Table Of ContentC H A P T E R
1
Why Water-Quality Indices
O U T L I N E
1.1. Introduction 3 1.4. The First Modern WQI:Horton’s Index 5
1.1.1. Water Quantity and WaterQuality 3
1.5. More on theBenefits of WQI 6
1.2. Water-Quality Indices (WQIs) 4
1.6. WQIs Based on Bioassessment 6
1.2.1. ANovel Idea? 4
1.3. Back toWater-Quality Indices(WQIS) 5
1.1. INTRODUCTION
been exceedingly felt so that, by now, water
1.1.1. Water Quantity and Water quality has acquired as much importance as
Quality waterquantity.
What is water quality? This question is
Ofallnaturalresources,water isunarguably immensely more complex than the question:
the most essential and precious. Life began in What is waterquantity?
water, and life is nurtured with water. There Wecansay:thisreservoircontains2millionm3
are organisms, such as anaerobes, which can of water or the present flow in this river is
survive without oxygen. But no organism can 15m3sec(cid:1)1. Expressing water quantity is as
survive for any length oftimewithout water. simpleasthis.
The crucial role of water as the trigger and But how do we express water quality of the
sustainer of civilizations has been witnessed samestream?Thequalitymaybegoodenough
throughout human history. But, until as late as fordrinkingbutnotsuitableforuseasacoolant
the 1960s, the overriding interest in water has in an industry. It may be good for irrigating
been vis a vis its quantity. Except in manifestly some crops but not good for irrigating some
undersirable situations, the available water othercrops.It may be suitable for livestock but
was automatically deemed utilisable water. not for fish culture. Whereas water quantity is
Onlyduringthelastthreedecadesofthetwen- determined by a single parameter e the water
tieth century the concern for water quality has mass e water quality is a function of anything
3
WaterQualityIndicesDOI:10.1016/B978-0-444-54304-2.00001-4 Copyright(cid:1)2012ElsevierB.V.Allrightsreserved.
4
1. WHYWATER-QUALITYINDICES
andeverythingthewatermighthavepickedup Diewert and Nakamura, 1993). Most countries
duringitsjourneyfromthecloudstotheearthto have their ’consumer price index’ in which, on
the water body: in dissolved, colloidal, or sus- thebasisofanintegrationofthepricesofcertain
pended form. Given the fact that water is commodities,asinglevalueisobtainedtodeter-
a ’universalsolvent’,it picks upa lot! minewhetherthemarketis,overall,cheaperor
One way to describe the quality of a given costlier at any given instant compared to any
water sample is to list out the concentrations otherpast instant.
of everything that the sample contained. Such The commodities for such indices are
alistwouldbeaslongasthenumberofconstit- selected on the basis of their ‘driver power’ e
uents analyzed and that may be anything from in other words, the ’power’ or the ’reach’ of
the 20-odd common constituents to hundreds! the commodity vis a vis influencing the prices
Moreover, such a list will make little sense to of several other commodities. If a shampoo
anyone except well-trained water-quality becomes costlier or cheaper, it will not affect
experts. the prices of other commodities significantly,
How to compare the quality of different whereas any change in the price of cement or
water sources? It can’t be done easily by petroleum would.
comparing the list of constituents each sample Then we have share-market indices such as
contains. For example, a water sample which the Dow Jones Index of the New York stock
contains six components in 5% higher-than- exchange and the Sensex of Mumbai’s stock
permissible (hence objectionable) levels: pH, exchange. These indices are also composed of
hardness, chloride, sulphate, iron and sodium thepricesofcertainsharesofhighdriverpower
may not be as bad for drinking as another (such as cement). In time these indices have
sample with just one constituent e mercury e becomemeasuresnotmerelyofthestocktraded
at5% higher-than-permissible. at the respective exchanges but also of the
Water-quality indices seek to address this economies of the respective countries. When
vexing problem. indices of economically advanced countries
such as the Dow Jones Index suffer a slump, it
impactsthestockexchangesofmostothercoun-
1.2. WATER-QUALITY INDICES
tries as well, often taking their indices down
(WQI )
S with it.
