Ship weather routing, which involves suggesting low-emission routes, holds potential for contributing to the decarbonisation of maritime transport. However, including because of a lack of readily deployable open-source and open-language computational models, its quantitative impact has been explored only to a limited extent.

As a response, the graph-search VISIR (discoVerIng Safe and effIcient Routes) model has been refactored in Python, incorporating novel features. For motor vessels, the angle of attack of waves has been considered, while for sailboats the combined effects of wind and sea currents are now accounted for. The velocity composition with currents has been refined, now encompassing leeway as well. Provided that the performance curve is available, no restrictions are imposed on the vessel type. A cartographic projection has been introduced. The graph edges are quickly screened for coast intersection via a

The resulting VISIR-2 model has been employed in numerical experiments within the Mediterranean Sea for the entirety of 2022, utilising meteo-oceanographic analysis fields. For a 125 m long ferry, the percentage saving of overall CO

VISIR-2 serves as an integrative model, uniting expertise from meteorology, oceanography, ocean engineering, and computer science, to evaluate the influence of ship routing on decarbonisation efforts within the shipping industry.

As climate change, with its unambiguous attribution to anthropogenic activities, rapidly unfolds

Zero-emission bunker fuels are projected to cost significantly more than present-day fossil fuels

The variability in percentage savings reported in the literature can be attributed to the diversity of routes considered, the specific weather conditions, and the types of vessels analysed. Additionally, reviews often use a wide range of bibliographic sources, including grey literature, company technical reports, white papers, and works that fail to address the actual meteorological and marine conditions.

The VISIR (discoVerIng Safe and effIcient Routes, pronunciation: /vi′zi:r/) ship weather routing model was designed to objectively assess the potential impact of oceanographic and meteorological information on the safety and efficiency of navigation. So far, two versions of the model have been released (VISIR-1.a –

To address all these requirements, we designed, coded, tested, and conducted extensive numerical experiments with the VISIR-2 model

The VISIR-2 model enabled systematic assessment of CO

VISIR-2 is a numerical model well-suited for both academic and technical communities and facilitating the exploration and quantification of ship routing's potential for operational decarbonisation. In the European Union, there may be a need for independent verification of emissions reported in the monitoring, reporting, and verifying (MRV) system (

The remainder of this paper includes a literature investigation in Sect.

This compact review of systems for ship weather routing will be limited to web applications (Sect.

FastSeas (

The Avalon web router (

GUTTA-VISIR (

OpenCPN (

A review of ship weather routing methods and applications was provided by

A specific weather routing model was documented by

A stochastic routing problem was addressed in

The impact of stochastic uncertainty on WAPS ships was addressed by

A few open web applications exist, mainly for sailboats and with limited insight into their numerical methods. Case study results from weather routing systems developed in academia have been published, but (with the exception of

From this assessment, the lack of an open-source and well-documented ship weather routing model, for both motor vessels and sailboats, with flexible characterisation of vessel performance appears as a gap which the present work aims to close.

This section includes a revision of the vessel kinematics of VISIR, as given in Sect.

For a graph-search model such as VISIR to deal with waves, currents, and wind for various vessel types, several updates to its approach for velocity composition were needed. They included both generalisations and use of new quantities, addressed in this subsection, as well as a new numerical solution, addressed in Appendix

As in

Making reference to Fig.

Angular configuration with (

This implies that, to balance the cross flow from the currents, the vessel must head into a direction

Furthermore, we here define leeway as a motion, caused by the wind, transversal to the ship's heading. From the geometry shown in Fig.

Upon expressing the magnitudes

Equations (

Within this formalism, if the vessel is a sailboat (or rather a motor vessel making use of a WAPS), just one additional condition should be considered. That is, given the wind-magnitude-dependent no-go angle

In analogy to Eqs. (

Finally, the components of the effective flow

The graphical construction in Fig.

However, Eqs. (

As it reduces the ship's speed available along its course (Eq.

We note that Eq. (

As

Equations (

Finally, given that

From Eq. (

The CO

Graph preparation is crucial for any graph-search method. Indeed, graph edges represent potential route legs. Therefore, the specific edges included within the graph directly influence the route topology. In addition, as will be shown in Sect.

The structure of the mesh is also the most fundamental difference between a graph-search method (such as Dijkstra's or

To efficiently address all these aspects, VISIR-2's graph preparation incorporates several updates compared to its predecessor, VISIR-1b, as outlined in the following sub-sections.

Graph stencil for

The nautical navigation purpose of a ship routing model necessitates the provision of both length and course information for each route leg. On the other hand, environmental fields used to compute a ship's SOG (Eq.

