Open Journal of Marine Science, 2011, 3, 73-81
doi:10.4236/ojms.2011.13008 Published Online October 2011 (http://www.SciRP.org/journal/ojms)
Parametric Linear Stochastic Modelling of Benue River
Flow Process
Otache Y. Martins, Isiguzo E. Ahaneku, Sadeeq A. Mohanned
Department of Agricultural & Bioresources Engineering, Federal University of Technology, Minna, Nigeria
Received May 11, 2011; revised May 30, 2011; accepted July 2, 2011
Abstract
The dynamics and accurate forecasting of streamflow processes of a river are important in the management
of extreme events such as floods and droughts, optimal design of water storage structures and drainage net-
works. In this study, attempt was made at investigating the appropriateness of stochastic modelling of the
streamflow process of the Benue River using data-driven models based on univariate streamflow series. To
this end, multiplicative seasonal Autoregressive Integrated Moving Average (ARIMA) model was developed
for the logarithmic transformed monthly flows. The seasonal ARIMA model’s performance was compared
with the traditional Thomas-Fiering model forecasts, and results obtained show that the multiplicative sea-
sonal ARIMA model was able to forecast flow logarithms. However, it could not adequately account for the
seasonal variability in the monthly standard deviations. The forecast flow logarithms therefore cannot read-
ily be transformed into natural flows; hence, the need for cautious optimism in its adoption, though it could
be used as a basis for the development of an Integrated Riverflow Forecasting System (IRFS). Since fore-
casting could be a highly “noisy” application because of the complex river flow system, a distributed hydro-
logical model is recommended for real-time forecasting of the river flow regime especially for purposes of
sustainable water resources management.
Keywords: Stochastic Process, Water Resources, Dynamics, River Flow, Modeling
1. Introduction
Inherent in the principles of water resources management
is the judicious utilization and conservation of the avail-
able water resources. One of the ways to enhance this is
the proper estimation of water demand both quantita-
tively and qualitatively. Within this overall management
system, the hydrologist is often required to estimate the
magnitude of extreme events, whereas operation of some
of the design works is often dependent on reliable esti-
mates of flow for an ensuing period of time. Since river
is an essential component of the hydrologic cycle, its
flow forecasting provides a veritable, and basic informa-
tion on a wide range of problems related to the design
and operation of the entire river system. A very common
constraint encountered in the context of water resources
planning is inadequacy of streamflow records. The
available streamflows, known as historical records, are
often quite short, generally sometimes less than a quarter
of a century in length. Thus, a system designed on the
basis of the historical record only faces a chance of being
inadequate for the unknown flow sequence that the sys-
tem might experience. The historical record comprising a
single short series does not cover a sequence of low
flows as well as high flows. Hence, the reliability of a
system has to be evaluated under these conditions which
are not possible with historical records alone.
Statistically, the historical record is a sample out of a
population of natural streamflow process. Thus, the gen-
erated flows are neither historical flows nor a prediction
of future flows but rather are representative of likely
flows in a stream or river. Streamflow, being a natural
phenomenon, has a random component, though not fully
random since it has been observed that it exhibits het-
eroscedastic behavioural pattern. Forecasting river flow
in general or after heavy rainfall event is important for
public safety, environmental issues, and water manage-
ment. For these purposes, mathematical models have
been developed based either on physical considerations
[1-4] or on statistical analysis [5-7]. Conventional mod-
els for streamflow forecasting typically involve a number
of physical variables that function as inputs. A physical

