INTRODUCTION
One of the most efficient overland flow routing methods, is TimeArea
method (Ponce, 1989; Maidment, 1993; Singh, 1996). In this method, with
omitting the storage effects, watershed is divided into some subareas.
This is performed by constructing isochrones. An isochrone is actually
a contour which passes through points of the same travel time to the outlet
of the basin. Histogram of subareas, named TimeArea Histogram (TAH) is
the base of the TimeArea method as a RainfallRunoff model.
Time to equilibrium is actually the time of wave translation. All of
the area behind the wave front is in equilibrium state, which in hydraulics
of surface flow means the considered area is in steady state. For a long
duration rainfall, time to equilibrium at outlet is equal to time of concentration.
The latter is familiar to hydrologists. Time of concentration according
to definition, is the time which a drop of water needs to reach to the
outlet from farthest point of watershed (Ponce, 1989; Chow et al.,
1988). For constructing TAH, time to equilibrium must be divided into
some equal parts, say Δt. This Δt will be used as the time difference
between isochrones.
Researches on isochrone delineation methods are very limited. Henceforth
almost all of the available isochrone mapping methods are empirical, approximate
and without a well defined hydraulic basis. This issue introduces errors
in hydrograph calculation, which are not clear in origin and magnitude.
In the present research this shortcoming of TimeArea method was investigated.
Shokoohi and Saghafian (2007) had performed a research on precision of
isochrone mapping methods in one dimensional (1D) flow. They showed that
all of the methods define travel time (time to equilibrium for any desired
point; t_{e}) as a function of travel Length (L) at a power (t_{e}
≈ L^{β}). For kinematic wave, β was obtained as
5/3 and for all of the other methods the power was in a range as 0.5 to
1.5. Quoted from Shokoohi and Saghafian (2007) TimeArea method could
give results as precise as analytical method if β = 1.5. In this
research isochrones arranged from upstream to downstream. These conclusions
have a conflict with that of kinematic wave theory.
Regarding the above literature a new research was conducted on 1D (parallel
flow). The achieved results confirmed new concept of reordering isochrones
and then finding a robust hydraulic based method for isochrone mapping.
A noticeable outcome of the present study was the point that the best
result was achieved by application a value of β which comes from
kinematic wave theory.
There are two methods for studying spatial phenomena; Lumped modeling
and Distributed modeling. In Lumped modeling, watershed is considered
as a black box. In this box, rainfall is accepted as input and yield runoff
as output at outlet. In this method which can be categorized as conceptual
modeling, rainfall is related to runoff by an InputOutput model such
as Unit Hydrograph model, which supposes a uniform rainfall throughout
the watershed.
Lumped models without considering the spatial distribution of important
properties of watersheds, take an average value for each parameter (Beven
and O`Connell, 1982). In distributed modeling, in contrast to lumped modeling,
modeller tries to describe and distinguish spatial pattern of all of the
parameters contributing in hydrologic processes (Maidment, 1992). Almost
in all of the situations, problem is solved by dividing solution domain
into some squares (Raster System) or triangles TIN (Triangulated Irregular
Network) System.
To achieve high precision, one needs to use distributed model which always
suffers lack of one of the most important features of any efficient mathematical
model; meaning simplicity (Beven, 1985). A distributed model often needs
many geomorphologic related parameters such as slope, plant coverage,
distribution of channel network, drainage density and infiltration capacity
which are not inherently uniform (Beven, 1985). Simplicity of a model
is an important property which makes it to be capable of forecasting reasonably.
This property makes a model feasible and costeffective. Simplicity of
rainfallrunoff models, especially when used in real time flood forecasting
framework, is an important feature of such models.
These advantage and disadvantage encourage engineers to use intermediate
models. In this relation Kite et al. (1996) mentioned that when
one wants to use a model to simulate a big and high extent watershed,
Lumped models are not feasible and distributed models are not applicable.
Quoting from Kite (1996) there are many benefits in application of intermediate
models. For example such a model can obey the most important physical
rules and meanwhile remains simple. Kite believed that intermediate models
are capable of simulating watershed behaviour at any point and meanwhile
do not need the excess amount of data which are necessary for distributed
models (Kite et al., 1996).
Actually, TimeArea method is the most efficient semidistributed model
which has been developed in 1940`s. This method is known as a hydrologic
watershed rainfallrunoff model. After developing and applying in Clarke
conceptual model, it was used by specialists in very limited area. The
main cause of this limitation was shortcoming of isochrone deriving methods.