Leg courses of a ship route originate from graph edge directions. To determine them, we consider the Cartesian components of the edges in a projected space. As seen from Fig.

For a given sea domain, a graph is typically computed once and subsequently utilised for numerous different routes. However, in VISIR-1, the edge direction was recalculated every time a graph was utilised. In VISIR-2, the edge direction is computed just once, during graph generation, after which it is added to a companion file of the graph (

Finally it should be noted that in a directed graph, such as VISIR-2's, edge orientation also matters. Each edge acts as an arrow, conveying flow from “tail” to “head” nodes. Edge orientation refers to the assignment of the head or tail of an edge. Orientation holds a key to understanding vessel performance curves, explored further in Sect.

A further innovation regarding the graph involves an edge pruning procedure. This eliminates redundant edges that carry very similar information. Indeed, some edges have the same ratio of horizontal to vertical grid hops (see the

Edge count and minimum angle with due north. The total number of edges in the first quadrant is

This benefits both the computer memory allocation and the computing time for the shortest path. The latter is linear in the number of edges; cf.

Bathymetry field from EMODnet represented in shades of grey, with contour lines at depths at

The minimal safety requirement is that navigation does not occur in shallow waters. This corresponds to the condition that vessel draught

However, for a specific edge, UKC could still be positive and the edge cross the shoreline. This is avoided in VISIR by checking for mutual edge–shoreline crossings. Given the burden of this process, in VISIR-1b a procedure for restricting the check to inshore edges was introduced. In VISIR-2, as envisioned in

Various bathymetric databases can be used by VISIR-2. For European seas, the EMODnet dataset (

The bathymetry dataset, if detailed enough, can even be used for deriving an approximation of the shoreline. From Fig.

Such a pseudo-shoreline is the one used in VISIR-2 for checking the edge crossing condition specified in Sect.

For computing shortest paths on a graph, its edge weights are preliminarily needed. Due to Eqs. (

Temporal grid of VISIR-2. The upper horizontal axis (

The time at which edge weights are evaluated is key to the outcome of the routing algorithm. In

The numerical environmental field and the graph grid may possess varying resolutions and projections. Even if they were identical, the grid nodes might still be staggered. Moreover, it is necessary to establish a method for assigning the field values to the graph edges. For all these reasons, spatial interpolation of the environmental fields is essential.

Spatial interpolation in VISIR-2.

VISIR-2 first computes an interpolant using the

The two interpolation schemes were tested with various non-convex functions, and some results are reported in Sect. S0 of the Supplement. We observed that both options converge to a common value as the

However, setting Sint

The resulting edge-representative environmental field value affects the edge delay (Eq.

Even before the spatial interpolation is performed, the so-called “sea-over-land” extrapolation is applied to the marine fields. This step, which is needed for filling the gaps in the vicinity of the shoreline, is conceptually performed as in

Since wave direction is a circular periodic quantity, we calculate its average using the circular mean (

A major improvement made possible by the Python coding of VISIR-2 is the availability of built-in, advanced data structures such as dictionaries, queues, and heaps. They are key in the efficient implementations of graph-search algorithms

Nonetheless, Dijkstra's original algorithm exclusively accounted for static edge weights

We note the generality of the pseudo-code with respect to the edge weight type (

The shortest-distance and the least-time algorithms invoked for both motor vessels and sailboats are identical. Differences occur at the post-processing level only, as different dynamical quantities (related to the marine conditions or the vessel kinematics) have to be evaluated along the optimal paths. Corresponding performance differences are evaluated in Sect. S1 of the Supplement.

_DIJKSTRA_TDEP.

GET_TIME_INDEX.

As previously mentioned, Dijkstra's algorithm can recover the optimal path in the presence of dynamic edge weights if Eq. (

At the heart of the VISIR-2 kinematics of Sect.

In VISIR-1 the forward speed resulted, for motor vessels, from a semi-empirical parameterisation of resistances

In VISIR-2 new vessel models were used, and just two of them are presented in this paper: a ferry and a sailboat. However, any other vessel type can be considered, provided that the corresponding performance curve is utilised in the

The interpolating function was either a cubic spline (for sailboats) or the outcome of a neural-network-based prediction scheme (for the ferry). The neural network features are provided in Appendix

The ferry modelled in VISIR-2 was a medium-size Ro-Pax vessel whose parameters are reported in Table

Principal parameters of the ferry.