OTACHE ET AL. 74
variable that is not very useful for forecasting on its own
can often be useful when used in conjunction with other
variables. Given the number of physical variables that
could be considered as potentially relevant, it is apparent
that a very large number of different combinations of
both variables and mathematical relationships that link
them together are available when developing a stream-
flow forecasting model. Determining an appropriate
model structure by trial-and-error process is therefore not
always practical [8].
The non-practical determinate nature of model struc-
ture for streamflow/river flow forecasting can really be
appreciated in a wider context considering the fact that
river flow is usually treated as a random process, purely
stochastic. The justification is that river flow is a func-
tion of precipitation and other processes which, at pre-
sent level of knowledge, seem to evolve randomly in
time and space. Even if the underlying phenomena and
their interactions were thoroughly understood, it would
not be possible to describe mathematically the rate of
discharge in a natural water course without involving
unsystematic or unknown effects [9]. Considering the
issues involved in river flow studies within the premise
of a wider hydrological horizon, it is pertinent to appre-
ciate the following seemingly, contemporaneous para-
doxes:
1) In the face of the stifling dearth of long and con-
tinuous data availability, can realistic generalizations be
made from forecasting the dynamics of the Benue River?
2) Considering the complex nature of river flow and
the significant variability it exhibits in both time and
space, what is the appropriateness of using stochastic
method for modelling the Benue River flow process?
To this end, the objective of this study is to model the
streamflow process of the Benue River with Autoregres-
sive Integrated Moving Average (ARIMA) models, fo-
cusing on short term forecasting for the purposes of
evaluating suitability of particular model type as a pre-
liminary step towards developing an enhanced “River
Flow Forecasting System” for the river.
2. Materials and Methods
1) Hydrology of the Benue River
The Benue River is the major tributary of the Niger
River. It is approximately 1 400 km long and almost
navigable during the rainy season (between July and Oc-
tober). Hence, it is an important transportation route in
the regions it flows through. Its headwaters rises in the
Adamawa Plateau of the Northern Cameroon, flows into
Nigeria south of the Mandara Mountains through the
east-central part of Nigeria before entering the Niger
River at Lokoja (Figure 1a). The wide flood plain is
used for agriculture, with main crops being sugar cane
and rice. There is only one high-water season because of
its southerly location; this normally occurs from May to
October, while on the other hand, the low-water period is
from December to June. Figure 1b explains the hydro
logical flow regime of the Benue River in line with the
general climatic pattern. There are definite wet and dry
seasons which give rise to changes in river flow and sa-
linity regimes. The flood of the Benue River (upper,
middle, and downstream) lasts from July to October, and
sometimes up to early November.
2) Data Base Management
In this study, historical time series for gauging stations
at the base of the Benue River (i.e., Lower Benue River
Basin) at Makurdi (7°44′ N, 8°32′ E) was used. A total of
26 years (1974–2000) water stage and discharge data
were collected and used. The daily flow data were ag-
gregated to monthly and annual data series by taking the
average of each month’s flow and calendar year. Simi-
larly, the annual maximum and minimum daily average
discharges were obtained according to the water year, i.e.,
months of April to March for the streamflow process.
3) Model Formulation and Forecast Strategy
The possibility of fitting a multiplicative seasonal
ARIMA model to the logarithms of the monthly flows was
examined. The forecasts from this model were compared
to forecasting using a conventional Thomas-Fiering model.
Comparison of forecast errors was also performed to
bring to the fore the suitability of either of the models for
forecasting the streamflow process of the river. Model
formulation and development was patterned after Box
and Jenkins [1], Carlson et al. [11] and McKerchar and
Delleur [12].
a) Thomas-Fiering Model
Thomas and Fiering [13] described a linear stochastic
model for simulating synthetic flow data. On a monthly
basis, this represents the means, standard deviations,
serial correlations between successive flows, and the
skewness. This model uses a linear regression relation-
ship to relate the flow
1t
Q

in the (t+1)th month, (t be-
ing from the start of the generated sequence) to the flow
t
in the t(th) month. If Q
j
Q and
1
j
Q

be the mean
monthly discharges during months j and j+1, respectively,
within a repetitive annual cycle of 12 months,
j
b be the
regression coefficient for estimating the flow in the
(j+1)th month from the jth month, and tt be a normal de-
viate with zero mean and unit variance, the Tho-
mas-Fiering equation will be