Indices have also been used in ecology to
Water-quality indices aim at giving a single represent species richness, evenness, diversity
value to the water quality of a source on the etc. Accordingly, we have the Shannon Index,
basisofoneortheothersystemwhichtranslates theSimpsonIndex,andsoon.Innumerousother
the list of constituents and their concentrations fields e such as of medicine, sociology, process
present in a sample into a single value. One safety,etce indicesareextensivelyused.
can then compare different samples for quality It can be said that indices (the singular of
on thebasisof theindexvalueof each sample. indices is index) are composite representations
ofaconditionorsituationderivedfromacombi-
1.2.1. A Novel Idea? nation,doneincertainways,ofseveralrelevant
butnoncommensurateobservedfacts/measure-
Theconceptofusinganindextorepresentin ments.Thecombinationleadstoasingleordinal
a single value the status of several variables is numberthatfacilitatesunderstandingandinter-
not a novel idea; it has been well-entrenched pretation of the overall import of the facts that
in economics and commerce (Fisher, 1922; have contributed to that number.
I.WATERQUALITYINDICESBASEDPREDOMINANTLYONPHYSICO-CHEMICALCHARACTERISTICS
5
1.4. THEFIRSTMODERNWQI:HORTON’SINDEX
Environmental indices e of which water- 1.4. THE FIRST MODERN WQI:
quality indices form a major component e are HORTON’S INDEX
used as communication tools by regulatory
agencies to describe the ‘quality’ or ‘health’ of Horton (1965) set for himself the following
aspecificenvironmentalsystem(e.g.,air,water, criteria when developing the first-ever modern
soil and sediments) and to evaluate the impact WQI:
ofregulatorypoliciesonvariousenvironmental
1. Thenumberofvariablestobehandledbythe
management practices (Song and Kim, 2009;
indexshouldbelimitedto avoid makingthe
Pusatli et al., 2009; Sadiq et al., 2010).
indexunwidely.
Environmental indices have also been used in
2. The variablesshouldbe ofsignificance in
life-cycle assessment (Weiss et al., 2007; Khan
mostareas.
et al., 2004) and to characterise different types
3. Onlysuchvariablesofwhichreliabledataare
of environmental damages, including global
available, orobtainable,shouldbeincluded.
warming potential (Goedkoop and Spriensma,
2000). Horton selected 10 most commonly
measuredwater-quality variablesforhis index,
including dissolved oxygen (DO), pH, coli-
1.3. BACK TO WATER-QUALITY
forms, specific conductance, alkalinity, and
INDICES (WQI )
S chloride. Specific conductance was intended to
serve as an approximate measure of total dis-
WQIs may have gained currency during the solved solids (TDSs), and carbon chloroform
last3decadesbuttheconceptinitsrudimentary extract (CCE) was included to reflect the influ-
form was first introduced more than 150 years ence of organic matter. One of the variables,
agoein1848einGermanywherethepresence sewage treatment (percentage of population
or absence of certain organisms in water was served),wasdesignedtoreflecttheeffectiveness
used as indicator of the fitness or otherwise of of abatement activities on the premise that
a water source. chemical and biological measures of quality
Since then various European countries have are of little significance until substantial prog-
developedandapplieddifferentsystemstoclas- ress has been made in eliminating discharges
sify the quality of the waters within their of raw sewage. The index weight ranges from
regions. These water classification systems 1 to 4. Notably, Horton’s index did not include
usually areof two types: any toxic chemicals.
