**1. Introduction**

238 Autonomous Underwater Vehicles

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Wastewaters are often discharged into coastal waters through outfall diffusers that efficiently dilute effluent and usually restrict any environmental impact within a small area. However, predicting this impact is difficult because of the complexity of the hydrodynamic processes that mix the wastewater and also because of the variability in oceanic conditions. Despite great improvements over the years in the understanding of these mixing processes, since models are now available that can make reasonable predictions under steady-state conditions (Hunt et al., 2010), many aspects remain unknown and unpredictable. For this reason, much effort has been recently devoted to improve ways of monitoring and characterizing sewage plumes under a variety of oceanographic conditions.

#### **1.1 MARES AUV**

Autonomous Underwater Vehicles (AUVs) have been used efficiently in a wide range of applications. They were first developed with military applications in mind, for example for mine hunting missions. Later on, scientists realized their true potential and started to use them as mobile sensors, taking measurements in difficult scenarios and at a reasonable cost (Bellingham, 1997; Bellingham et al., 1992; Fernandes et al., 2000; Nadis, 1997; Robinson et al., 1999; Yu et al., 2002). MARES (Modular Autonomous Robot for Environment Sampling) AUV has been successfully used to monitor sea outfalls discharges (Abreu et al., 2010; Abreu & Ramos, 2010; Ramos & Abreu, 2010; 2011a;b;c) (see Fig. 1). MARES is 1.5 m long, has a diameter of 8-inch and weighs about 40 kg in air. It features a plastic hull with a dry mid body (for electronics and batteries) and additional rings to accommodate sensors and actuators. Its modular structure simplifies the system's development (the case of adding sensors, for example). It is propelled by two horizontal thrusters located at the rear and two vertical thrusters, one at the front and the other at the rear. This configuration allows for small operational speeds and high maneuverability, including pure vertical motions. It is equipped with an omnidirectional acoustic transducer and an electronic system that allows for long baseline navigation. The vehicle can be programmed to follow predefined trajectories while

**2. Geostatistical analysis**

Wackernagel, 2003).

**2.2 Ordinary kriging**

2003; Webster & Oliver, 2007):

true value and the estimate:

is

**2.1 Stationary random function models**

Based on Geostatistics Using an Autonomous Underwater Vehicle

The most widely used geostatistical estimation procedures use stationary random function models. A random function is a set of random variables that have some spatial locations and whose dependence on each other is specified by some probabilistic mechanism. A random function is stationary if all the random variables have the same probability distribution and if any pair of random variables has a joint probability distribution that depends only on the separation between the two points and not on their locations. If the random function is stationary, then the expected value and the variance can be used to summarize the univariate behavior of the set of random variables. The parameter that is commonly used to summarize the bivariate behavior of a stationary random function is its covariance function, its correlogram, and its variogram. The complete definition of the probabilistic generating mechanism of a random function is usually difficult even in one dimension. Fortunately, for many of the problems we typically encounter, we do not need to know the probabilistic generating mechanism. We usually adopt a stationary random function as our model and specify only its covariance or variogram (Isaaks & Srivastava, 1989; Kitanidis, 1997;

<sup>241</sup> Mapping and Dilution Estimation of Wastewater Discharges

Ordinary kriging method is often referred with the acronym BLUE which stands for "Best Linear Unbiased Estimator". "Linear" because its estimates are weighted linear combinations of the available data; "Unbiased" since it tries to have the mean error equal to 0; and "Best"because it aims at minimizing the variance of the errors. Let us then see how the concept of a random function model can be used to decide how to weight the nearby samples so that our estimates are unbiased. For any point at which we want to estimate the unknown value, our model is a stationary random function that consists of *n* random variables, one for the value at each of the *n* sample locations, *Z*(**x**1), *Z*(**x**2),..., *Z*(**x***n*), and one for the unknown value at the point we are trying to estimate *Z*(**x**0). Each of these random variables has the same probability law; at all locations, the expected value of the random variable is *m* and the variance is *σ*2. Every value in this model is seen as an outcome (or realization) of the random variable. Our estimate is also a random variable since it is a weighted linear combination of the random variables at the *n* sampled locations (Cressie, 1993; Goovaerts, 1997; Isaaks & Srivastava, 1989; Kitanidis, 1997; Stein, 1999; Wackernagel,

*Z*ˆ(**x**0) =

*E* [*ε*(**x**0)] = *m*

*n* ∑ *i*=1

The estimation error is defined as the difference between the random variable modeling the

The estimation error is also a random variable. Its expected value, often referred to as the bias,

 1 − *n* ∑ *i*=1 *wi* 

*wi* · *Z*(**x***i*). (1)

. (3)

*<sup>ε</sup>*(**x**0) = *<sup>Z</sup>*(**x**0) <sup>−</sup> *<sup>Z</sup>*ˆ(**x**0). (2)

collecting relevant data using the onboard sensors. A Sea-Bird Electronics 49 FastCAT CTD had already been installed onboard the MARES AUV to measure conductivity, temperature and depth. MARES' missions for environmental monitoring of wastewater discharges are conducted using a GUI software that fully automates the operational procedures of the campaign (Abreu et al., 2010). By providing visual and audio information, this software guides the user through a series of steps which include: (1) real time data acquisition from CTD and ADCP sensors, (2) effluent plume parameter modeling using the CTD and ADCP data collected, (3) automatic path creation using the plume model parameters, (4) acoustic buoys and vehicle deployment, (5) automatic acoustic network setup and (6) real time tracking of the AUV mission.

Fig. 1. Autonomous Underwater Vehicle MARES.

#### **1.2 Data processing**

Data processing is the last step of a sewage outfall discharge monitoring campaign. This processing involves the ability to extrapolate from monitoring samples to unsampled locations. Although very chaotic due to turbulent diffusion, the effluent's dispersion process tends to a natural variability mode when the plume stops rising and the intensity of turbulent fluctuations approaches to zero (Hunt et al., 2010). It is likely that after this point the pollutant substances are spatially correlated. In this case, geostatistics appears to be an appropriate technique to model the spatial distribution of the effluent. In fact, geostatistics has been used with success to analyze and characterize the spatial variability of soil properties, to obtain information for assessing water and wind resources, to design sampling strategies for monitoring estuarine sediments, to study the thickness of effluent-affected sediment in the vicinity of wastewater discharges, to obtain information about the spatial distribution of sewage pollution in coastal sediments, among others. As well as giving the estimated values, geostatistics provides a measure of the accuracy of the estimate in the form of the kriging variance. This is one of the advantages of geostatistics over traditional methods of assessing pollution. In this work ordinary block kriging is used to model and map the spatial distribution of temperature and salinity measurements gathered by an AUV on a Portuguese sea outfall monitoring campaign. The aim is to distinguish the effluent plume from the receiving waters, characterize its spatial variability in the vicinity of the discharge and estimate dilution.
