Monitoring networks aims at capturing the spatial and temporal variability of
one or several environmental variables in a specific environment. The optimal
placement of sensors in an ocean or coastal observatory should maximize the
amount of collected information and minimize the development and operational
costs for the whole monitoring network. In this study, the problem of the design
and optimization of ocean monitoring networks is tackled throughout the
implementation of data assimilation techniques in the Shallow water HYdrodynamic
Finite Element Model (SHYFEM). Two data assimilation methods – nudging and
ensemble square root filter – have been applied and tested in the Lagoon of
Venice (Italy), where an extensive water level monitoring network exists. A
total of 29 tide gauge stations were available, and the assimilation of the
observations results in an improvement of the performance of the SHYFEM model, which went from an initial root mean square error (RMSE) on the water level of
5.8

Ocean and coastal monitoring networks are fundamental for tracking contaminants
in the water, assessing environmental change and water quality, observing sea
level rise, and developing strategies for managing resources in a changing
climate

Oceanographic models are increasingly used in coastal systems to describe sea
dynamics induced by tide and atmospheric and terrestrial forcing, thus complementing the collected information retrieved by direct observations

There is however another point of view. If only observations were available,
the best distribution of the monitored variable over the system could be given
only by data interpolation (DI) of the observation points to the other areas.
The direct observations of the sea conditions are considered to represent the
true state at the monitoring point. However, the spatio-temporal interpolation
of such true values is not meant to correctly describe the variability of the
investigated state variable over the whole system. This is especially true in
coastal systems that are characterized by complex small-scale and
high-frequency dynamics. In this case the resulting picture of interpolated
values may show non-coherent features and inconsistency between data points.
When an oceanographic model is available, the interpolation of these
observations can be carried out by the model and much better representation of
the environment can be achieved. In this context, models are used to connect sparse observations (in space and time) or synthesize them through data
assimilation (DA) techniques

Validated ocean circulation models and DA can also assist in the network design of a
new observing system or in optimizing an existing observatory

In this study, we show how data assimilation techniques are implemented in the
Shallow water HYdrodynamic Finite Element Model (SHYFEM) for optimizing the tide
gauge network of the Lagoon of Venice (Italy). Since one limitation of the
observing system evaluation procedure is that it depends on the properties of
the DA employed for the evaluation

The numerical experiments consisted of simulating the circulation in the Lagoon
of Venice using the open-source SHYFEM hydrodynamic model

The Coriolis term and pressure gradient in the momentum equation, and the
divergence terms in the continuity equation are treated semi-implicitly. Bottom
friction and vertical eddy viscosity are treated fully implicitly for stability
reasons due to the shallow nature of the lagoon, while the remaining terms
(advective and horizontal diffusion terms in the momentum equation) are treated
explicitly. At the boundaries, water levels are either prescribed at the open
boundaries or the free-slip condition is implemented at solid (closed)
boundaries. A detailed description of the model equations is given in

The nudging method is a flexible assimilation technique that is computationally
more economical than other assimilation methods like variational data
assimilation. First used in meteorology

The ensemble square root filter (hereinafter referred to as EnSRF) is a
more complex assimilation method, widely used in environmental sciences

The formulation of the EnSRF is slightly different from the EnKF and avoids the
perturbation of the observations. Using the notation of

In the traditional Kalman filter formulation, the covariance matrix

In the ensemble methods, using Eq. (

After some eigenvalue and singular value decompositions

The approximation of the covariance matrix with the ensemble perturbations
(Eq.

Starting from the DA run with the assimilation of all stations (

The optimization procedure is easily and efficiently parallelized since all simulations within each iteration step are independent of each other. Similarly, all members of each DA-EnSRF process are independent and can be carried out simultaneously on different processors.

The Lagoon of Venice (Fig.

