Modelling the Effect of Salinity on the Multivariate Distribution of a Water Quality Index

Salinity is a tracer of water and nutrient flows an d is employed in studies on coastal environment to establish the coastal or remote orig ins of pollutants and nutrients. The present articl e analyses the effects of salinity on the multivariat e distribution of the Trophic Index TRIX. The approach taken is based on a multivariate nested mo del with salinity as an explicative variable. We believe that our results are of use in discerning t hose areas of the North-West Adriatic Sea influence d by the Po River.


INTRODUCTION
The use of indicators to characterize the status of a specific environment is of considerable importance since these indicators play a fundamentally important role in linking policy objectives and targets, in communicating data priorities to different nations and in reporting complexity in simple ways that both policy makers and the public can understand.
In this study we are going to analyze the index used to evaluate the trophic state of the coastal environment: the trophic index TRIX. This topic is of considerable importance since the eutrophication due to overloading with nitrogen and phosphorus nutrients has produced changes in the structure and functioning of marine ecosystems, together with a reduced degree of biodiversity and falling income from fishery, mariculture and tourism.
The TRIX index is routinely used by Italian authorities to monitor the trophic condition of the Adriatic Sea. In particular, upper limits have been established for the yearly average value of the index for the whole coastline. In order to draw up suitable environmental policies, one has to remember that the trophic conditions in the Northern Adriatic Sea are mainly due to the contribution of the Po River, even though in such a complex system, local contributions are also of some importance.
It is important that regional authorities recognize the existence of the two sources of contributions, since they wield the power to intervene at the local level, whereas the effects due to the Po River are beyond their control. Detailed studies of nutrient flows are useful in determining the sources of those nutrients that contribute towards eutrophication and the ways in which nutrients are transported to the ecosystem and they may also suggest methods of combating this problem. Such studies require a wide range of approaches, technology and technical skills and are best addressed by the cooperation between multidisciplinary teams of scientists.
In this context, studies of the coastal water environment [1,2] indicate that salinity is a very peculiar tracer that can be used to interpret the coastal or remote origins of pollutants and nutrients. In the said works, "tracers" are used on the basis of deterministic models.
The aim of the present work is to set out the problem from a statistical point of view involving: a multivariate approach, which enables us to study the entire area using a single model; statistical methodology, which allows us to verify whether salinity has diverse effects on the trophic condition in the studied area.

THE INDEX AND THE DATA
The index TRIX was proposed in [3] as a means of evaluating the trophic state of coastal waters: In (1), n is the number of considered components, X i is the measured value of component i, while U i and L i are the upper and lower limits of each component. The n=4 components are: chlorophyll-a (X 1 ), oxygen saturation (X 2 ), mineral nitrogen (X 3 ) and total phosphorus (X 4 ). Their ranges are currently standardized to 3 log units, i.e. logU i −logL i =3 and k=10 is an expansion factor for ranging the index out from the interval [0,1] to [0,10]. Variables X 1 and X 2 are direct expressions of productivity, while variables X 3 and X 4 are nutritional factors.
The data set consists of the weekly point measurements, taken from 1998 to 2002, of the four variables contributing towards the TRIX index and of salinity. Measurements come from thirteen monitoring sites along the Emilia-Romagna coast. This information has been kindly provided by the Emilia-Romagna Region and the Regional Agency for Environmental Protection (ARPA).
Five monitoring points are situated at a distance of 500 metres from the coastline: Lido di Volano (identification number: 1), Porto Garibaldi (id.no. 2), Cesenatico (id.no. 3), Rimini (id.no. 4) and Cattolica (id.no. 5); together they constitute the inshore subgroup. The other eight monitoring sites, which constitute the offshore subgroup, are situated at three kilometres out to sea from Lido di Volano (id.no. 6), Porto Garibaldi (id.no. 7), Cesenatico (id.no. 10) and Cattolica (id.no. 11) and at ten kilometres out to the sea from Lido di Volano (id.no. 8), Porto Garibaldi (id.no. 9), Cesenatico (id.no. 12), Cattolica (id.no. 13). The map of the monitoring network is shown in Fig. 1. Marine trophic conditions are evaluated by the public authorities according to the yearly average of the TRIX index. Since at present data sets are only available for the last few years, our work is based on monthly averages of the index.
The monthly averages at each site are well approximated by the normal distribution. Means, variances and the results of the Kolmogorov-Smirnov normality tests of the monthly data are given in Table 1.
In this coastal zone the water circulation pattern is influenced by several factors such as bottom bathymetry, surface winds, inputs from the Po River, etc.. This is also evident from the correlation matrix corresponding to the monthly TRIX average, reported in Table 2, featuring groups of highly correlated sites such as the four sites out to sea along the southern stretch of coastline (sites 10÷13) and the four sites out to sea along the northern part of the coast (sites 6÷9). Therefore, we may reasonably assume correlation among sites and thus adopt a multivariate modelling approach.