1. thoseconcernedwiththeamountofpollution Theindexscoreisobtainedwithalinearsum
aggregation function. The function consists of
presentand
2. those concerned with living communities of the weighted sum of the subindices Ii divided
by the sum of the weights Wi and multiplied
macroscopic ormicroscopic organisms.
by two coefficients M and M , which reflect
1 2
Rather than assigning a numerical value to temperatureandobviouspollution,respectively:
represent water quality, these classification P
n WI
systems categorised water bodies into one of QI ¼ Pi¼1 i iM M
W 1 2
several pollution classes or levels. By contrast, i¼1 i
indices that use a numerical scale to represent
gradations in water-quality levels are a recent Horton’s index is easy to compute, even
phenomenon, beginning with Horton’s index though the coefficients M and M require
1 2
in 1965,detailed below. some tailoring to fit individual situations. The
I.WATERQUALITYINDICESBASEDPREDOMINANTLYONPHYSICO-CHEMICALCHARACTERISTICS
6
1. WHYWATER-QUALITYINDICES
indexstructure,itsweights,andratingscaleare 3. Enforcement of standards
highlysubjectiveastheyarebasedonthejudge- Indicesmaybeappliedtospecificlocations
ment of the author anda fewof his associates. to determine theextentto whichlegislative
Horton’spioneeringefforthasbeenfollowed standardsand existing criteria arebeing met
up by several workers who have striven to orexceeded.
develop less and less subjective but more and 4. Trendanalysis
moresensitiveandusefulwater-qualityindices. Indices may beappliedto water-quality
dataat different points in time to determine
the changesin thequality (degradation or
1.5. MORE ON THE BENEFITS
improvement)which haveoccurredoverthe
OF WQI
period.
5. Public information
Theformulationanduseofindiceshavebeen Indexscorebeinganeasy-to-understand
strongly advocated by agencies responsible for measureofwater-qualitylevel,indicescanbe
water supply and control of water pollution. usedtokeepthepublicinformedoftheoverall
Oncethewater-qualitydatahavebeencollected waterqualityofanysource,orofdifferent
through sampling and analysis, a need arisesto alternativesources,onaday-to-daybasisjust
translateitintoaformthatiseasilyunderstood. asSensexscoretellsinonewordwhetherthe
OncetheWQIsaredevelopedandapplied,they stocks,byandlarge,wentupordown.
serve as a convenient tool to examine trends, to 6. Scientific research
highlightspecificenvironmentalconditions,and The inherent quality ofan index e which
to help governmental decision-makers in evalu- translates a largequantity of datato a single
atingtheeffectivenessofregulatoryprogrammes. scoreeis immensely valuable in scientific
WQIs, of course, are not the only source of research, forexample,in determining the
information that is brought to bear on water- efficacy of differentecorestoration measures
relateddecisions.Manyotherfactorsareconsid- or water-treatment strategies with reference
eredbesidesindicesandthemonitoringdataon toawaterbody,theimpactofdevelopmental
which the indices arebased. activities on waterquality,etc.
Indeed,nearlyallthepurposesforwhichone
monitors water quality e assessment, utilisa-
tion,treatment,resourceallocation,publicinfor- 1.6. WQIS BASED ON
mation,R&Dandenvironmentalplanningeare BIOASSESSMENT
allservedbyindicesaswell.Inaddition,indices
make the transfer and utilisation of water-
On the basis of parameters that are incorpo-
quality data enormously easier and lucid. To
rated in an index to judge water quality, the
wit,water-quality indiceshelp in:
WQIs can be loosely classified into ‘indices
1. Resourceallocation predominantlybasedonphyscio-chemicalchar-
Indices maybe applied inwater-related acteristics’and‘indicesbasedonbioassessment’.
decisionsto assist managersin allocating The first modern WQI, the Horton’s index,
funds anddetermining priorities. described in Section 1.4, belonged to the first
2. Ranking of allocations category, so are all the indices described in Part
Indices maybe applied toassist in I of this book. Just as Horton’s index has one
comparing waterquality atdifferent parameter (out of 10 chosen by him) which
locationsorgeographical areas. requires bioassessment e coliforms e several
I.WATERQUALITYINDICESBASEDPREDOMINANTLYONPHYSICO-CHEMICALCHARACTERISTICS
7
REFERENCES
other indices described in Part I have one or Khan,A.A.,Paterson,R.,Khan,H.,2004.Modificationand
more ‘biological’ parameters but they are all applicationoftheCanadianCouncilofMinistersofthe
EnvironmentWaterQualityIndex(CCMEWQI)forthe
predominantly based on physico-chemical
communication of drinking water quality data in
parameters. Incontrast, theWQIsbased onbio-
Newfoundland and Labrador. Water Quality Research
assessment, described in Part II, are predomi- JournalofCanada39(3),285e293.
nantly based on sampling, identification, and Pusatli, O.T., Camur, M.Z., Yazicigil, Z.H., 2009. Suscepti-
enumerationofbiologicalorganisms. bilityindexingmethodforirrigationwatermanagement
planning: applications to K. Menderes river basin
Turkey. Journal of Environmental Management 90 (1),
References 341e347.