The city of Venice is located in the centre of the lagoon and is composed of
more than a hundred islands linked by bridges. The elevation of these islands is
extremely low, subjecting them to flooding during storms, which in turn
threatens the unique cultural heritage of this city and affects its everyday
life. The northern Adriatic Sea is frequently affected by storm surge events,
mainly triggered by strong south-easterly wind

The Lagoon of Venice has two tide gauge networks for supporting the local
real-time storm surge prediction and warning system. They are managed by the
Institute for Environmental Protection and Research – National Centre for
Coastal Zone and Characterization Marine Climatology and for Operational
Oceanography (ISPRA, Unit for Tides and Lagoons,

In this study, we collected all the available data from both the ISPRA and CPSM
monitoring networks over a 1-month period (November 2013) with the highest
number of stations without missing data. The selected dataset consists of
quality-controlled 10

In order to investigate at which degree the observations represent the state
variable over the whole system, a field approximation through optimal
interpolation (OI) of the data has been performed. OI is a commonly used and
fairly simple method to perform interpolation of sparse data also in
data assimilation. OI was first described in

Bathymetry and unstructured mesh of the Lagoon of Venice. The red dots mark the tide gauge monitoring station.

Spatial distribution of

A snapshot on 4 November 2013 at 14:00

The water circulation in the Lagoon of Venice, induced by tide and wind was
simulated by the unstructured model SHYFEM applied over a spatial domain that
represents the entire Lagoon and its adjacent shore. The model adequately
reproduces the complex geometry and bathymetry of the Lagoon of Venice using
unstructured numerical meshes composed of triangular elements of variable form
and size, going down to a few metres in the channels (Fig.

The application of the SHYFEM model to the Lagoon of Venice has been validated
in previous work reproducing correctly tidal propagation, storm surge, water
flows at the lagoons' inlets, and water temperature and salinity variability

In this study, hydrodynamics in the lagoon were simulated using 10

In order to apply the nudging DA method, a value for the relaxation parameter

The EnSRF needs an ensemble of model states that should ideally represent the
error of the simulation. In the present case the ensemble of the model states
is created varying the boundary condition. We used 60 perturbations for the
sea-level boundary condition (member 0 is unperturbed) taken from a Gaussian
distribution with a zero mean and a standard deviation set to 30

After several preliminary numerical tests, the best cut-off distance for the
local analysis was set to 0.1 geographical degrees (about 10

The EnSRF assimilates water level from the selected stations considering them
independent (the

In the exposition of the results, we defined the model run without data
assimilation as the control simulation, while, for both the DA schemes,
the base run accounts for the assimilation of all the 29 monitoring stations.
All mentioned parameters (

When entering a shallow basin, such as the Venice lagoon, the tidal wave is
deformed, either damped or amplified, according to a relationship between
local flow resistance and inertia and the characteristics of the
incoming tidal wave

Does the interpolation of the observations provide a realistic spatial
representation of the water level variability over the lagoon domain?
To answer this question, in Fig.

Observed, interpolated, and simulated water levels at station 12. In this computation, station 12 was not included in the DI and DA algorithms.

Observed and simulated vertically integrated current velocity at the Lido inlet. The ADCP was located close to tide gauge number 15.

Root mean square errors (in

Statistical analysis of simulated current velocity at the Lido
inlet. Results are given as RMSE (root mean square error,

In order to establish which method better represents the water level
variability over the lagoon, we need to evaluate the capacity of each approach
to describe the parameter at locations not included in the computation. Thanks
to a large number of available tide gauges in the Lagoon of Venice, the model
skill assessment (in terms of the root mean square error, RMSE) is determined
by re-running DI and DA experiments removing one station from the assimilation
and comparing the water level in this station with the modelled one.
The evaluation procedure was repeated for each monitoring station and the results
are reported in Table

By contrast, both DA methods strongly improved the model skills in all parts
of the lagoon. The average RMSE resulted in 2.1 and 3.2

Additionally, in a multivariate analysis approach we tested the capability of
the applied DA-driven simulations in reproducing the current velocities recorded
by an acoustic Doppler current profiler (ADCP) mounted on the bottom of the Lido
inlet, close to station no. 15 shown in Fig.

The next step is to use DA methods to find the minimum number of stations – and
their distribution – that correctly represent the state variable in the
investigated system. The optimization procedure of this tide gauge network,
composed of

Root mean square error of the water levels as a function of the number of tide gauge stations interpolated or assimilated. The RMSE value with zero considered stations for DA also indicates the error of the base simulation when no DA methods are applied.

The evaluation procedure allows finding the minimum number of tide gauges for a successful description of the water level in the lagoon. However, the optimization criterion (the RMSE threshold) is arbitrary and may differ for different environments, state variables and monitoring networks. In the present case, we can see that using both DA-nudging and DA-EnSRF, the RMSE does not change too much passing from 29 to 10–12 assimilated stations. Even if the EnSRF has an average RMSE higher than the DA-nudging, the RMSE of the EnSRF has a slower increase with the reduction of the stations. The initial decrease in the RMSE is probably due to the fact that observations have errors, and tide gauges close to each other can provide slightly different data. The EnSRF considers the observation error in the observation covariance matrix, but it is difficult to find the right value and normally the nominal instrument error is used.

