The consequence of this is that sea based mines have been developed that recognise the magnetic anomaly caused by the signature and detonate. Equally, there is considerable activity in the measurement of magnetic anomalies – both from detection sub-sea and airborne – in an effort to establish which platform is in the vicinity. To overcome this, countermeasures, such as de-perming and degaussing, are often used to reduce signature. Nevertheless, magnetic anomaly detection (MAD) is still widely used and this has led to much interest in the solution of inverse problems to determine the platform causing the anomaly. Broadly stated, an inverse problem can be described as determination of an unknown source from a known response and there are many classic methods that can be applied .
In this paper, a method normally used in design synthesis ,  has been applied to the MAD inverse problem. Pareto methods are commonly used when solving design optimization problems with multiple, usually competing, objectives. For example, produce the maximum level of central field maintaining the best possible homogeneity over a defined volume using less than a pre-defined maximum current limit in a fixed number of coils. Continuous design variables, such as current in and position of each coil, result in a multi-dimensional design space to search, subject to the constraint on the current limit. The two objectives – homogeneity over the largest volume and maximum field – are likely to be in conflict. Homogeneity will tend to spread the position of the coils and make the current in each similar, whereas maximum central field will tend to concentrate the coils towards the centre with variable currents in each coil. One method of solving this multi-objective problem is to assign a weight to each objective and add all the weighted objectives to form a single objective – leading to only one optimal design. In contrast, Pareto methods produce a multidimensional optimization space with a family of optimal designs that are on the Pareto optimal “front”. These are designs for which the Pareto algorithm can find no other designs for which all of the objectives are improved. A simple two objective result is shown in figure 1. In this case, minimization of both objective functions leads to a family of six Pareto optimal designs. Some are better at minimizing one objective, others the second objective, while some offer different levels of compromise. The infeasible solutions also shown on the optimization space map are designs that have been investigated but fail one or more of the constraints applied. Feasible solutions satisfy the constraints but are “dominated” by the Pareto optimal designs.
In this paper, the same technique has been extended to measured induced signatures. Field values on a two dimensional grid below the vessel (as would typically be obtained from a range) are used in the objective functions. Two branches of the research are reported here: identification of a known source topology (a hollow steel tube with hemispherical end caps, but unknown dimensions and wall thickness) and an unknown topology (signatures from models of either a surface ship or a submarine). In all cases, the parameterized model that is used in the Pareto optimization is represented by a 2D thin-plate surface in a non-linear magnetostatic simulation using the Opera software , . The 2D thin-plate surface simplifies model and mesh construction by eliminating the requirement to handle high-aspect ratio volume structures (such as deck and bulkhead plates). It also gives considerable benefit in reducing simulation requirements (memory and run time) for the same accuracy, compared to the equivalent model constructed with the actual volumes.
2. Inversion study with a known topology
• 25 ≤ length ≤ 60 metres
• 3 ≤ radius ≤ 5 metres
• 100 ≤ thickness of steel ≤ 200 mm
Although this inversion is not dissimilar to the previous work undertaken on buried objects, the behaviour is not entirely similar. For the integral objectives, there were only 4 Pareto optimal designs found when the Earth’s field was applied in XZ and all were in the same cluster very close to the true dimensions for length and radius, as shown in figure 3a, where the parameter values have been normalized against the real values. However, finding the thickness of the steel was more varied, although the mean value of the 4 designs agreed to about 5%, as shown in table 1. As shown in figure 3b, when the Earth’s field was applied in YZ and integral objectives were used, only a single Pareto optimal design was found. This again showed very good agreement in length and radius prediction and was 4% in error for the steel thickness. Consequently, there is no need to identify clusters in each set of results and determine which cluster exists in both.
For the YZ applied field, as shown in figure 4(b), there is a cluster of designs within 1% of the correct values for both length and radius, comprising six designs: 1, 3, 4, 6, 7 and 11. There are also three clusters each containing 2 similar designs for length and radius: 10 and 13, 9 and 14, 2 and 8. These clusters tend to over predict one of the length or radius parameters and under predict the other one. However, when the field was applied in XZ, as discussed in the previous paragraph, no designs with the values in the small clusters emerged. Consequently, it is still possible to identify which of the clusters in the YZ applied field results is showing the approximate correct values. As with all other optimizations, the prediction of steel thickness is much more varied. However, the average of the six designs in the main cluster is 157 mm – only 4.7% in error.
Tables 1 and 2 also show that the cluster around the correct dimensions tends to dominate the overall results. Although smaller clusters may form, they do not provide sufficient weight, when the average value of all designs is taken, to disturb the average value. In three of the four optimizations, only the cluster in the vicinity of the actual values emerged, while in the Discrete Values YZ Applied Field results, the average length for the six designs in the dominant cluster is 45.47 metres and, for the radius, the average 3.994 metres. These are not significantly better than the mean for all designs (45.92 and 3.995 metres). Consequently, in the subsequent inversion from signatures of an unknown topology, it was decided to use average values from all the Pareto optimal designs rather than identify clusters. It was also decided to only use the integral objectives, as these perform more reliably (as shown from the standard deviation values in table 2).