1 kn

A vessel's seakeeping model was used at the ship simulator at the University of Zadar, as documented in

In a given sea state, the sustained speed is determined by the parameter

The

Ferry performance curve: in

Any sailboat described in terms of polars can in principle be used by VISIR-2. For the sake of the case study, a Bénétau First 36.7 was considered. Its hull and rigging features are given in Table

The modelling of the sailboat STW was carried out by means of the WinDesign velocity prediction program (VPP). The tool was documented in

Principal parameters of the sailboat (First 36.7).

Sailboat performance curve: forward speed

The outcome corresponds to Eqs. (

Further innovations brought in by VISIR-2 concern the visualisation of the dynamic environmental fields and the use of isolines.

To provide dynamic information via a static picture, the fields are rendered via concentric shells originating at the departure location. The shape of these shells is defined by isochrones. These are lines joining all sea locations which can be reached from the origin upon sailing for a given amount of time. This way, the field is portrayed at the time step the vessel is supposed to be at that location. Isochrones bulge along gradients of a vessel's speed. Such shells represent an evolution of the stripe-wise rendering introduced in VISIR-1.b

In addition to isochrones, lines of equal distance from the route's origin (or “isometres”) and lines of equal quantities of CO

Software modularity has been greatly enhanced in VISIR-2. While in VISIR-1 modularity was limited to the graph preparation, which was detached from the main pipeline

VISIR-2 workflow. Modules enclosed within thicker frames are intended for direct execution by the end user, while the other modules can be modified for advanced usage. The data flow occurs along the wavy arrow, with routine calls along the dashed line.

VISIR-2 modules with their original names, purpose, and references within this paper. Module nos. 1–5 represent the core package and require the visir-venv virtual environment. Module no. 6 runs with visir

The modules can be run independently and can optionally save their outputs. Through the immediate availability of products from previously executed modules, this favours research and development activities. For operational applications (such as GUTTA-VISIR) instead, the computational workflow can be streamlined by avoiding the saving of the intermediate results. VISIR-2 module names are Italian words. This is done for enhancing their distinctive capacity; cf.

A preliminary graphical user interface (GUI) is also available. In the version of VISIR-2 released here, it facilitates the ports' selection from the World Port Index (

VISIR-2 was developed on macOS Ventura (13.x). However, both path parameterisation and the use of virtual environments ensure portability, which was successfully tested for both Ubuntu 22.04.1 LTS and Windows 11, on both personal computers and two distinct high-performance computing (HPC) facilities.

Validating a complex model like VISIR-2 is imperative. The code was developed with specific runs of VISIR-1 as a benchmark. The validation of VISIR-1 involved comparing its outcomes with both analytical and numerical benchmarks and assessing its reliability through extensive utilisation in operational services

Previous studies have compared VISIR-1 to analytical benchmarks for both static wave fields (“Cycloid”;

VISIR-2 route durations compared to analytic oracles (Cycloid and Techy), as referenced in the main text.

1 nmi

Additionally, in

VISIR-2 vs. LSE durations. The relative error

For both analytical and numerical benchmarks, distinct from the scenario discussed in Sect.

During the tests mentioned earlier, the vessel's STW remained unaffected by vector fields. In instances where there was a presence of current vector fields (Techy oracle), they were merely added to STW, without directly impacting it. Therefore, the enhanced capability of VISIR-2 to accommodate angle-dependent vessel performance curves (cf. Eqs.

To achieve this objective, the OpenCPN model was utilised. This model can calculate sailboat routes with or without factoring in currents and incorporates shoreline knowledge, though it does not consider bathymetry. For our tests, we provided VISIR-2 with wind and sea current fields identical to those used by OpenCPN (further details are provided in Sect.

VISIR-2 vs. OpenCPN comparison. Durations

VISIR-2 routes with wind and currents vs. OpenCPN: graphs of variable resolution, indexed by

VISIR-2 routes exhibit topological similarity to OpenCPN routes, yet for upwind sailing, they require a larger amount of tacking (see Fig.

Downwind routes all divert northwards because of stronger wind there (Fig.

The disparities in duration between OpenCPN and VISIR-2 routes could be attributed to various factors, including the interpolation method used for the wind field in both space and time and the approach employed to consider currents. Delving into these aspects would necessitate a dedicated investigation, which is beyond the scope of this paper.

Numerical tests have been integrated into the current VISIR-2 release

The computational performance of VISIR-2 was evaluated using tests conducted on a single node of the “Juno” HPC facility at the Euro-Mediterranean Center on Climate Change (CMCC). This node was equipped with an Intel Xeon Platinum 8360Y processor, featuring 36 cores, each operating at a clock speed of 2.4 GHz, and boasting a per-node memory of 512 GB. Notably, parallelisation of the cores was not employed for these specific numerical experiments. Subsequently, our discussion here is narrowed down to assessing the performance of the module dedicated to computing optimal routes (“

In Fig.