Sadiq,R.,Haji,S.A.,Cool,G.,Rodriguez,M.J.,2010.Using
Diewert,W.E.,Nakamura,A.O.,1993.Indices.In:Essaysin penalty functions to evaluate aggregation models for
Index Number Theory, vol. 1. North Holland Press, environmental indices. Journal of Environmental
Amsterdam.71e104. Management91(3),706e716.
Fisher, I., 1922. The Makingof Index Numbers.Houghton Song, T., Kim, K., 2009. Development of a water quality
Mifflin,Boston,MA. loadingindexbasedonwaterqualitymodeling.Journal
Goedkoop,M.,Spriensma,R.,2000.TheEco-Indicator99-a ofEnvironmentalManagement90(3),1534e1543.
Damage Oriented Method for Life Cycle Impact Weiss, M., Patel, M., Heilmeier, H., Bringezu, S., 2007.
Assessment,MethodologyReport.http://www.pre.nl Applying distance-to-target weighing methodology to
Horton, R.K., 1965. An index number system for rating evaluate the environmental performance of bio-based
water quality. Journal of Water Pollution Control energy, fuels, and materials. Resources, Conservation
Federation.37(3),300e306. andRecycling50,260e281.
I.WATERQUALITYINDICESBASEDPREDOMINANTLYONPHYSICO-CHEMICALCHARACTERISTICS
C H A P T E R
2
Approaches to WQI Formulation
O U T L I N E
2.1. Introduction 9 2.5. Assignment ofWeightages 15
2.1.1. Indices for ‘Water Quality’and
2.6. Aggregation ofSubindices toProduce
‘Water Pollution’ 9
aFinal Index 15
2.2. The Common Steps 10 2.6.1. Linear Sum Index 16
2.6.2. Weighted Sum Index 16
2.3. Parameter Selection 10
2.6.3. Root Sum PowerIndex 17
2.4. Transformation of theParameters of 2.6.4. Multiplicative FormIndices 17
Different Unitsand Dimensions to 2.6.5. MaximumOperator Index 17
aCommon Scale: Making Subindices 11 2.6.6. Minimum Operator Index 18
2.4.1. DevelopingSubindices 12
2.7. Characteristics of Aggregation Models 18
2.4.2. Different Types of Subindices 12
2.7.1. Ambiguity and Eclipsing 18
2.4.2.1. LinearFunctionSubindices 12
2.7.2. Compensation 18
2.4.2.2. SegmentedLinearFunction
2.7.3. Rigidity 18
Subindices 13
2.4.2.3. NonlinearFunction 13
2.4.2.4. SegmentedNonlinear
Function 13
2.1. INTRODUCTION
numbers decrease with the degree of pollution
2.1.1. Indices for ‘Water Quality’ (decreasing scale indices). One may classify the
formeras‘waterpollutionindices’andthelatter
and ‘Water Pollution’
as ‘water-quality indices’. But this difference is
Water-quality indices can be formulated in essentially cosmetic; ‘water quality’ is a general
two ways: one in which the index numbers term of which ‘water pollution’ e which indi-
increasewiththedegreeofpollution(increasing cates ‘undesirable water quality’ e is a special
scale indices) and the other in which the index case.
WaterQualityIndicesDOI:10.1016/B978-0-444-54304-2.00002-6 9 Copyright(cid:1)2012ElsevierB.V.Allrightsreserved.
