A Linear Unbiased Minimum-Error-Variance Algorithm for Marine Oil Spill Estimation
Kufre J. Bassey
Compared with other sources of pollution in the oceans, the risk of crude oil spillage to the sea presents the major threat for the marine ecology. There is no doubt the fact that there are sophisticated instruments like photoacoustic instrumentation for measuring oil in water bodies. Since oil start spreading immediately it enters water, it follows that spilled oil in water bodies is a state with incomplete information which therefore requires state estimation. This study uses Bayesian Statistics in deriving a linear unbiased minimum-error-variance algorithm for estimating the state of oil in water bodies. This is essential especially when chemical dispersant is a control alternative, given that this substance may have its own effect on marine biomass.
Received: February 25, 2011;
Accepted: June 22, 2011;
Published: September 06, 2011
Petroleum products that enter the marine environment have distinct effects,
according to their composition, concentration and the element that are considered
(Steinberg et al., 1997). The severity of these
effects varies from disastrous to no detectable impacts. Akpovire
(1989) and Yapa et al. (1994) highlighted
possible causes of oil spills to include oil transport; oil exploration; inland
navigation and oil storage facilities. Review of global oil spills incidence
shows that marine oil spills also play a significant role in the world economic
and environmental devastation (Cekirge et al., 1992).
As a case in point when oil is spilled on water, the transport and fate of
the spilled oil are governed by the physical, chemical and biological processes
that depend on the oil properties, hydrodynamics, meteorological and environmental
conditions (ASCE Task Committee on Modeling of Oil Spills
of the Water Resources Engineering Division, 1996). The processes include
advection, turbulent diffusion; surface spreading; evaporation; dissolution;
emulsification; hydrolysis; photo-oxidation; biodegradation and particulation
On the sea surface, the oil spreads to form a thin film called oil slick. The
movement of the slick is governed by the advection and turbulent diffusion due
to current and wind actions (Maged and Ibrahim, 1999).
The slick spreads over the water surface due to a balance between gravitational,
inertial, viscous and interfacial tension forces and the composition of the
oil changes from the initial time of the spill (Tkalich
and Chao, 2001).
The physical and chemical changes which spilled oil undergo are sometimes collectively
known as weathering. During this process, light fractions of the oil evaporate
water soluble components dissolve in the water column and the immiscible components
become emulsified and dispersed in the water as droplets. Thus, the severity
of environmental impacts resulting from this weathering process varies from
disastrous to no detectable effect, depending upon the quantity of hydrocarbons
released, the physio-chemical properties of oil, the nature of the incident
and its proximity to the shoreline (Lyman et al., 1990;
This threat of economic and environmental devastation by petroleum products
that enters marine environments has lead to the development of a number of clean-up
alternatives which may be grouped as: (i) recovery of oil from the sea surface
with mechanical devices (ii) dispersion of oil into the water column with chemical
dispersants and (iii) sinking of oil with heavier-than-water materials. Among
these three categories, only category one does not require the addition of chemical
to the water, yet it is not always effective, especially for longer range planning
(Dewling and McCarthy, 1981; Etkin,
1999). A significant problem in solving environmental issues is related
to the issues of integrated environmental management (Magram,
2009). Thus, the aim of this study is to develop a linear unbiased minimum-error-variance
algorithm for estimating the state of oil in water bodies which will serve as
a guide for optimal decision strategy with respect to oil spill clean-up alternatives.
CHEMICAL DISPERSANT AS CONTROL TOOL
On like other water disinfection methods (Ibeto et al.,
2010), chemical dispersants are mixtures of solvents, surfactants and other
additives that are applied to oil slicks to reduce the oil-water interfacial
tension (Clayton et al., 1993). The reduction of
interfacial tension between oil and water by addition of a chemical dispersant
promotes the formation of a larger number of small oil droplets when surface
waves entrain oil into the water column. These small submerged oil droplets
are then subjects to transport by subsurface currents and other weathering process
such as evaporation, dissolution, biodegradation, etc. It is evident in the
literature that the presence of dispersant enhanced crude oil biodegradation
(Zahed et al., 2010). The use of dispersant in
oil spill cleanup has also shown a significant reduction in the overall cleanup
cost (Etkin, 1998). Nevertheless, recent studies suggest
that toxicity from physically and chemically dispersed oil appears to be primarily
associated with the additive effects of various dissolved-phase Polynuclear
Aromatic Hydrocarbons (PAH) with additional contributions from heterocyclic
(N, S and O) containing polycyclic aromatic compounds. Additional toxicity may
be coming from the particulate or oil droplet, phase but a particular concern
stems from potential synergistic effects of exposure to dissolved components
in combination with chemically dispersed oil droplets (Humphrey,
STATISTICAL ESTIMATION TECHNIQUE
In statistics, estimation theory is a basic tool used in system and control
theory. When spill occurs in soil or sea, it constitutes part of the events
happening in that system. These events generate messages which require observations
at a receiver (Cheng et al., 2008). The messages
and the observations are stochastically related. The objective in the event
of spill is to determine a strategy or a rule which forms a best guess of the
message based on the observation. Let M be the message space which is a subset
of a Euclidean space. The message of oil spill therefore involves selecting
a message, such as the amount of oil m in the observed location from M. The
problem is to determine the value of m from the observation in some optimal
fashion. It is known that the moment oil is spilled into water bodies, it starts
spreading immediately. Thus, a good estimation of the amount of oil in the system
requires some certain level of observation, y which can only be done through
a noisy channel. In this case, statistical estimation technique is required
in trying to determine the value of m.