 
12
2
11 1
1
tjjtjtj j
QQ bQQtr 
 
 (1)
If an average first-order serial correlation coefficient r1
is used to replace the 12 monthly rj values, it can easily
be shown using the relationship

OTACHE ET AL. 75
(a)
(b)
Figure 1. (a) Map of Nigeria showing Benue River and its traverse; (b) general hydrological year flow regime

OTACHE ET AL. 76
t
tj
QQ
y



j
t
a 
(2)
That is, the model (1) is the first-order case of the
general non-seasonal autoregressive model
1
p
tjtj
j
yy 



 (3)
where yt, yt-j, σj and фj represents the transformed (i.e.,
standardized) series; transformed series at the previous
time step, standard deviation of each month, and autore-
gressive parameter value of order one for each month,
respectively. In the application of the Thomas-Fiering
model, negative values are sometimes generated. It is
recommended that these values be retained and used to
derive the subsequent values in the sequence, and when
once the generated sequence is completed, all the nega-
tive values in the generated sequence be replaced by zero.
Similarly, if there is no occurrence of flow for a particu-
lar month, then generation of flow for such a month may
not be carried out. Since there is flow all year round in
the Benue River, this procedure was ignored.
b) AR IMA Analysis of the Monthly flow data
To be able to identify the most suitable model to fit the
flow series, serial correlations were calculated for possi-
ble differencing schemes d = 0, 1, 2 and D = 0, 1, 2,
where d and D stand for non-seasonal and seasonal dif-
ferencing, respectively. Figure 2 shows the autocorrela-
tion function plots for these differencing schemes.
To account for runoff phenomenon in the streamflow
data, the prospect of seasonal differencing seem more
promising since seasonality cannot really be accounted
for by non-seasonal differencing, nor is an integrated
moving average scheme expected to account for the
non-seasonal autoregressive behaviour. Thus considering
this factor, a multiplicative ARIMA model




12
1, 0,21,1,1, was examined. This model has the
form




 
1212 2
11,121,121 2
111 1
tt
BBz BBB     a
(4)
where,
i

,
i

, and
i

stand for non-seasonal autoregres
D = 0, d = 0 D = 1, d = 0
D = 0, d = 1 D = 1, d = 1
Figure 2. Estimated autocorrelations for logarithmic differenced monthly flow series.

OTACHE ET AL. 77
sive, seasonal autoregressive, and seasonal moving av-
erage parameters, respectively; while z
t
and a
t
are loga-
rithmic transformed series and model random shocks,
respectively.
c) Flow Forecasting
The ARIMA model was used to forecast flows for one
to 24-month ahead. With reference to an origin at time t
(here, t = 288), the model was used to make minimum
mean square error forecasts of z
t+L
for, where L is
the lead time. The values forecasted for z
t+L for an origin
at t with lead time L will be written as
1L

ˆ
t

Z
L. Diagnos-
tic verification of the adequacy of the model was done by
evaluating the autocorrelation function for the residuals
by modifying the model to take into account any
non-random features. Figure 3 shows the residual auto-
correlation function for model

12
in re-
spect of parameter estimation (Table 1) and the final
parameter values (Table 2) as well as the corresponding
diagnostic check for model adequacy (Table 3). At 5%
level of significance, the autocorrelation plot of the
model residual reflects that the residual series may be
considered random (Figure 3).