Paying attention to its power and capabilities were commenced after fast
development of computer science and GIS (Geographical Information System)
software. One of the most important advantages of TimeArea method is
including two important geomorphologic properties of watershed; shape
and drainage pattern of basin in its simulation. TimeArea method uses
these two watershed properties in determining shape and peak discharge
of flood hydrograph (Anonymous, 2000). It must be stated that TimeArea
method success in rainfallrunoff simulation is mainly dependent on precision
of isochrone mapping. According to available reports, nowadays in 40 to
60% of Corp of Engineers (USA) projects, TimeArea method is used as rainfallrunoff
model (Kull and Feldman, 1998).
There are very limited researches in this domain and unfortunately almost
all of the available isochrone mapping methods are experimental, approximate
and do not have a well defined hydraulic base. To beginning the discussion
and meanwhile illustrating importance and depth of the conducted research,
a short review of theoretical basis of TimeArea and isochrone mapping
methods is represented in the following section.
MATERIALS AND METHODS
General equation of TimeArea method which gives a net hydrograph due
to an effective rainfall is as follows:
Where:
j 
= 
Time step 
Q 
= 
Discharge 
I 
= 
Effective rainfall intensity 
A 
= 
Area between two consecutive isochrones 
According to Eq. 1 , for applying TimeArea method,
one should first delineate isochrones. Main purpose of this article is
investigation on the inherent errors of the available experimental methods
and meanwhile introduces a substitute hydraulic based method. In the next
section these methods with a review on kinematic wave theory as analytical
solution to the problem are presented. It is illustrated that in all of
the methods, travel time to the basin outlet can be presented as the travel
length at a power (Shokoohi and Saghafian, 2007).
Equal velocity method: Equal velocity method was introduced by
Pilgrim (1977) at first. In this method TimeArea Histogram (TAH) is obtained
by supposing uniformity of velocity throughout the watershed (Pilgrim,
1977). In this technique one can write:
Where:
t_{w} 
= 
Travel time of any point to the basin outlet 

= 
Travel length 
All of the methods which Maidment (1992 and 1993), Ajward and Muzik (2000)
and Melesse et al. (2003) developed for flow tracing and also the extension
which works in ILWIS (GIS) (Donker, 1993), are constructed on the basis of this
concept.
Laurenson method: The basic assumption of this method is the proportionality
of travel time to this proportion;
in which L is travel length and S is average slope of flow path. On the
other hand average slope of any route is equal to length of route divided
by height difference between start and end point of the route. Then one
can write:
In this method for isochrone delineation, at first step some points are
determined on elevation contours and then ordered according to ^{
1.5}. By new ordered points one can draw isochrone lines.
Empirical method of HEC1: HEC1 has a preassumption of TimeArea
function and takes it as a parabola. This curve is used in absence of
any physiographic data (Anonymous, 1991). The mentioned equation is as
follows:
A = 1.414t^{1.5} ... (0≤
t < 0.5) 
(4) 
A = 1 1.414(lt)^{1.5}
... (0.5≤ t < 1) 
(5) 
In which A = Dimensionless area, meaning proportion of considered area
to total area and t = Dimensionless time, meaning proportion of the considered
area cumulative time to the time of concentration. If above equations
are solved for
in a rectangular plane (one dimensional/ parallel flow) one can write:
tw ≈ ^{
0.67} 
(6) 
Kinematic wave modeling: Kinematic wave equation is a simplified
form of Saint Venant equations.
If all terms of momentum equation except S_{0} (bed slope) and
S_{f} (energy line slope) are omitted in the Saint Venant equations,
it is reduced to a uniform flow equation. A general form of resistant
equation can be used as follows:
In which Q is the discharge, A_{x} is the flow cross sectional
area and α and β are coefficients. For Manning equation, α
and β are given as follows:
In which n and P are Manning`s roughness coefficient and wetted perimeter,
respectively. By inserting resistance equation into the continuity equation,
one obtains:
Which,
i 
= 
Excess rainfall intensity, 
h 
= 
Flow depth, 
q 
= 
Unit width discharge and 
c 
= 
Kinematic wave celerity. 
Derivation of dimensionless relations for 1D flow: Consider a
rectangular plane of length L and slope S_{0 }on which a net rainfall
with intensity of I and long duration falls. The latter specification
about rainfall (t_{r} ≥ t_{e}) is for guarantying the
whole plane to reach to equilibrium state (Mahmood and Yevjevich, 1975).
According to Eq. 9, if t_{0} is the time which
a wave needs to translate a length x_{0} (from upstream) when
the entire plane behind the wave front is in steady state, one can write:
Discharge at the end of plane at time t_{0}, will be equal to:
To delineate isochrones by applying kinematic wave theory, it is enough
to obtain x_{0} in Eq. 9, for any desired number
of time interval say N. Henceforth isochrones are consecutive lines which
their time difference is .
For example, x_{j} corresponding to the jth isochrone
is derived from Eq. 10 and then Eq. 11
is solved for x_{j} = x_{0}.