Definition: Estimation is an aspect of inference whereby some information (statistic) are obtained from a sample which helps in determining (estimating) the corresponding parameter of the population from which the sample was drawn. A function of the elements of the sample used in making a good guess of the unknown parameter is called an estimator. Thus, the numerical value of the estimator obtained from the sample data is referred to as an estimate.
Estimation is a necessary condition for optimal prediction (Tzimopoulos
et al., 2008). The focus of this study is to estimate a time-varying
state of spilled oil in water bodies using a linear unbiased minimum-error-variance
sequential-state estimation algorithm known as Kalman filter and the performance
index of the system after a control has been applied using maximum likelihood
The algorithm: Consider the development of a linear unbiased minimum-error-variance recursive algorithm for estimating the state xk of a state-space model at time k given the values of the observed outputs yk-1 = (y0, y1,
, yk-1). This state is called the state of oil in water bodies which is described by the following linear vector model:
where, A (k), C (k), H (k) and G (k) are unknown and thus assumed time-varying coefficient matrices governed by a Gauss-Markov process, ek is an input or message white noise process with zero-mean and unit covariance (E [ek e'k] = I), lk is a zero-mean measurement white-noise process with unit covariance (E [lk l'k] = I). The initial state x0 is uncorrelated with the white noises and has a known mean μ0 and covariance V0 with the following assumption:
Equation 1 is called the message model given that it describes the basic information that is to be determined. Thus the state in question xk is observed by means of noisy mechanism defined as measurement model and given by Eq. 2. Let us denote by Vm, Vx and Vx, the covariance of the message white-noise process, the measurement white-noise process and the variance of the initial state x0, respectively. And also with the assumption that e and l are uncorrelated so that:
Hence, sequential observations should be conducted in the system up to the
decision time k since the oil starts spreading immediately it is released. In
what follows, a set of sequential observations Yk = (y1,
y2,..., Yk) will be expected. The aim is to estimate x
at k. So, to obtain a linear minimum-variance estimator, a Gaussian amplitude
distribution for x, e and l which makes the Kalman filter the best (minimum-error-variance)
linear filter is assumed (Kalman and Bucy, 1961; Van
Trees, 1968; Anderson and Moore, 1979). To derive
the Kalman filter algorithm for this kind of estimation, it is noted that getting
measurement for the state of oil in water x at time k is conditioned on the
observations up to time k (i.e., Yk) which is a chance event. Therefore,
Bayes technique is employed and the maximum a posteriori procedure which
gives the estimate of xk|yk as the value of x at time
k which maximizes the probability p (xk|yk) in every realization
of x is utilized. This is done by first expressing:
Then by setting Yk to denotes all observation up to time (k-1) and that of time k, Eq. 5 becomes:
One can write:
which follows from the joint probability law. Putting Eq. 8 in Eq. 6 gives:
Equation 9 can be written following the probability law as:
But the objective is to determine xk given Yk. Thus, the probability p (yk|xk) needs to be examined. To do this, the expectation of (yk|xk) is first considered:
then its variance:
Next, the expectation of the density p (xk|yk-1) is considered as follows:
is the unbiased estimator of x at time y and the variance:
Again, one can write:
where, Vk-1 = var
Putting Eq. 13 and 18 in Eq.
10 it becomes:
And then, applying the maximum a posteriori procedure it gives:
It then follows from matrix inversion lemma (Hannan et
al., 1980; Sage and Melsa, 1971; Hastings-James
and Sage, 1969) that:
and is called the Kalman gain. The filtering error is given by:
If Eq. 22 is substituted into Eq. 25 and Eq. 2 recalled, the following is obtained:
and the error variance given by:
Finally, the Kalman filter algorithm derived above can be summarized as:
||Backward: Message model: xk = A (k) xk-1+ek
||Given a sequence of observation yk-1 = (y0, y1,
yk-1), for k>0, compute Eq. 14
||Forward: Message model: xk-1 = xk+ek
To implement the Kalman filter, compute the sequence Vk from Eq.
27 and the corresponding sequence F (k) using Eq. 24.
can then be done recursively using Eq. 14 as successive observations
In disaster emergency management of chemical pollutant like oil spill, spill
volume or amount is the most difficult to determine or estimate (El-Shenawy
et al., 2010). For example, in the case of a vessel accident, the
exact volume in a given compartment may be known before the accident but the
remaining oil may have been transferred to other ships immediately after the
accident. Sometimes the exact character or physical properties of the oil lost
are not known and this leads to different estimations of the amount lost. For
optimal response to oil spill cleanup in marine environment, there is need to
have a good estimate of the state variable for a better strategic decision-making.
When the state of a system is unknown or partially known, there are two main
situations that are responsible: first, restrictions on the system observability
and secondly, restrictions on the system controllability. Consequently when
the system variables are subject to theoretical observability and controllability
restrictions, an estimation problem results. The control problem can be solved
by reconstructing the system state variables. This can be carried out by using
the Kalman filter. Least squares estimators are therefore recommended because
they are minimum variance linear unbiased estimators which also permit easy
computation and readily modify to yield recursive estimators of the Kalman filter
form when new information is available. This can be achieved through the following
||Determine the parameter of the system at each time period
by least-squares method based on all available data
||Determine the control policy for the model estimated at step 1
||Repeat step 1 and 2 at each succeeding sampling interval
The updating of the parameter estimates allows the accuracy of the control
to be improved at each step.
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