Anatomy of a Hydrological Model: A case study of the Athi and Tana River
Basins
Balla Maggero
Marine & Oceanography Services. Ngong road 30259 Nairobi. e-mail: nambuye@yahoo.com
Abstract: An algorithm is presented for reconstructing historical rainfall from few observation sites within
the Athi-Tana river basins (coastal Kenya). The Kalman filter Hydrologic Atmosphere Model (KHAM)
algorithm is based on the state-space model that constitutes two modules; the first one for determining
the state of the hydrologic phenomena and the system noise, and the second one for determining
observation noise. Using observed data from only six locations as input parameters in the KHAM
algorithm, both spatial and temporal distribution of the hydrologic state is estimated for the whole basin.
On the other hand, it is shown that the KHAM can be a suitable tool for rainfall and flow prediction in the
ungaged basins. Verification of this results is made with the actual observations.
Keywords: state-space model, system noise, observation noise, Kalman filter,
1 Introduction
Inundation of vast alluvial lowlands is a complex
process involving local generation of runoff from
rainfall or precipitation, flow from surrounding
highlands, overbabank, spilling along floodplains
channels, and tidal and storm influences near
coastlines. The interplay of these processes
depends upon the timing of hydrologic events
and their alteration by engineered structures. A
contributing factor is the intricate geometry of
the low-lands themselves, composed of tidal
channels, levees, distributory channels, scroll
bars, and floodplain lakes. Predicting areas of
inundation is limited by inaccuracies of
hydrologic routing in areas of low relief and of
poor spatial resolution of precipitation data.
To asses, forecast and predict hydrologic
responses, it is vital to understand how the
amount, rate and duration of precipitation are
distributed in both space and time. As such it is
also a key to improving the specifity, accuracy
and reliability of weather/climate forecasts.
Precipitation climatology may come to be
important as are temperature anomalies in the
current debate about climate change and its
impact on critical human infrastructures because
of those systems’ sensitivity to rapid change
(Shuttleworth,1996). Outside of the developed
world, hydrologic data are the newly endangered
species because of the declining number of
ground-based observation sites.
It’s obvious that it’s impossible to create models
that would reproduce to the perfection the
behaviour of nature. For a selected study basin,
the accuracy of collected data has the same
importance as the model itself. Even when the
fastest computers are used, their solutions are
only approximation; they cannot cope with the
complexity of the world and especially in the ability
to capture all the details of the hydrologic system
to be modelled. Therefore, data assimilation
methods to integrate all the current and potentially
available information hold great promise. These
methods in hydrology uses algorithms that have
been tailored for signal processing, oceanography
or meteorology. The models generate an effective
simulation for exploring wetland processes and
provide quantitative predictions and field
interpolation via model physics, which ultimately
increase our insight on the processes on the
ground (Troch et al., 2003). The Kalman filter is a
one most well-known sequential data assimilation
scheme. It has been commonly used in stream
flow modelling (e.g. Chui and Chen, 1999). Also,
some recent studies in assimilating remotely
sensed rainfall observations with Kalman filtering
reveal that short-term corrected forecast
represents improvement over raw model results
within a limited spatial domain The objective of this
article is to introduce the standard Kalman Filter in
its original form (Brown, and Hwang, 1996), herein referred to as the Kalman Hydrologic
Atmospheric Model (KHAM) and presents some
preliminary results for the Tana-Athi river delta
using six gage points.
2. The Probabilistic Model
The KHAM method employs a Kalman filter in a
linear fashion. It is a mathematical procedure that
provides an efficient computational (recursive)
method for least square estimation of a system.
It does so in a predictor-corrector style (see fig.
1), predicting short-term (from observed to
unobserved in both space and time) changes in
the state using a state space model. The time
update projects the current state estimate ahead
in time. The measurement update adjusts the
projected estimate by an actual measurement at
that time. The state-space constitute two
modules, the first being the dynamical (system)
model, which expresses the state of hydrologic
phenomena, and the other is the observation
equation.
measurements in y. The weights are determined
by the error covariance for the model predicted
onto the measurement, the measurement error
covariance and the innovation : y - Hf.