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2. APPROACHESTOWQIFORMULATION
2.2. THE COMMON STEPS 2.3. PARAMETER SELECTION
Thefollowingfourstepsaremostoftenasso- Aswe have elaborated in Chapter1, awater
ciated with the development of any WQI; sample may have hundreds of constituents,
depending on the sophistication being aimed including elements in neutral or ionic form
at,additional steps may also betaken: (metals, non-metals, metalloids); organics
(pesticides, detergents, otherorganics of indus-
1. Parameter selection.
trialornaturalorigin);anionssuchascarbonate,
2. Transformationoftheparametersofdifferent
bicarbonate,sulphate,nitrate,nitriteetc.Itmay
units and dimensions to acommon scale.
also have suspended solids which, in turn,
3. Assignment ofweightages to allthe
constitute a bewildering range of chemicals. It
parameters.
may also have radioactivity, and may have
4. Aggregation of subindices toproduce afinal
colourandodour.Thenitmayhavepathogenic
index score.
bacteria, fungi, helminthiccystsetc.
Of these, steps 1, 2 and 4 are essential for all AWQIwouldbecomeunwieldyifeachand
indices. Step 3 is also commonly taken through every possible constituent is included in the
some indices may be formed without this step index. Instead, one needs to choose a set of
as well. parameters which, together, reflect the overall
Water-quality indices make it very easy for water quality for the given end use. It is an
a lay person to judge whether a water source exercise similar to the one involved in the
is usable or not and how one source compares selection of a few dozen or a few hundred
to another, but the development of WQI is shares out of thousands to construct a share
by no means an easy task. It, in fact, is market index. To reduce the number of shares
fraught with several complications and to be incorporated in the index, yet keep the
uncertainties. index representative of the overall stock
Aswemayseefromthefollowingdiscussion, market situation, those shares are picked up
and from the numerous examples given in the which have a high ‘driver power’ e i.e shares
following chapters on how different WQIs ofwhosemovementinfluencesa largenumber
have been developed for different needs, it of other shares. In this manner, the share price
would be clear that a great deal of subjective indexismade‘sensitive’tothestockmarketas
opinion and judgement is associated with each a whole even as it is computed on the basis of
step,particularlysteps1and3.Thereisnotech- thepricesofonlyafractionofthelistedshares.
nique or device by which 100% objectivity or Water-quality indices are also made on the
accuracy can be achieved in these steps. Even basis of a few parameters chosen for their
parameter selection through statistical analysis ‘forcing’ or ‘driver power’. But it is here that
of past data (detailed in Chapter 4), though subjectivity creeps in. Different experts and
apparently objective, is fraught with inherent end users may have different perceptions of
uncertainties and incompleteness. the importance of a parameter vis a vis a given
One can only try to reduce subjectivity and end use. For example, a medical expert may
inaccuracy by involving large number of perceivewatercarryingafaintodourbutother-
experts in collecting opinion, and doing it by wisefreefromharmfulconstituentsasgood.In
well-developed opinion-gathering techniques his/her opinion, odour may be a parameter of
such as Delphi (Abbasi 1995, Abbasi and very little significance. But to others even the
Arya 2000). faintest odour in their drinking water may be
I.WATERQUALITYINDICESBASEDPREDOMINANTLYONPHYSICO-CHEMICALCHARACTERISTICS
11
2.4. TRANSFORMATIONOFTHEPARAMETERSOFDIFFERENTUNITSANDDIMENSIONSTOACOMMONSCALE
totally unacceptable. They will advocate that correlationscanoccurbysheerchanceandneed
odour must be included as a key parameter in nothaveacause-effectlink.Forexample,dataon
anyWQIdealingwithdrinking water. chloride may strongly correlate with data on
Even the water-quality standards, on which abundance of a fish species but the two may, in
much of our decisions on fitness or otherwise reality,havenolinkwhatsoever.
