webster, mathematical treatment of soil information

8/6/2019 Webster, Mathematical treatment of soil information

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l ' ' t . Anzah1 k1einersorgfHltig gep1ant wurden. In Austra ~ e n e ~ n e . d hme entwickelt worden, In den Vereinigtet; S t a ~ t e n , 5 ~ n m f a ~ : : r ~ : a ; r ~ t : a ; ~ : : ~ ; . ~ ~ ~ ~ . : systeme in Betrieb, denen jedoch noch d ~ e w e ~ t r e L : h e n d Struktur feh1t FUr di e Erstellung der Struktur e ~ n e s hltierten I n f o r m ~ t i o d i ~ S s i s t g ~ ~ d ~ r ; a ~ ~ ~ f ~ ~ ~ ! : ~ r : ~ n : ~ ~ ~ ~ ~ ~ i ~ ~ : ~ ; t e 1 1 t . Anforderungen an eo. d' ErfordernisseIn vie1en bestehenden Informationssystemen s ~ n ~ e s e nicht genUgend berUcksichtigt.

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Information. Soil Survey of England andStation, Harpenden, Herts: AL5 2JQ, UK.INTRODUCTI ON

Soil scientists have been describing soil conditions mathefo r a long tlme. The work on chemical equillbrJ"a, hydraulic

soil strength and ionic diffusion, for example, could notMathematics has also been used to relate

performance of plants In fjeld and pot experJHowever, i t is only fairly recently that mathematics has beento provide quantitative descrlptions, generallzatlons and pre-

(in a spatial sense) about the soil of whole regions, whetherI territories or large regions within them or small admlnistra

areas or farms. The reasons fo r the noveltytedium of collecting sound data, of handltng them inand the length or complexity of th e mathematical tech

to have impact. However, computers enable us tothe last two problems, and now that this is realized soilare increasingly willing to put effort Into satisfactory samp

their data. Therefore we see soil Information systems, accepting reliable Information from numerous sampling sites

and presenting transformations, summaries, displays and predlc-(output) to the user. Between Input and output, data need to be

I scre"ened and analysed to provide sensible and usable results,here that mathematics plays it s part.Statements and predictions about the soil of areas must be

inst a background of variation, and be taken Into account In15. This 15 the province of statist ics, and most of the mathe

applicable to regional soil Information are In someStatistical methods provide close and usable mathe

descriptions of the real world of the soil.Clearly, a short review cannot cover al l aspects of mathe

applications. This paper wIll concentrate therefore on threein which there has been significant progress or promise In recentnamely regional classification and it s Implications for predlc

analysis, and a particular class of models - the


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spline functions.REGIONAL CLASSIFICATION AND VARIATION

5011 variation across the land surface has been expressedtraditionally by means of 5011 maps. Most large-scale mapsdiVided Into parcels, within anyone of which th e 5011 is of th ekind. Distant parcels possessing similar soil are grouped Intoor mapping units. In 50me cases th e mapping units attemptthe known distribution of particular types of soi l .darles are drawn along distinct changes in surface features, eitherthe 5011 i tself or of associated physiography, vegetation or land use,In both there is the hope. expressed or Implied, thatnot used fo r classification will vary less within the classes thanthe land as a Whole, and that the classification will be useful forgeneralizing about them and predicting their values at previously unvisited or unrecorded sites.

Although soil maps had been made for agricultural a d ~ i s o r y and development work fo r decades, It was engineers who f i rs t examinedsuch maps with mathematical rigour. Thornburn and Larsen (1959) and~ I o r s e and Thornburn U961} sampled parts of Illinois that had beenmapped fo r agricultural purposes. They expressed measurements ofseveral mechanical properties using the classical statist icalof mean and variance for each class. They found substantial dibetween class means, and thus, since variation within the mappingwas less than that over the landscape as a whole, theymapping units as sampling strata to reduce th e size ofcut costs. Table I summarizes Morse and Thornburn's results fo rproperties in Livingston County. Two features should be noted.approximately half the total variance is attributable tobetween soil types and half to variation within types. Second, thewithin-groups standard deviation is approximately 20% of the mean Inmost instances.

