D renderng to select color values for a textured surface. For a gven set of ponts n space, a Vorono dagram s a decomposton of space nto cells, one for each gven pont, so that anywhere n space, the closest gven pont s nsde the cell. Ths s equvalent to nearest neghbor nterpolaton, by assgnng the functon value at the gven pont to all the ponts nsde the cell. The fgures on the rght sde show by color the shape of the cells. Nearest neghbor s the most basc technque and as t requres the least processng tme whch s major advantage among all the nterpolaton algorthms because t takes only one pxel nto consderaton.e. the nearest one to the nterpolated pont. Ths has the results nto smply makng each pxel larger. Although nearest neghbor method s very effcent because of ts tme effcency, the qualty of mage s comparatvely very poor Natural neghbor nterpolaton Natural neghbor nterpolaton s a method of spatal nterpolaton, developed by Robn Sbson (1980, 1981). The method s based on Vorono tessellaton of a dscrete set of spatal ponts. Ths has advantages over smpler methods of nterpolaton, such as nearest-neghbor nterpolaton, n that t provdes a smoother approxmaton to the underlyng "true" functon. The ponts used to estmate the value of an attrbute at locaton x are the natural neghbors of x, and the weght of each neghbor s equal to the natural neghbor coordnate of x wth respect to ths neghbor. If we consder that each data pont n S has an attrbute a (a scalar value), the natural neghbor nterpolaton s gven below. f(x) n 1 w ( x) a Where f(x) s the nterpolated functon value at the locaton x. The resultng method s exact and f(x) s smooth and contnuous everywhere except at the data ponts.

7 Reference # Page 6 of The Movng Average Method The Movng Average method assgns values to grd nodes by averagng the data wthn the grd node's search ellpse. To use Movng Average, a search ellpse must be defned and the mnmum number of data to use, specfed. For each grd node, the neghborng data are dentfed by centerng the search ellpse on the node. The output grd node value s set equal to the arthmetc average of the dentfed neghborng data. If there are fewer, than the specfed mnmum number of data wthn the neghborhood, the grd node s blanked. Ths process nvolves calculatng a new value for each known locaton based upon a range of values assocated wth neghborng ponts. The new value wll usually be ether an average or weghted average of all the ponts wthn a predefned neghborhood. In some textbooks you wll fnd that they use the term movng wndow. Applyng a movng average nvolves makng a number of decsons about the shape, sze and character of the neghborhood about a pont. The most common shape for a neghborhood s a crcle, snce ponts n all drectons have an equal chance of fallng wthn the radus of a gven pont. However, n the raster world a square or rectangular wndow of cells s often used. The sze of the neghborhood s determned by the user and s usually based upon assumptons about the dstance over whch local varablty n the data s mportant. Applyng a movng average nvolves makng a number of decsons about the shape, sze and character of the neghborhood about a pont. The most common shape for a neghborhood s a crcle, snce ponts n all drectons have an equal chance of fallng wthn the radus of a gven pont. However, n the raster world a square or rectangular wndow of cells s often used. The sze of the neghborhood s determned by the user and s usually based upon assumptons about the dstance over whch local varablty n the data s mportant Inverse Dstance Weghtng (IDW) It s an exact method that enforces the condton that the estmated value of a pont s nfluenced more by nearby known ponts than by those farther away. The IDW s smple and ntutve determnstc nterpolaton method based on prncple that sample values closer to the predcton locaton have more nfluence on predcton value than sample values farther apart. Usng hgher power assgns more weght to closer ponts resultng n less smoother surface. On the other hand, lower power assgns low weght to closer ponts resultng n smoother surface. Major dsadvantage of IDW s bull's eye effect (hgher values near observed locaton) and edgy surface. The process s hghly flexble and allows estmatng dataset wth trend or ansotropy, n search neghborhood shapng (Burrough and McDonnel, 1988; Lu, 1999).

