The Dictionary of Human Geography

correspond to it. At one extreme, one macro-
state (all 100 people in one region) has only
one associated microstate, whereas the macro-
state with 10 people in each region corres-
ponds to a large number of different
microstates. The number of microstates corre-
sponding to a macrostate is denoted here by
W, and finding the entropy measure is a com-
binatorial calculation, given by

W¼N!=

Y

i

ni!,

the factorial of the total number of individuals
N, divided by the product of the factorials for
eachni(the number in each region). An alter-
native, but equivalent, measure is that used in
information theory. lwh

Suggested reading
Wilson (1970).

entropy-maximizing models Statisticalmo-
delsfor identifying the ‘most likely’ spatial
allocation pattern in a system subject to con-
straints. The approach was introduced into geo-
graphical modelling by A.G. Wilson (1967) as
thebasisforamorerigorousinterpretationofthe
gravity model, and has been extensively used
since for spatial interaction modelling in
urban regions and for modelling inter-regional
flows of traffic and commodities. It is based on
the concept ofentropy, a measure of the uncer-
tainty or ‘likelihood’ in a probability distribu-
tion. A journey-to-work model illustrates the
method. For a city divided intokzones, we wish
to calculate the best estimate of interzonal com-
muting flowsTij, without knowing the detailed
information of each individual movement.
Assume that there areNtotal commuters. Any
specific trip distribution patternTij, known as a
‘macrostate’(seeentropy),canarisefrommany
different sets of individual commuting move-
ments, or ‘microstates’, and entropy measures
thenumber ofdifferentmicrostates thatcan give
rise to a particular macrostate. In the absence
of detailed microstate data, we assume that
each microstate is equally probable, and that
themacrostatewiththemaximumentropyvalue
is the most probable or most likely overall pat-
tern. Additional information is also normally
available,notably thenumberofcommutersori-
ginatingfromeachzone,thetotalnumberofjobs
available in each zone, and estimates ofthe aver-
age or total travel expenditure for the city,C
(usually based on survey data). The entropy-
maximizingmethodthenconsistsofmaximizing
theentropymeasuresubjecttotheseconstraints.

This maximization is a non-linear optimization

problem, and must be solved by iterative search

methods. These models not only fit empirical

trip-distributions well, but also facilitate easy

calculation of the effects of new housing or jobs

(by altering the constraints), and so have been

widely used in more general urban models.

Wilson and his Leeds colleagues have extended

the model in many ways, making it dynamic,

linking it to industrial and urbanlocation the-

ory, and including several types of disaggrega-

tion (e.g. by mode of travel). lwh

Suggested reading

Gould (1972); Thomas and Huggett (1980);

Wilson (1970).

environmental audit Environmental audit-

ing originated in the 1970s as a management

tool to evaluate how well a corporation was

complying with the complex array of environ-

mental legislation that was emerging as a

result of theenvironmental movement.Itis

an important part of environmental manage-

ment systems (EMS).

Environmental audits are an ‘official exam-

ination and verification ofaccounts and records

to assess how close the situation comes to meet-

ing a set requirement’ (Thomas, 2005, p. 236).

They are a one-off collection of data, and there-

fore differ from ongoing monitoring activities.

Environmental audits may be internal to a cor-

poration, produced externally, undertaken vol-

untarily or to fulfil legal requirements, and may

checkcompliance withregulationsorfocusona

company’s systems to achieve and maintain

compliance (de Moor and de Beelde, 2005).

Environmental audits may be undertaken for

governments, aid agencies, financial institu-

tions and community groups. Internal environ-

mental auditing helps companies meet

regulations and their own environmental goals

in a cost-effective manner. Mandatory, external

auditing verifies and encourages compliance

with environmental regulations. Who under-

takes the environmental audit and who has

access to the resulting document are questions

abouttheregulationofindustryandcommerce.

Environmental auditing is situated within

larger debates about how to achieve and main-

tain high environmental quality. pm

environmental determinism A type of

reasoning that holds that the character and

form of asociety,cultureorbodycan be

explained by the physical conditions within

which it has developed. Determinism is a form

of explanation that finds no place for other

Gregory / The Dictionary of Human Geography 9781405132879_4_E Final Proof page 196 1.4.2009 3:17pm

ENTROPY-MAXIMIZING MODELS