Kantey and Williams (1962) in South Africa were th eworkers to estimate variances fo r their own soil maps, and soany quantitative assessment of their quality. They too were mainlyconcerned to show that soil mapping could lead

In Britain the stimulus fo r quantitative measurement oftlon came from the Royal Engineers, whose prime task was prediction

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soil conditions at unvisited sites and the confidence that could beplaced in it . To assess the likelY value of 5011 maps as aids to prediction, trials were carried out In both temperate and desert regions.The land was classified, actually into types of terrain rather thanspecific soil classes, largely on It s appearance on ai r photographs,and then sampled. The resul ts of that wor!, have been summarized byBeckett et al . (1972), and Webster and Beckett (1970). Table 2,from Webster and Beckett (196B) gives examples of the resul ts . As inthe mapping in Illinois, approximately half the total variance in themechanical properties of the soii was attributable to differences be-tween classes (land facets) at the local level. The remainder withinclasses was sufficiently small to alloW worthwhile prediction. Forchemical properties, however, classification was, less profitable, andfo r some, e.g. available phosphorus and potassium, the within-classvariance was only very slightly less than the to ta l , and waS certainly'.oo large for prediction.

Similar studies carried out since confirm that thisgeneral. Variance of physical and mechanical propertiesunits at series level or thereabouts is likely to be about halftotal In an area in which there ar e a number of such uni ts . InInstances it is somewhat less, even for some chemical properties (e.Corcoran et a1. 1977). However, sincerange fluctuation {Beckett and Webster 1971; Webster and CuanaloIt cannot be appreciably diminished by more painstaking mapping.scientists must also be prepared fo r situations whereproduces l i t t le benefit for many properties. This is the caseof the Australian Capital Territory (Webster and Butler 1976), andarises partly because of lack of correlation among solipartly because different properties vary spatially on quitescales (Fig. I) .Although we must credit engineers for stimulating th e 1of soil scientists In mathematical description of soilan earlier paper of Youden and Mehllch (1937) deserves aattention than It has received. These workers wished tothe soil of a part of Broome County, New York,tions, and to sample in a way that would allow this and at the sametime reveal regional differences. They carried ou t a multi-stage

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sampling In which the stages were different sample spacings arranged ona roughly logarithmic scale. Nine first-stage units were spaced approxl-

1. 6 km apart. Two second-stage units 305 m apart were choseneach first stage unit, two third-stage units 30.5 m apart werewithin each second-stage unit, and two fourth-stage units 3.05 m

apart were chosen wIthin each third-stage unit. The pH of th e soileach sampling point was measured, and th e variance attributable tospacing estImated. Table 3 gives components of variance for pH atThe largest component derives from the largest spacing, but.

InevltablYJ there Is a substantial variance within th e smallest3.05 m, and It is not the smallest component.Table 3. Components of variance of soil pH at 0-15 cmin Culvers gravelly s i l t loam, Broome County, New York

Stage Spacing Component1 1.6 km 0.028192 305 m 0.023403 30.5 m 0.005524 3.05 m 0.01391

This sampling strategy was applied In th e Australian Capital(Webster and Butler 1976). The results for four of the pro

of Interest ar e shown graphically In Figure 1. The patterns offo r al l four, and In part explain why general

classif ication was unprofitable.MULTIVARIATE ANALYSIS

The classical multivariate methods, principal component analydiscriminant analysis and canonical analysis, were

in th e decade forty to fifty years ago. They could not beto study problems In soi1 science where they were most wantedthere were no computers to handle th e large matrices involved.

arrive there was no sudden rush to apply the classiInstead, It was th e newly developing techniques fo r

classifIcation that caught the Imagination. It seemedlast was a means of resolving the age-old arguments about

In which soi 1 should be classif ied. In th e event, these


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numedcal techniques enabled sets of 5011 speciments, profi les or sampling sites to be classified swiftly and repeatably. However, theresults ar e not necessarily better than tradItional methods fo r eXpressing relationships or for predicting properties other than thoseused to create th e classif ications. They are also sensitive to smallchanges In technique and data. When the classical methods were appliedIt became clear why stable classif ication of soil is 50 diff icul t toattain . This section reviews briefly what a t present seem to be th etwo most important forms of analysis and some results of their applicat ion.Principal Axis Methods