1, 0,21,1,1,
Figure 3. Residual autocorrelation function for ARIMA




12
1, 0,21,1,1, model.
Table 1. Estimation of ARIMA model parameters.
Iteration SSE
1

1,12
 **
1

2

1,12
**
0 64.5264 0.100 0.100 0.100 0.100 0.100
1 54.8937 0.012 0.055 -0.050 0.089 0.145
2 50.5089 -0.056 0.152 -0.151 0.071 0.295
3 45.4580 -0.162 0.171 -0.301 0.037 0.386
4 42.0178 -0.275 0.149 -0.451 -0.007 0.424
5 39.6586 -0.394 0.124 -0.601 -0.055 0.447
6 36.2933 -0.244 0.075 -0.507 -0.071 0.479
7 36.1324 -0.094 0.072 -0.359 -0.035 0.480
8 35.9804 0.056 0.069 -0.212 0.003 0.481
9 35.8173 0.206 0.066 -0.064 0.041 0.482
10 35.6239 0.356 0.063 -0.083 0.081 0.484
11 35.3723 0.506 0.059 0.231 0.120 0.486
12 34.9998 0.656 0.054 0.377 0.161 0.488
13 34.2944 0.806 0.043 0.521 0.204 0.494
14 32.1154 0.956 -0.002 0.642 0.253 0.519
15 28.0190 0.943 -0.102 0.544 0.269 0.699
16 27.2837 0.946 -0.181 0.486 0.266 0.700
17 27.2450 0.954 -0.179 0.483 0.270 0.719
18 27.2367 0.957 -0.173 0.484 0.274 0.730
19 27.2331 0.958 -0.168 0.485 0.276 0.738
20 27.2317 0.959 -0.166 0.486 0.277 0.742
21 27.2317 0.960 -0.166 0.486 0.278 0.743
22 27.2312 0.960 -0.166 0.486 0.278 0.745
** is seasonal autoregressive parameter; is seasonal moving average parameter
1,12

1,12


OTACHE ET AL. 78
Table 2. Final model parameter estimates.
Parameters Statistics
Type Coef SE Coef T P
AR 1 0.9600 0.0284 33.83 0.000
SAR 12** -0.1657 0.0748 -2.22 0.028
MA1 0.4859 0.0667 7.29 0.000
MA 2 0.2782 0.0646 4.31 0.000
SMA 12* 0.7447 0.0544 13.69 0.000
Constant 0.000318 0.001155 0.28 0.783
** Seasonal autoregressive parameter, * seasonal moving average parameter
Table 3. Modified Box-Pierce (L jung-Box) Chi-Square statistic.
Lag 12 24 36 48
Chi-Square 10.2 29.0 46.3 60.4
Critical value 12.6 28.9 43.8 58.1
DF 6 18 30 42
In terms of the forecasting function, the general
ARIMA model can be written in three alternative forms:
as a difference equation, an infinite sum of the current
and weighted previous values of shocks at, and an infi-
nite sum of weighted previous observations plus the cur-
rent value of at. Conditional expectation of any of these
forms supplies a forecasting function. In this regard, the
difference equation was used. By recalling that
 

tt
ZL Z


L
, using square brackets to signify condi-
tional expectation, noting that


1
11
0,1, 2,
ˆ1, 2,
ˆ10,1,2,
0
tjtj
tt
tjttj tj
tj
zz j
zzjj
aazzj
a











 







1
,2,j




t
a
(5)
and taking expectation of the model, which has the
general form

 
1212 2
11,121,121 2
111 1
t
BBz BBB    ,
the forecasting function can be obtained according as:
111,12 1211,1213
11 221,1212
1 1,121321,1214
tL tLtLtL
tL tLtLtL
tL tL
zz zz
aa aa
aa




 
 
 
 