Regarding the fact that the plane totally will be translated in a time
equal to t_{e}, one can write:

Fig. 1: 
Isochrones relative position corresponding to different
exponents 
Saghafian and Shokoohi (2006) obtained the following equation for calculating
rising limb of a flood hydrograph, by applying TimeArea method and arranging
isochrones from upstream to downstream:
Where, q_{(tk)} = Discharge at time t_{k} at the end
of the plane.
By using such a methodology, the Eq. 13 when arranging
isochrones from downstream to upstream is changed as follows:
As it is seen in all of the methods, t_{w} (wave travel time)
is proportional to L (travel length) at a power. To envelope all of the
possible conditions, isochrones for a range of 0.5 to 1.5 were obtained
and rising limb of hydrograph was calculated.
Figure 1 shows 10 consecutive isochrones for different
methods (powers). It is obvious that the less the power of L, the more
will be the distance between isochrones toward the upstream edge of the
plane. In TimeArea method this means that the less the power of L, the
less will be the contribution of watershed in supplying discharge at the
start of the storm.
To determine the errors of the methods in comparison with the analytical
solution, relative error was obtained by the following equation:
In which RE is the relative error, q is the discharge corresponding to
each exponent, q_{kw} is the KW discharge and t_{k} is
the time.
RESULTS AND DISCUSSION
To calculate a flood hydrograph due to a long duration and spatially
uniform storm, with unit intensity throughout a rectangular plane with
unit width, isochrones were delineated using Eq. 10
and 12 in first step. Then by applying TimeArea method
for 4 conditions of different powers and also by solving the problem with
kinematic wave equation independently, rising limb of the hydrograph were
calculated (Fig. 2). Ordinates shows relative discharges
(any discharge relative to equilibrium discharge; iL) and abscissa shows
relative time (any time relative to time to equilibrium).
As it is seen from Fig. 3 the less the power of travel
length, the closer were the results to analytical solution. This issue
means Laurenson method is not a reliable method for isochrone mapping.
Henceforth in parallel (1D) flows by choosing β = 1.67 ()
hydrograph derived by TimeArea method application coincides precisely
with that of kinematic wave. As it is seen, relative error of all of the
methods was high at the start of storm and reduced to zero at t = t_{e}.
Finally, regarding to the trend of errors it can be concluded that the
closest result to analytical one belongs to the case which uses travel
length at a power equal to 0.67.

Fig. 2: 
Comparison of hydrographs simulated by different isochrone
mapping methods and analytical solution in 1D flow 

Fig. 3: 
Relative errors of different isochrone mapping methods
respect to analytical method in 1D flow 
This study was performed to develop a robust hydraulic based method for
isochrone mapping. The past experiments for deriving isochrones obeyed
arbitrary rules and the recent researches gave results in opposite side
of kinematic wave model results.
It seems that outcomes of the present research could solve the most important
deficiency of TimeArea method as a popular semidistributed rainfallrunoff
model.
CONCLUSIONS
Hydrograph derived by kinematic wave model application revealed that
total discharge at outlet increases by time. This issue when using TimeArea
method as rainfall runoff model means basin contribution (the area responsible
of supplying discharge at outlet) must be low in the start of storm and
then increasing gradually. Regarding the basis of TimeArea method for
deriving TAH, considering arrangement of subareas from downstream to
upstream, the case where isochrones near the outlet are closer together
gave the best result. According to this result, the worst or the lowest
precision was obtained from the method which used ^{
1.5} and the best or the most precise method was that one which used
^{
0.5} (Fig. 2and 3).
TimeArea method is one of the most suitable method for overland flow
routing. Integrating within or linking with GIS to obtain necessary basic
data gives a special place to this method. The most sensitive component
of this method is isochrone mapping which divide the basin into some subbasins.
TimeArea method uses TAH and multiplying it in intensity of a rainfall
which is uniform in space and time to obtain flood hydrograph. Regarding
the methodology of TimeArea method where diffusion and attenuation is
not considered one can say that TimeArea actually is some kind of numerical
solution of kinematic wave equation. The most important deficiency of
TimeArea method up to now has been the use of experimental approach for
isochrones mapping. The present study, succeeded in developing a methodology
with a robust theoretical basis and compatible with GIS, regarding the
capability of the latter to release sophisticated maps and deriving physiographical
properties of basin, needed in TimeArea method.
The present research tried to find out the location of isochrones as
a proportion of travel time at a power. According to the results of the
present research Laurenson method as a popular one is not recommended.
Another popular method, which was introduced in HEC1, gave better result
up to middle of time to peak of hydrograph. However this method showed
considerable errors too.
The best method, regarding the results of the present research for calculating
rising limb of hydrograph in 1D flow, was the one which uses the power
derived from kinematic wave equation. However if one tends to utilize
one of the available methods for isochrone delineation in 1D flows, Hec1
approach is recommended.