The numerical tests utilise the number of degrees of freedom (DOF) for the shortest-path problem as the independent variable. This value is computed as

Profiling of computing time for the

The primary finding is a confirmation of a power-law performance for all three optimisation objectives of VISIR-2: distance, route duration, and total CO

Digging deeper into the details, we observe that the least-distance procedure within VISIR-2, while exclusively dealing with static edge weights, exhibits a less favourable scaling behaviour compared to both the least-time and least-CO

Lastly, it was found that peak memory allocation scales linearly across the entire explored range, averaging about 420

Fit coefficients of the

A more comprehensive outcome of the VISIR-2 code profiling, distinguishing also between the sailboat and the motor vessel version of the

A prior version of VISIR-2 has empowered GUTTA-VISIR operational service, generating several million optimal routes within the Adriatic Sea over the span of a couple of years. In this section, we delve into outcomes stemming from deploying VISIR-2 in different European seas. While the environmental fields are elaborated upon in Sect.

The fields used for the case studies include both static and dynamic fields. The only static one was the bathymetry, extracted from the EMODnet product of 2022 (

The kinematics of VISIR-2 presented in Sect.

To showcase some of the novel features of VISIR-2, we present the outcomes of numerical experiments for both a ferry (as outlined in Sect.

The chosen domain lies at the border between the Provençal Basin and the Ligurian Sea. Its sea state is significantly influenced by the mistral, a cold northwesterly wind that eventually affects much of the Mediterranean region during the winter months. The circulation within the domain is characterised by the southwest-bound Liguro–Provençal current and the associated eddies

We conducted numerical experiments using VISIR-2 with a graph resolution given by

In Fig.

Figure

The ferry's optimal routes between Porto Torres (ITPTO) and Toulon (FRTLN) with both waves and currents:

To delve deeper into the statistical distribution of percentage CO

The analysis of the CO

Average relative savings of the CO

Percentage CO

Metrics relative to ferry routes pooled on sailing directions (FRTLN

Fit coefficients of

Further comments regarding the comparison of the CO

The chosen area lies in the southern Aegean Sea, along a route connecting Greece (Monemvasia) and Türkiye (Marmaris). This area spans one of the most archipelagic zones in the Mediterranean Sea, holding historical significance as the birthplace of the term “archipelago”. The sea conditions in this area are influenced by the meltemi, prevailing northerly winds, particularly during the summer season. Such an “Etesian” weather pattern can extend its influence across a substantial portion of the Levantine basin

We performed numerical experiments with VISIR-2, with a graph resolution of

In Fig.

Moving to Fig.

In only five cases (constituting

The sailboat's optimal routes between GRMON and TRMRM, considering both wind and currents:

A statistical evaluation of the time savings resulting from the optimisation process for sailboat routes is illustrated in Fig.

It is important to acknowledge that, under excessively weak or consistently sustained upwind conditions, a sailboat route might become unfeasible. A quantitative overview of such “failed” routes is provided in Table

In Fig.

Turning to leeway, when not in combination with currents, it consistently extends the duration of routes, particularly, as indicated in the Supplement, when facing upwind conditions (more likely for westbound routes), as the speed loss is exacerbated due to a higher leeway speed (region with

As in our earlier comment in Sect.

When both effects, currents and leeway, are considered together, the distribution of duration changes in comparison to wind-only routes resembles the distribution for the case with currents only. However, due to the impact of leeway, it is slightly skewed towards longer durations.

Finally in Table

Metrics of the sailboat's optimal routes pooled on sailing directions (GRMON

Average time savings of the sailboat routes (in %), considering just wind (wi) or also various combinations of currents (cu) and leeway (le) for the sailboat routes between Monemvasia (GRMON) and Marmaris (TRMRM) as in Fig.

In this section, we critically compare the CO

Only a handful of peer-reviewed papers have reported results on emission savings through ship routing, yet a few of these findings are reviewed here for comparison with VISIR-2's results.

For the ferry case study examined in this paper, the CO

Applying the VOIDS model,

We note that, as for example in

VISIR possesses a capability to incorporate ocean currents into the voyage optimisation process. As shown in

In general, both average and extreme CO

VISIR-2 contributes to the existing body of literature by providing an open computational platform that facilitates the simulation of optimal ship routes in the presence of waves, currents, and wind. These simulations are designed to be transparent, with customisable sea domain and vessel performance curves, allowing for thorough inspection, modification, and evaluation. This addresses the concern raised by

Given its open-source nature, validated results, and numerical stability, VISIR-2 can have great utility across various fields. The fact that both motor vessels and sailboats are treated equally will make VISIR-2 suitable for use in weather routing of vessels with wind-assisted propulsion.