Considering the spatial interpolation method, the use of 10 stations has an RMSE comparable to the error of the control simulation. But we have to stress that in this case the spatial representation of the water level is clearly wrong. We should also mention that the model with the assimilation of only three stations gives a lower RMSE than DI with all 29 stations, apart from the fact that results are physically more coherent and consistent.

The resulting optimal distributions of the 10 tide gauge stations determined by
DA-nudging and DA-EnSRF are shown in Fig.

The optimal distribution of 10 tide gauge stations (marked with green
dots) according to DA-nudging

Same as Fig.

Ensemble weighted correlation (averaged over the simulation period) of the 10 monitoring stations selected using the EnSRF method.

However, the choice of which stations to keep in the monitoring network
depends also on many practical factors. As an example, the monitoring authority
would decide to keep some stations because of their strategic relevance,
maintenance costs, distance from the laboratory, or for continuing long-term
time series. The optimization method can be easily customized based on
predetermined specific constraints. As a realistic exercise, we fixed the
stations at the inlets (4, 12, 11) and in the main urban settlements (2, 6,
17, 19) in the monitoring network. The evaluation procedure is then repeated
using the DA-nudging method, keeping these 7 stations and the results are
presented in Fig.

The methodology presented in this study allows for the evaluation of existing
coastal observatories. Using a DA system, which is an observation-driven and
process-based method, the iterative optimization procedure establishes the
relevance of each single monitoring station for the description of the considered
environment. The example reported in this study describes the optimization of
an existing observatory with defined monitoring points. However, the methodology
could be applied also to design new monitoring networks. As described by

As indicated by

Additionally, as specified at Sect.

The combination of observations and numerical models is particularly important
in coastal regions with scarce monitoring resources. However, to reduce the model
error, the applied numerical models must correctly reproduce the complex
morphology of the coastline and the exchange processes between the shelf and the
open seas. The processes in such complex systems at the land–sea transition are
extremely dynamic and require a holistic approach in which all the hydrological
entities (river mouth, salt marshes, lagoons, swamps, coastal sea) should be
regarded as integral parts of the entire domain of computation. Moreover,
due to the complex geometry and morphology of the coastal regions, the numerical
models need to be able to represent hydrodynamic conditions with very high
resolution, in the horizontal, vertical, and temporal dimensions. With respect to
the above-cited requirement, unstructured models – as the one applied in this
study – realize a seamless transition between different spatial scales for
reproducing the coast–sea interactions, adopting a variable resolution of the
mesh elements

The model-driven optimization procedure was here applied using hindcast
simulations, but it can be also used in forecasting modelling for evaluating
the effect of the assimilated data on the predictions

In the case of the Lagoon of Venice tide gauge network, we demonstrated how numerical
models with data assimilation can play a valuable role in optimizing and
designing coastal observatories. The iterative optimization process was based
on the evaluation of the RMSE at the stations not assimilated. It is worth
noting that the existing monitoring network can be reduced by a factor of

The SHYFEM hydrodynamic model is open source (GNU General Public License as
published by the Free Software Foundation) and freely available through GitHub
at

GU conceived the idea of the study with the support of CF. GU developed the optimization procedure and the nudging data assimilation routines, and MB developed the ensemble square root filter data assimilation software. CF and MB performed the numerical simulations. All authors discussed, reviewed, and edited the different versions of the paper.

The authors declare that they have no conflict of interest.

The authors wish to thank the Tide Forecast and Early Warning Center of the City of Venice and the Italian Institute for Environmental Protection and Research (ISPRA) for providing tide gauge and current velocity data.

This research has been supported by the Venezia2021 research programme funded by the “Ministero delle Infrastrutture e dei Trasporti – Provveditorato Interregionale per le Opere Pubbliche del Veneto, Trentino Alto Adige e Friuli Venezia Giulia, già Magistrato alle Acque di Venezia”, provided through the concessionary Consorzio Venezia Nuova and coordinated by CORILA.

This paper was edited by David Ham and reviewed by Joseph Wallwork and one anonymous referee.