4. The KHAM Algorithm
Given an initial state estimate f and error
covariance estimate Pf, the KHAM algorithm
proceeds through the following steps whenever
measurement y is provided.
Figure 1. The Kalman filter cycle.
The dynamic model is governed by the linear
stochastic relation:
 (t +t) =  (t) +G (t)…………1
(t) is state vector (rainfall in this case) at time t.
The observation equation is expressed as;
y(t) = H(t) + v(t)……………..2
y is the observation vector at time t ,
where random variables (t) and v(t) are white
Gaussian noises with zero mean and
covariances Q and R, respectively. A, G and H
are matrices with appropriate sizes.
5. Study Area
3. The variance-minimizing analysis
The model is integrated forward in time and
whenever measurements are available, these
are used to re-initialise the model before the
integration continues. Neglecting the time index
the model forecast, and analysis are denoted as
f and a
respectively. The respective
covariances for model forecast, analysis and
measurements are denoted Pf , Pa and R. The
analysis equations is represented as:
 a =  f + PfHT(HPfHT + R)(y – H f)……..3
the analysis error covariance are given as:
Pa = Pf - PfHT(HPfHT + R)-1H Pf……..4
a
is determined as a weighted linear
combination of the model prediction f and
covariances PfHT corresponding to each
Figure 2. Map of Kenya and the location of the study area
the blue box inside the projected map.
within
The study area selected for this test of the model
and the analysis of rainfall prediction is the coastal
Athi-Tana Delta (Kenya) where the two rivers meet
the Indian ocean (see fig. 2). The hydrology of this
area can best be viewed by examining the
drainage patterns of these rivers that extent far
from the coastal hinterlands. The Tana discharges
an average of 4,000 million cubic metres of
freshwater and about 3 million tonnes of sediment
into the ocean annually. Before it enters the ocean
it gives off a branch which leads to the complex
system of tidal creeks, floodplains, coastal lakes
and mangrove swamps. Its delta covers some
1,300 km2 behind a 50m high sand dune system,
which protects it from the open ocean in
Ungwana Bay. In total, the high sediment loads
carried by the Tana and Athi rivers are
attributable to floods caused by both heavy rains
and poor land use practices. Such a high rate of
sediment discharge are a threat to marine and
coastal ecological biotopes.
estimation of rainfall and its consequent mapping.
The model is based on the dynamical assimilation
of sequences of gage observation from different
site points. The application of the KHAM for 4-year
run has shown a consistency. The assimilation
model falls within the range of values measured
6. Results
Figure 2. Autocorrelation of the resulting noises
on the ground. Since it is impossible to set up
uniform network of rain-gages, algorithms such as
the Kalman filter may play a vital role in producing
realistic rainfall distributions and filling missing
gaps. Therefore, the synergistic use of a modelbased information is now of paramount
importance. Indeed, , monthly spatial and temporal
distribution of past rainfall estimated is in good
agreement with observed data.
Figure 1 comparisons of monthly observed and model results for
the six station
Figure 3 showss the relation between the
observed and model results. Except for the third
station which is showing some differences
during the March to July, the rest depicts a close
fit, with the first , second and the fifth revealing
a small overestimation. The fourth and the sixth
station have an underestimation The observed
amplitude variations are not exactly fitted due to
considerable noises (residuals). These noises
should be random and with a normal distribution.
The autocorrelation results are illustrated in Fig.
4
7.0 Summery
The Kalman filter discussed here allows
Acknowledgment: This work was performed within the
framework of the Oceanography & Marine Service of the
Kenya Meteorological Department. The author would
like to thank Mr. Kibue, with whom they have had many
fruitful discussions, and Dr. Chadih for reviewing this
manuscript and many constructive comments.
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