Moreover, VISIR-2 can serve as a valuable tool for regulatory bodies seeking to make informed policies on shipping. As previously outlined, agencies like the European Maritime Safety Agency (EMSA), which oversee systems for monitoring, reporting, and verifying emissions, could utilise the model – provided that vessel performance curves and GHG emission rates are available – to calculate baseline emissions for various vessel and GHG types. With the inclusion of shipping in the EU-ETS, shipowners, ship managers, or bareboat charterers – whoever bears the fuel cost – are mandated to surrender allowances for their emissions. These stakeholders may find it beneficial to explore open-source solutions alongside existing commercial options.

VISIR-2 also has the potential to help reduce uncertainty regarding the effectiveness of weather routing in reducing CO

Furthermore, VISIR-2 could be used to generate a dataset of optimal routes for the training of artificial intelligence systems for autonomous vessels

There are several possible avenues for future developments of VISIR-2: the computer science, the algorithms, the ocean engineering, and the environmental fields.

First, as mentioned in Sect.

Further algorithmic work could address, for instance, construction of the graph directly in the projection space, facilitating the presence of more isotropic ship courses and perfectly collinear edges (cf. Sect.

Transitioning to upgrades in ocean engineering for VISIR-2, the focus could shift towards targeting large ocean-going vessels, contingent upon the availability of related performance curves. Safety constraints on vessel intact stability

In terms of environmental data, it should be feasible to extend beyond the reliance on surface currents alone by incorporating the initial few depth layers of ocean models. Furthermore, VISIR-2 currently operates under the assumption of having perfect knowledge of meteo-oceanographic conditions, which is provided through forecast fields for shorter voyages or analysis fields for longer ones. The latter corresponds to retracked routes as discussed in

This paper presented the development of VISIR-2: a modular, validated, and portable Python-coded model for ship weather routing. It provides a consistent framework for both motor vessels and sailboats by accounting for dynamic environmental fields such as waves, currents, and wind. The model can compute optimal ship routes even in complex and archipelagic domains. A cartographic projection, a feature that had been overlooked up to now, has been introduced in VISIR-2. The model provides, for vessels with an angle-dependent performance curve, an improved level of accuracy in the velocity composition with sea currents. It is found that heading and course differ by an angle of attack, which is given by the solution of a transcendental equation (Eq.

The validation of VISIR-2 included comparisons to oracles and two inter-comparison exercises. Differently from the few available ship weather routing packages or services, the VISIR-2 software is accompanied by comprehensive documentation, making it suitable for community use.

The computational performance of the VISIR-2 shortest-path module displayed a significant enhancement compared to its predecessor, VISIR-1 (Sect.

While the model is general with respect to vessel type, two case studies with VISIR-2, based on realistic vessel seakeeping models, were documented in this paper.

From nearly 6000 routes of a 125 m long ferry, computed considering both waves and currents in the northwestern Mediterranean, average CO

From close to 3000 routes of an 11 m sailboat, within the southern Aegean Sea, accounting for both wind and currents, an average sailing time reduction of 2.4 % was observed. When considering currents as a factor, the duration of optimal routes could further be reduced by 3.2 %. Additionally, confirming prior work by

In summary, this paper provides comprehensive documentation of the scientific hypotheses and decisions underpinning the development of an open-source ship routing model while also contributing to the quantification of achievable reductions in greenhouse gas emissions through voyage optimisation.

The angle

Approximate vs. exact solution of Eq. (

Both Eqs. (

The outcome for a sailboat is provided in Fig.

The case

For identifying the vessel performance curves from a LUT via a neural network, a multi-layer perceptron was used.

The models were built and trained via the

The source code of VISIR-2 is available from

Videos for this paper are available at

The supplement related to this article is available online at:

GM: conceptualisation, funding acquisition, methodology, project administration, supervision, validation, writing (original draft), writing (review and editing); MLS: data curation, investigation, software, validation, visualisation; LC: data curation, investigation, software, validation, visualisation; NP: investigation, resources; JO: investigation, resources.

The contact author has declared that none of the authors has any competing interests.

The authors are not liable for casualties or losses that may occur in using routes computed via VISIR-2 for navigation purposes.Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

Both ChatGPT-3 and Gemini were utilised to review sections of this paper for English-language accuracy.

This research has been supported by Interreg (grant nos. 10043587 and 10253074) and Horizon Europe (grant nos. 101093293 and 101138583).

This paper was edited by David Ham and reviewed by two anonymous referees.