8 Reference # Page 7 of 7 IDW combnes the noton of proxmty wth that of gradual change of the trend surface. IDW reles on the assumpton that the value at an unsampled locaton s a dstance-weghted average of the values from surroundng data ponts, wthn a specfed wndow. The ponts closest to the predcton locaton are assumed to have greater nfluence on the predcted value than those further away, such that the weght attached to each pont s an nverse functon of ts dstance from the target locaton. The general IDW predcton formula s Zˆ( u N 0 ) w Z( u ) 1 where: Z(u0) s the value beng predcted for the target locaton u0; N s the number of measured data ponts n the search wndow; w are the weghts assgned to each measured pont; and Z(u) s the observed value at locaton u. u=(x,y.) w 1/ d 1/ N 1 d d 2 2 ( x x ) ( y y ) 2 The Inverse Dstance Method s very smple and easy to use whch s one of ts bggest advantages. It s applcable for a wde range of data as the method often delvers reasonable results and does not exceed the range of meanngful values (Caruso et al., 1998) Splne Splne s determnstc nterpolaton method whch fts mathematcal functon through nput data to create smooth surface. Splne can generate suffcently accurate surfaces from only a few sampled ponts and they retan small features. Splne works best for gently varyng surfaces lke temperature. Splnes are pece-wse polynomal functons that are ftted together to provde a complete, yet complex, representaton of the surface between the measurement ponts. Functons are ftted exactly to a small number of ponts whle, at the same tme, ensurng that the jons between dfferent parts of the curve are contnuous and have no dsjunctons. Splnes are often useful for calculatng smooth surfaces from a large number of nput data ponts and often produce good results for gently varyng surfaces such as elevaton. Thn plate splnes are a specal form of splne, whch have partcular value n nterpolaton. These, n effect, replace the exact splne surface wth a locally smoothed average whch passes as close as possble through the data ponts. They can be

9 Reference # Page 8 of 8 used to remove artfacts (e.g. excessvely hgh or low predcted values) resultng from natural varaton and measurement error n the nput data. Splne creates a surface that passes through the control ponts and has the least possble changes n slope at all the ponts. Q( x, y) N 1 A d 2 logd a bx cy d 2 2 ( x x ) ( y y ) 2 Where x and y are the coordnates of the pont to be nterpolated, and x and y are the coordnates of control pont. Unlke the IDW method, the predcted values from Splne are not lmted wthn the range of maxmum and mnmum values of the known ponts. Instead of averagng values, the Splne nterpolaton method fts a flexble surface, as f t were stretchng a rubber sheet across all the known pont values. Clff face or a fault lne, are not represented well by a smooth-curvng surface. In such cases, you mght prefer to use IDW nterpolaton, where barrers can be used to deal wth these types of abrupt changes n local values. There are two types of Splne. A Regularzed Splne yelds a smooth surface and smooth frst dervatves. Wth the Regularzed opton, hgher values used for the [weght] parameter produce smoother surfaces. The values entered for ths parameter must be equal to or less than zero. Typcal values that may be used are 0, 0.001, 0.01, 0.1, and 0.5. A Tenson Splne s flatter than a Regularzed Splne of same sample ponts, forcng the estmate to stay closer to the sample data. It produces a surface more rgd accordng to the character of the modeled phenomenon. Wth the Tenson opton, hgher values entered for the [weght] parameter result n somewhat coarser surfaces, but surfaces that closely conform to the control ponts. The values entered must be equal to or greater than zero. The typcal values are 0, 1, 5, and Krgng Krgng, also known as the Theory of regonalzed varables, was developed by G. Matheron and D. G. Krgge as an optmal method of nterpolaton for use n the mnng ndustry. Burrough (1986)