MULTINORMALITY TEST
Let Y vi be the TRIX monthly average at site v (v=1,2,..,13) during month i (i=1,2,...,60). The multivariate process for the entire area under study may thus be considered to be as follows: with mean vector µ µ µ µ y and covariance matrix Σ Σ Σ Σ y where each component is normally distributed.
To verify whether y may be treated as a multinormal process, we adopt the test procedure proposed by [4] .
The normal data matrix X can be written also as Where x (j) (j=1,2,...,p) is the n-vector whose elements denote the observations regarding the j-variable.
Measures of multivariate skewness and kurtosis are, respectively, Where x and z are identically and independently distributed. If x ∼ N p (µ µ µ µ,Σ Σ Σ Σ) then, and x r denotes the r-th rows of X.
In [5] has been shown that b 1,p and b 2,p have the following asymptotic distributions p nb , and These statistics can be used to test the null hypothesis of multinormality: In our case y i (i=1,2,...,n) is the row vector of the TRIX monthly average in month i (p=13 and n=60).
The degrees of freedom for the chi-square are 455 and fixed α = 0.05 , the critical value is 506, therefore (1/6)nb 1,p = 495.58 is not significant. We may also note that the observed value 0.332 is not significant at the 5% level. The hypothesis of multinormality is therefore not rejected for the entire region under study; the multivariate distribution of the TRIX is suitably approximated by a multivariate Gaussian process.

THE PROPOSED MODEL
Salinity in coastal ecosystems is an important indication of inputs from coastal drainage basins and it can be used as a tracer of river plumes, nutrient and sediment loading.
In order to relate the multivariate distribution of the index (y∼N(µ µ µ µ y ,Σ Σ Σ Σ y )) to salinity, we adopt a nested link structure; that is, we assume that the mean µ µ µ µ y of the multivariate Gaussian process can be expressed as a linear function of the tracer and the maximum likelihood estimates of parameters are given in Table 3.

RESULTS
For the mean of the process the results confirm a common linear dependence pattern. Table 4 shows, for each site, the evaluation j μ obtained by The fit of the estimated model is good (Fig. 2) and the mean of the relative absolute differences between the sample means j μ (Table 1) and j μ is 1.07%. This means that the effect of salinity on the mean of the index distribution follows a common pattern (linear), albeit with a significant difference in intensity.
As far as the variance and correlation parameters are concerned, the mean of the relative absolute differences between the sample variances 2 j σ , Table 1 and the estimate parameters in Table 3 is 17.9%. The fit is also displayed in Fig. 3. While for the correlations, the overall difference is 27.4%.

CONCLUSION
An understanding of the connection between water quality in coastal areas and river catchments is important if we are to remedy current environmental effects and be in a position to plan future options so as to improve the coastal ecosystem.
The problem of how to define the areas influenced by fluvial discard is a substantial one for which there is no one single solution. The literature includes several contributions in which salinity is used, in deterministic models, as a tracer in order to distinguish the origin of nutrients and pollutants.
In this study, a multivariate model has been used to identify the effects of salinity on the multivariate distribution of the index TRIX. In particular, the effects of the tracer are relevant on the mean of the TRIX distribution, while the covariance matrix is substantially given by the sea current pattern.
Results have revealed two groups of monitoring sites with significant differences in the characteristics of the relationship. The differences can be seen as evidence of the existence of two different situations: the offshore monitoring points subject to the influence of the Po river and the inshore monitoring sites mainly influenced by local factors such as urban discharges or nutrients carried in to the sea by local rivers.