 
 

the model fit was done with 26 years of flow data). The
3. Results and Discussion
Figure 4 shows the behaviour of the ARIMA model
(6)
Equation (6) can be expanded for the respective lead
time (L) to make the forecasts with zt+L, the dependent
forecast variable as a function of L. Both the Thomas
Fiering and ARIMA models were used to make forecasts
of the monthly flow series. Subsequently, the forecasts
from the models were compared with the actual flows.
Because the last 2 years flow data was used for the com-
parison, the parameters were re-estimated for both mod-
els for the entire flow series shortened by 2 years (i.e.,
flow forecasts were considered from the aspect of
choosing a particular time origin and taking cognizance
of the behaviour of the forecast function as the lead time
L increases; that is, the long-term behaviour of the fore-
cast function should be a useful theoretical check on the
fit of a model. Taking the origin t = 288, forecasts for the
logarithms of the flow were made using both models.
forecast function; the forecasts are quite close to the
monthly means. Baring data quality problems, stationar-
ity issues, and model over-fitting, for an ideal forecast
function, this behaviour is to be expected. Forecasts in
the distant future for a trend-free series should be the
unconditional estimates of the means. Figure 5 indicates
that the forecasts are all within the bounds with respect
to actual flows. Based on this, and taking into considera-
tion the data size for model fitting, the ARIMA model
has reproduced the monthly means well. Figure 6 illus-
trates the standard errors of the forecasts. This figure
compares the monthly standard deviations of the loga-
rithms of the monthly flow with the standard errors for
forecasts of the two models under discourse, respectively,
for 124L

. As L becomes large (say, greater than 4),
the error of a forecast for Thomas-Fiering
model, tends closely to that of the historic flow, whereas
the ARIMA model deviates away significantly. The be-
haviour of the Thomas-Fiering model in this regard is
further explained by Figure 7, where it was used to
simulate the flow regime for 26 years. It was able to re-
produce the flow dynamics clearly well. This attribute
reinforces its suitability to be used for long-term flow
forecasting of the Benue River. The failure of the
ARIMA model to account for the seasonal pattern in the
standard deviations is a major limitation of the model. In
standard

OTACHE ET AL. 79




12
1, 0, 21,1,1, modelFigure 4. Forecasting for flow logarithms of ARIMA .




12
1, 0, 21,1,1, moFigure 5. Forecast pattern of the ARIMA del.
Figure 6. Monthly standard deviation of logarithms and forecast errors for Thomas-Fiering and ARIMA



12
1, 0, 21,1,1,
models.

OTACHE ET AL. 80
Probability of Exceedance or Equalled (%)
Figure 7. Long-term flow-duration curve of Tynthetic flow simulation).
articular, it leads to problems in transforming forecasted
. Conclusions
ased on the results of analysis done, it is evident that
homas-Fiering model (S
p
flow logarithms into natural flows.
4
B
autoregressive and ARIMA models have an important
place in stochastic hydrology. Specifically, logarithms of
monthly flows may be represented either with a
low-order autoregressive model (if the series are first
standardized) or with a multiplicative seasonal ARIMA
model of the order



12
1, 0,21,1,1, . The stochastic
models (Thomas-Fier

ing and ARIMA


12
1, 0,21,1,1, )
may be used for forecasting of the Benuey
flow, though the former performed relatively better than
the later. The ARIMA model was able to forecast flow
logarithms, but because it did not adequately account for
the seasonal variability in the monthly standard devia-
tions, the standard errors associated with the forecasts
may not be physically correct. Also, the logarithms can-
not be correctly transformed into natural flows; thus giv-
ing concern for cautious optimism. However, the sto-
chastic modelling does show that the ARMA type mod-
els could be used as preliminary models which may form
the basis for understanding the dynamics of the stream-
flow process. For the purposes of developing real-time
Integrated River flow Forecasting System for the Benue
River within the overall context of water resources man-
agement strategy, consideration should be given to dis-
tributed river flow hydrological models that incorporate
hydroclimatic forcing. It suffices to note also that the
appropriateness of the stochastic process for every flow
series may be debated in the context of nonlinear deter-
minism and chaos, according to which seemingly com-
plex and irregular behaviours could be the outcome of
simple deterministic systems with only a few nonlinear
initial conditions. On this basis, nonlinear deterministic
methods could be viable complement to linear stochastic
ones for studying river flow dynamics if sufficient cau-
tion is exercised in their implementation and interpreta-
tion of results.
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