of a water source depend, are not common to
all countries. Further, as new research
brings to light new facts on the beneficial or 2.4. TRANSFORMATION OF THE
harmful effects of a constituent, or gives PARAMETERS OF DIFFERENT
new information on the concentration beyond UNITS AND DIMENSIONS TO
which a constituent becomes harmful or the A COMMON SCALE: MAKING
concentration below which a constituent ceases SUBINDICES
to be helpful, the standards are continuously
revised. Different water-quality parameters are
The criteria of ‘acceptability’ also vary from expressed in different units. For example,
region to region. In regions well-endowed temperature is expressed in degrees celsius
with water resources e for example the State or fahrenheit, coliforms in numbers, electrical
of Kerala, India e drinking water containing conductivity in micro-mhos, and most chemi-
more than 500 mg/L of total dissolved solids cals in milligramme per litre (or microgram
(which is the BIS (Bureau of Indian Standards) per ml). Further, the ranges of levels to which
limit for ideal drinking water) may be consid- different parameters can occur vary greatly
ered unfit for drinking because the state may from parameter to parameter. For example,
find ample drinking water in an alternative dissolved oxygen would rarely be beyond the
source meeting with the BIS standard. On the range 0e12 mg/L but sodium can be in the
other hand, people in arid or semi-arid range0e1000mg/Lorbeyond.Toxicelements
regions, for example in the Indian states of such as mercury rarely occur above 1 mg/L
Rajasthan, Tamil Nadu and elsewhere, level, whereas acidity/alkalinity, hardness,
routinely drink water with TDS well above chloride and sulphate nearly always occur at
500 mg/L. levels above 1 mg/L. Yet again, watercontain-
Therefore, parameter selection is as fraught ing10mg/Lofchlorideisasfitfordrinkingas
withuncertaintyandsubjectivityasitiscrucial water containing fifteen times higher chloride.
to the usefulness of any index. Enormous care, But a water sample containing 0.001 mg/L of
attention, experience, and consensus-gathering mercury is acceptable while a water with
skills are required to ensure the most represen- even twice this concentration of mercury
tative parametersareincludedin a WQI. is not.
In an attempt to reduce the subjectivity in Inotherwords,differentparametersoccurin
parameter selection, statistical approaches have differentranges,areexpressedindifferentunits,
beenattempted,asdescribedinChapter4.While andhavedifferentbehaviourintermsofconcen-
in theory such approaches are objective e trationeimpact relationship. Before an index
because they select parameters on the basis of canbeformulated,allthishastobetransformed
considerations such as frequency of occurrence intoasinglescaleeusuallybeginningwithzero
of different parameters, the number of other and ending at 1. Some index scales have the
parameters to which they seem to correlate, range 0e100. But this, again, makes only
etcetheycanleadtoerroneousresultsbecause a cosmetic difference.
I.WATERQUALITYINDICESBASEDPREDOMINANTLYONPHYSICO-CHEMICALCHARACTERISTICS
12
2. APPROACHESTOWQIFORMULATION
2.4.1. Developing Subindices oramultiplicationoperation,inwhichaproduct
isformedofsomeorallthesubindices,orsome
Subindices, one for each parameter selected
other operation; some of the common aggrega-
for the index, are developed so that different
tion methods aregiven in Section 6.
parameters, their units and the range of
The overall process: calculation of subindices
concentrations (from highly acceptable to
andaggregationofsubindicestoformtheindex
highly unacceptable) are all transformed onto
can beillustrated as in Figure2.1.
a single scale.
If we consider a set of n pollutant variables
2.4.2. Different Types of Subindices
denoted as (x , x , x . x, x ), then for each
1 2 3 i n
pollutant variable xi, a subindex Ii is computed Subindices can be classified as one of four
using subindexfunction fi, (xi): general types:
I ¼ FðxÞ (2.1) 1. Linear
i i i
2. Nonlinear
In most indices, different mathematical 3. Segmented linear
functions are used to compute different 4. Segmented nonlinear
pollutantvariables,yieldingthesubindexfunc-
tionsf (x ),f (x ).f (x ).Suchfunctionsmay
1 1 2 2 n n 2.4.2.1. Linear Function Subindices
consist of simple multiplier, or the pollutant
The simplest subindex function is the linear
variable raised to a power, or some other func-
equation:
tional relationship.
Once the subindices are calculated, they I ¼ axþb (2.3)
usually are aggregated together in a second
whereIisthesubindex,xthepollutantvariable
mathematical step to form thefinal index:
and a,b the constants.