10 Reference # Page 9 of 9 notes that the orgns of the method le n the recognton that the spatal varaton of many geographcal propertes s too rregular to be modeled by a smooth mathematcal functon. The bass of krgng les n estmatng the average rate at whch the dfference between values at ponts change wth dstance between ponts. Ths s the most complex of all the exact nterpolaton methods and untl recently was absent from vrtually all commercally avalable GIS software. Krgng was developed n the 1960s by the French mathematcan Georges Matheron. The motvatng applcaton was to estmate gold deposted n a rock from a few random core samples. Krgng has snce found ts way nto the earth scences and other dscplnes. It s an mprovement over nverse dstance weghtng because predcton estmates tend to be less bas and because predctons are accompaned by predcton standard errors (quantfcaton of the uncertanty n the predcted value). The basc tool of geostatstcs and krgng s the semvarogram. The semvarogram captures the spatal dependence between samples by plottng semvarance aganst separaton dstance Geostatstcs assumes spatal data analyss. The most common spatal tool s varogram defned by formula gven below. 1 ( h) 2N( h) N ( h) z( u ) z( u h) 1 N(h)s number of data pars at dstance h, z(u)s value at locaton u=(x,y) and z(u+h) value at locaton u+h Calculaton of expermental varogram s necessary nput for dfferent geostatstcal nterpolaton or smulaton technques, lke krgng gven below. n z( u ) z( u ) o 1 uo s ponts estmated by krgng, s- weght coeffcent for krgng calculated from matrx equaton and z u ) s hard data of varable. ( The krgng ncludes several nterpolaton technques lke Smple, Ordnary, Unversal and other krgng nterpolatons. For more detaled nformaton the classcal textbooks by Creesse (1993) and Journel & Huıjbregts (1978) are the mportant references. 2

11 Reference # Page 10 of Smple Krgng Model assumptons of Z(u) are gven below. Z(u) = μ + ε(u) E[ε(u)] = 0 E(Z(u)) = μ constant and known Smple krgng predctor for the pont uo s gven below. n Z (u 0 ) = λ Z(u ) λ = Г 1 γ γ = (γ(u 1 u o ),.., γ(u n u o )) λ = (λ 1, λ 2, λ 3,., λ n ) γ(u 1 u 1 ).. γ(u 1 u n ) Г =..... γ(u n u 1 ).. γ(u n u n ) Ordnary Krgng Model assumptons of Z(u) are gven below. Z(u) = μ + ε(u) E[ε(u)] = 0 E(Z(u)) = μ constant and unknown Ordnary krgng predctor for the pont uo s gven below. n Z (u 0 ) = λ Z(u ) λ = (γ + 1(1 1 Г 1 γ)/1 Г 1 1) Г 1 1=(1,,1) n dmensonal unt vector.

12 Reference # Page 11 of Unversal Krgng Model assumptons of Z(u) are gven below. Z(u) = μ(u) + ε(u) E[ε(u)] = 0 b μ(u) = j=0 β j f j (u) and f 1 (u), f 2 (u),, f b (u) Unversal krgng predctor for the pont uo s gven below. n Z (u 0 ) = λ Z(u ) λ = (γ + F(F Г 1 F) 1 (f F Г 1 γ)) Г 1 f 1 (u 1 ) f b (u 1 ) F = f 1 (u n ) f b (u n ) f 1 (u) = ( f 1 (u 0 ),, f b (u 0 )

13 Reference # Page 12 of Applcaton of spatal nterpolaton methods on an example Sample data gven n Table 1 was produced randomly wth space fllng expermental desgn to mplement the nterpolaton methods dscussed above. Column x and column y show the coordnates and the column z shows the random varable (or attrbute) of the pont. Table 1. Sample data of 100 spatal ponts Pont x y z Pont x y z Pont x y z Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont Pont sample grds are randomly selected from grds.

14 Reference # Page 13 of 13 Map 1. Dsplay of 100 observaton pont from a 100x100 grd map Table 2. Sample statstcs Statstcs Value Mean 140,58 Medan 136,5 Varance 2493,317

15 Reference # Page 14 of 14 Map 2. Dsplay of trend surface nterpolaton map (100x100 grd) Table 3. Statstcs of trend surface nterpolaton Statstcs Value Mean 139,76 Medan 139,7 Std.Dev. 6,74 Predcton value (10000 grds)

16 Reference # Page 15 of 15 Map 3. Dsplay of nearest pont nterpolaton map (100x100 grd) Table 4. Statstcs of nearest pont nterpolaton Statstcs Value Mean 141,84 Medan 137 Std.Dev. 51,26 Predcton value (10000 grds)

17 Reference # Page 16 of 16 Map 4. Dsplay of lnear movng average nterpolaton map (100x100 grd) Table 5. Statstcs of lnear movng average nterpolaton Statstcs Value Mean 142,31 Medan 141,7 Std.Dev. 21 Predcton value (10000 grds)

