• Entropy is a mathematical quantity
describing disorder or
• Boltzmann introduced the concept
of entropy to measure the disorder
in a thermodynamic system.
• Shannon used the concept of
informational entropy to measure
the uncertainty associated with
• According to Wilson, four ways
to view the concept of entropy
can be defined:
- Entropy as a measure of system
properties (eg order and
irreversibility, complexity and
- Entropy as a measure of
information, uncertainty, or
- Entropy as stastistics of a
probability distribution for
measure of information or
- Entropy as the negative of
Bayesian log likelihood function
for measure of function.
• It is generally defined as an
average value of information rate
to eliminate uncertainty, which is
given by finite number of
• Shannon defines the entropy as:
H = - ∑i=1P(si)log P(si)
• Where S is the system with finite
number of possible events si,
P(si) represents the probablity of
event si and the summation is
over the ranges 1,2,3…n.
• Maximum Entropy when all the
probablities are equal and the
most uncertain situation occurs.
• Mininum entropy when H=0
Entropy in case of two possibilities with
probablities p and (1-p)
• The entropy units are in bits and
the binary logarithm is used.
• If the common logarithm is
used, then the units are dits.
• And it is nits in case natural
logarithm is used.
• This was defined by Batty,from the
information theory basis presented by
• Where,xi represents the spatial interval
size.This spatial component is implemented
and this equation is more applicable in
spatial analysis, such as comparison
between different regions.
ENTROPY AS A USEFUL PART OF A
SPATIAL VISUALIZATION AND
• This figure shows the dependency between
the number of observation and the
outcoming kriging estimators of the spatial
phenomenon. Here ,there are number of
observations that gives us no useful
information for the outcoming spatial
• This figure shows the dependency of the
outcoming amount of information, measured
with the help of the entropy function on the
number of modelled points.
• The last part is the spatial visualization of
the natural phenomena.In this fig., the
function of spatial entropy is the convex
function and therefore there exists points
where the next added interval to the
visualization of the spatial phenomenon
gives us much less information like the
• In this there are 4 GRID layers of the same
climatic phenomemon but with different
numbers of intervals,increasing from upper
left to lower right picture.Each layers
contains various amount of
information,depending on various factors.
• The information theory,entropy and its
spatial form is widely used in geographical
• Entropy concepts can be used to investigate
channel networks. Much of the work
employing entropy concepts in hydrology
have been done with reference to the