One of th e major problems in 5011 systematics is to be ableto quantitative relations among Individuals when each is characterized by more than two or three measurements. Means ar e needed fo rcompressing the information into few variates with the least loss ofinformation, and this is t h ~ aim of what Is now known as ordination.Several methods have been proposed, and th e f irst application-to soliwas by Hole and Hlronaka (1960) who used the one devised by Bray andCUrtis l1957} for ecological work. Since then, however, most ordination, not only of soil bu t In other branches of science also, has beenby vector methods, of which principal component analysis Is the mostrl gorous.

Principal component analysis (PCA) has been used in severalstudies (e.g. Cuanalo and \4ebster 1970; Hebster and Burrough 1974;Burrough and Hebster 1976; Lamp 1972; Norris 1971) to elucidate relations among Individuals and population structure in general. It hasalso been applied profitably to agronomic problems (Kyuma and Kawaguchi1973) and to th e Interpretation of engineering tests (Leflaive et al .1973)

PCA may be regarded geometrically as the rigid rotation ofconfiguration of points (individuals) in multld.lmensional space (withone dimension fo r each property). Relations between individualsrepresented by their separating distances in the full space, andmethod assumes that this Is appropriate and that i t is possible tomeasure th e characters of interest. The assumption isWe m ~ y prefer other measures of relationship and In some Instancesto use qual itatlve characters. Gower (1966) Introduced hi s =-"'==

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coordinate analysis (PCO) to cope with this sltuat '. 'I Ion. In this methodSimi arlt ies, 5, between Individuals can be calculated by almost any ofth.e commonly used formulae (see Sneath and Sakal 1973 fo r their defini-tions) and converted to distances by d = V2{I-S) U ' ,I dl I sing thiS measure nn v duals ar e represented in a space, with n- l dimensions and theirc o o ~ d l n a t e s found relative to the principal axes of th e con;,gUratlAs In PCA, projection onto .the plane defIned on.by th e longest axes dis-plays relations In th e most informative way. Despite it hh ' s somew a teavler computing several workers have found PCO advant (R 6 ageous e, gayner 19 6, 1969; Campbell et al . 1970; Webster and Butler 1976. .Wi 11 iams and Rayner 1977; Banfield and Bascomb 1976). '

Two examples will i l lustrate the heuris t ic value of thesemethods. Figure 2 from a study of so i 1 in the River i na of New SputhWales (Norris 1971) is' perhaps th e more typical.paT components The fi rs t two princla ~ c o u n t for half the total variance a m o ~ g fifty o'riglnal

The distribution Is thus fairly f la t , and the project-ionwell represents relationships in th e whole 'h space. The f i ture alsoow the three main soil groups ar e separated Ea hfa l I c group occupies

r Y compact part of the plane, though there is a l i t t le overlthe red brown earths and th e tranSitional red brown e a r t h s ~ P third feature of th h hI h. e grap , w c we must now realize Is quite nor-

, IS the lack of clusterIng. There is evidently no 'natural 'ng I th 'n . n. . S sense, and though we may classify th e population byg diVISions through the configuration such divisions are b darbitrary. oun to

Figure 3, from the study by Webster and Butler (1976) In th eIan Capital Territory, also shows a fair ly ~ v e n scatter of points

f P J a n ~ of the f irst two prfncipal coordinates. However, it dlfrom Figure 2 in two respects. First, only 26% of th e variation

accounted for. This Is unusually small: the distribution is morehyperspherlcal than most. Second th 'th ' er e IS considerable overlap:- e portions of the SPace OCCUPied by each class of so i I. This was: s.o true for the other soil c lassif ications that were examined and Is

f;cet of th e lack of correlation in th e area, d l s c u s s e ~ he analysis shows quIte clearly that classif ication is Un-

be profitable Tn any general sense.PCA is also valuable as a means of data reduction. The first


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webster, mathematical treatment of soil information

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