I ¼ gðI1,I2.InÞ (2.2) With this function, a direct proportion exists
between the subindex and the pollutant vari-
The aggregation function, (2.2), usually able. The linear indices are simple to compute
consists either of a summation operation, in and easy to understand but have limited
whichindividualsubindicesareaddedtogether, flexibility.
FIGURE 2.1 The index develop- Information flow
mentprocess.
Parameter x1 Subindex I1 Aggregation
=
Water quality Parameter x2 Subindex I2
=
Data =
Index I
Parameter xn Subindex In
=
I.WATERQUALITYINDICESBASEDPREDOMINANTLYONPHYSICO-CHEMICALCHARACTERISTICS
13
2.4. TRANSFORMATIONOFTHEPARAMETERSOFDIFFERENTUNITSANDDIMENSIONSTOACOMMONSCALE
2.4.2.2. Segmented Linear Function plotted on a graph sheet. Such nonlinear func-
Subindices tionsareof two basic types:
A segmented linear function consists of two 1. animplicitfunction,whichcanbeplottedon
or more straight line segments joined at break
a graph but for whichno equation isgiven
points(thresholdlevel).Itoffersmoreflexibility 2. an explicit function,for which a
than linear function and is especially useful for
mathematical equation isgiven.
incorporating administratively recommended
limits, such as Bureau of Indian Standards Implicit functions usually arise when some
(BIS) limits, WHO limits etc. One of the impor- empirical curve has been obtained from
tant types of segmented linear functions is the a process under study. For example, Brown
step function, which exhibits just two states et al. (1970) proposed an implicit nonlinear
and therefore is called a dichotomous function. subindexfunction for pH.
The segmented linear function subindices may Explicit nonlinear functions automatically
also consist of a staircase of steps, giving lead to nonlinear curves. An important general
a multiple-state function. For example, Horton nonlinearfunctionisoneinwhichthepollutant
(1965) has used subindex functions containing variableisraisedtoapowerotherthanone,the
three, four, and five steps. In Horton’s dis- power subindex function:
solved oxygen subindex, one finds I ¼ 0 for x
I ¼ xc (2.5)
less than 10% saturation, I ¼ 30 for x between i
10% and 30% saturation, and I ¼ 100 for x wherecs 1
above 70% saturation. Walski and Parker (1974) used the following
Mathematically, the general form of generalparabolicform inevolvingthesubindi-
segmented linear function can be formulated cesfor temperatureandpH:
as follows.
Given that x and I coordinates of the break b
points are represented by (a1, b2), (a2, b2),., Ii ¼ (cid:2)a2ðx(cid:2)aÞ2þb, 0(cid:3)x(cid:3)2a (2.6)
(a, b), any segmented linear function with m
j j
segments can be presented by the following Another common nonlinear function is the
general equation: exponential function, in which pollutant vari-
ablex isthe exponentof aconstant:
b (cid:2)b
I ¼ iþ1 iðx(cid:2)aÞþb, a (cid:3)x(cid:3)a (2.4) I ¼ Cx (2.7)
i a (cid:2)a i i i iþ1 i
iþ1 i
Theconstantusuallyselectediseither10ore,
where,i ¼ 1,2,3,.,m: the base of the natural logarithm. If a and b are
Although segmented linear functions are constants, the general form of an exponential
flexible,theyarenotideallysuitedtosomesitu- function iswrittenasfollows:
ations, particularly those in which the slope
I ¼ aebx (2.8)
changes very gradually with increasing levels i
of pollution. In these instances, a nonlinear
functionis usually more appropriate.
2.4.2.4. Segmented Nonlinear Function
Segmented nonlinear function consists of
2.4.2.3. Nonlinear Function line segments similar to the segmented linear
When a causeeeffect relationship does not function; however, at least one segment is
vary linearly, it leads to a curvature when nonlinear.Usually,eachsegmentisrepresented
I.WATERQUALITYINDICESBASEDPREDOMINANTLYONPHYSICO-CHEMICALCHARACTERISTICS