18 Reference # Page 17 of 17 Map 5. Dsplay of movng surface nterpolaton map (100x100 grds) Table 6. Statstcs of movng surface nterpolaton Statstcs Value Mean 144,19 Medan 141,6 Std.Dev. 34,99 Predcton value (10000 grds)

19 Reference # Page 18 of 18 Map 6. Dsplay of ordnary krgng nterpolaton map (100x100 grd) Table 7. Statstcs of ordnary krgng Statstcs Value Mean 141,92 Medan 142 Std.Dev. 23,98 Predcton value (10000 grds)

20 Reference # Page 19 of 19 Map 7. Dsplay of unversal krgng (lnear trend) nterpolaton map (100x100 grd) Table 8. Statstcs of unversal krgng (lnear trend) Statstcs Value Mean 144,48 Medan 143,94 Std.Dev. 26,58 Predcton value (10000 grds)

21 Reference # Page 20 of 20 Map 8. Dsplay of unversal krgng (quadratc trend) nterpolaton map (100x100 grd) Table 9. Statstcs of unversal krgng (quadratc trend) Statstcs Value Mean 144,74 Medan 143,98 Std.Dev. 34,82 Predcton value (10000 grds)

22 Reference # Page 21 of Conclusons Interpolaton s a method that predcts the values at locatons where no sample values are avalable. Spatal nterpolaton assumes the attrbute data are contnuous over space. Ths allows the predcton of the attrbute at any locaton wthn the data boundary. Another assumpton s the attrbute s spatally dependent, ndcatng the values closer together are more lkely to be smlar than the values farther apart. These assumptons allow for the spatal nterpolaton methods to be formulated. In many areas spatal nterpolaton methods have been used successfully such as geology, envronment, mnng, clmatology, bology, forestry, agrculture, engneerng etc. Spatal nterpolaton provdes comprehensve nformaton about the spatal attrbutes (or varables) of any nterested subject for ponts or grds.

23 Reference # Page 22 of 22 References Azpurua, M. & Dos Ramos, K. (2010). A Comparson of Spatal Interpolaton Methods for Estmaton of Average Electromagnetc Feld Magntude, Progress In Electromagnetcs Research M, Vol. 14, Burrough, P. A., & McDonnell, R. A. (1998). Prncples of Geographcal Informaton Systems. Oxford: Clarendon Press. Caruso, C. & Quarta, F. (1998). Interpolaton methods comparson. In Computers & Mathematcs wth Applcatons 35 (12), pp DOI: /S (98) Cresse, N.A.C. (1993) Statstcs for Spatal Data, New York: A Wley-Interscence publcaton. De Floran, L., Magllo,P. & Puppo, E. (2000). Compressng Trangulated Irregular Networks, Kluwer Academc Publshers. Garnero G. & Godone D. (2013). Comparsons between Dfferent Interpolaton Technques, The Internatonal Archves of the Photogrammetrc, Remote Sensng and Spatal Informaton Scences, Volume XL-5/W3. Husman, O. & De By R.A. (2001). Prncpals of geografc nformaton systems, ITC textbook. Journel, A., G. & Huıjbregts C. J. (1978) Mnng Geostatstcs, London, Academc Press. L, J. & Heap, A., A Revew of Spatal Interpolaton Methods for Envronmental Scentsts. Geoscence Australa, Canberra. Naoum, S.& Tsans, I.K. (2004). Rankng Spatal Interpolaton Technques Usng A GIS-Based DSS. Global Nest: the Int. J. Vol 6, No 1, pp Owen, S.J. (1992). An Implementaton of Natural Neghbor Interpolaton n Three Dmensons, Thess, Brgham Young Unversty. Sbson, R (1980). A vector dentty for the Drchlet tesselaton. In Mathematcal Proceedngs of the Cambrdge Phlosophcal Socety, 87, pages Sbson, R. (1981). A bref descrpton of natural neghbor nterpolaton. In V Barnett, edtor, Interpretng Multvarate Data, pages Wley, New York, USA. Watson, D.F. (1995). Natural Neghbor Sortng. The Australan Computer Journal, vol. 17, no. 4, Thessen, A. H. (1911). Precptaton averages over large areas. In Mon. Wea. Rev. 39 (7), pp DOI: /

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