3. Inversion study with unknown topologies
However, when MAD results are obtained, the type of vessel creating the signature is often unknown. Is it a small vessel nearby or a larger vessel at some distance? Is it a surface ship or a submarine? To investigate the possibility of using Pareto optimization as an inversion tool in this situation, it was decided to investigate models that might, at least, be able to predict overall dimensions of an unknown topology. Two unknown topologies were chosen: a surface ship, as shown in figure 5, and a submarine, shown in figure 6. The finite element models for these vessels have been cut away to reveal some detail of their internal structure. As can be seen, the ship contains both internal decks and bulkheads, while the submarine is double hulled. The overall dimensions of these vessels are also significantly different, as shown in table 3.
a. Hollow Ellipsoid
The ellipsoid is characterized by 4 dimensions: 3 half-axis lengths and the steel thickness. The ranges for the dimensions are shown in table 4.
b. Hollow spheres
5 hollow steel spheres were defined situated on the X-axis at centres x = -30, -15, 0, 15, 30 metres, each with their radius as a variable in the range 0.5 to 15 metres. Hence, a five dimensional design space is searched to obtain the Pareto optimal designs. This proved quite unsuccessful. For example, figure 7 shows the configuration obtained from the average parameters using the signature field from the ship model.
5 dimensions were used to define the geometry of the two prisms, shown in figure 8. These were:
• The lengths of the two prisms (2 parameters)
• The radii of the circumscribed circles of the equilateral triangles defining the prisms (2 parameters)
• The vertical height of the hull prism centre line (1 parameter)
Table 6 shows the range for each parameter
When it comes to assessing the performance of a prospective transformer, three tests are commonly carried out. These can be reproduced in the virtual world using simulation to calculate the parameters that we see here.
1) Open Circuit test
In the Open circuit test the secondary windings are not connected to load. When there is no load at the secondary windings of the transformer, no current flows through the ideal transformer windings. The impedances R1 and L1 are usually much smaller than the impedances R3 and L3 so the transformer equivalent circuit can be approximated.
Once the model is solved:
- Lorentz forces are calculated on the primary and secondary windings using Integrals over Conductors;
- Eddy Current and Iron losses can be calculated.
- No load losses can be calculated as the sum of the eddy current, iron and resistive losses; and
- The active power Pm is obtained from the iron loss calculation on the core. The apparent power (V1 I1) is calculated from the rms values of the voltage and current in the primary. Hence, the reactive power Qm, the iron loss resistance R3 and the inductance L3 are calculated.
2) Short Circuit test
During this test, one phase of the transformer secondary is short-circuited and the primary is connected to a power supply. The current in the primary is increased, until the current in the secondary reaches the rated value. This allows the short-circuit impedance of the transformer to be calculated by the simulation.
Once the model is solved:
- Lorentz forces and Eddy Current and Iron losses can be calculated.
- The active power Pm is obtained from the resistive loss calculation in the windings. The apparent power (V1*I1) is calculated from the rms values of the voltage and current in the primary. Hence, the reactive power Qm, the resistance R1 + R’2 and the inductance L1 + L’2 are calculated from the equations.
3) Inrush test
In this test either the voltage or the frequency is ramped up to the rated value at the rated current. Here we are interested in the voltages and currents in the circuits and the forces on the coils.
Once the model is solved:
- Lorentz forces can be calculated on the primary and secondary windings, and Eddy current and Iron losses can be calculated, as can the efficiency of the transformer.
The final simple approximation was more complex than the previous ones and, to some extent, is a closer approximation to the shape of a real vessel, as shown in figure 11. Twelve parameters were used to describe this approximation.
• The vertical height and the radius of the circumscribed circle for each prism (6 parameters)
• The length of each prism (3 parameters)
• The length, width and height of the cuboid (3 parameters)
Table 8 shows the range for each parameter
The integrals used in the magnetron work were subtly different from those used in the signature work reported here. For the signatures, a typical objective function to be minimized, f_x, was:
where S is the area of the grid over which samples have been measured. In the magnetron work, the objective function was defined as
The second of these expressions is usually to be preferred, as it will tend to minimize the error between calculated and measured results at every sample point on the grid. However, it has the disadvantage that the measured result at every sample point must be accessed every time the objective function is calculated. In the first expression, the integral of the measured results need only be evaluated prior to the optimization and its value used. However, the first expression is potentially dangerous as the integral of the calculated results may be similar to the measured, while the discrete values have many large positive and negative errors that are tending to cancel. The first expression is probably successful in the signature modelling, as the calculated signature fields from the models produced during the optimization always tend to have the same shape over the grid space as the measured results. Consequently, the possibility of significant cancelling errors at the discrete points is much reduced.
The results presented in section 3 show the fine line that it is necessary to draw between a sufficient number of variables to reasonably capture possible geometry and choosing too many variables. As could be seen from the three prism and cuboid results, in the time / maximum designs limit imposed on all four simple models for effective comparison, it was not possible to adequately sample twelve dimensional space to obtain a statistically significant set of Pareto optimal designs. For the submarine, the magnitude of the standard deviation on the 12 variables ranged from 30 to 80%. For the ship, the range was even larger (20 to 110%), although the majority were around 30 to 50%. With only 6 and 10 Pareto optimal designs for the submarine and ship respectively, a dominant cluster is not emerging.
However, it is also shown that choosing a topology that has some similarity to the expected type is of benefit. While the dimensional space of the ellipsoid and spheres are smaller, they are not sufficiently close to the real topology for good results. In this respect, the two prism model is the most successful with only a five dimensional design space and a topology that has the fundamentals of the ship or submarine (hull and superstructure or sail).
Further work should investigate whether the objective functions used could be improved (as discussed) and whether a better, low dimensional space model than the two prisms can be discovered.
For inversion when the topology is complex and unknown, the results have also shown that it is possible to predict overall dimensions of a platform using Pareto optimization. The model chosen should be sufficiently simple in variable dimension space to allow a dominant cluster of results to emerge but have some characteristics that reflect the type of topology for the expected sources. In this respect, a design consisting of two equilateral triangle cross-section hollow steel prisms requiring five variables for its definition was the most successful model examined so far. Prediction of length, beam and vertical height to within 15% of true dimensions, for both a 95 metre long surface ship and a 41 metre long submarine, was obtained. The same ranges for the variables were used in both cases.
The benefit of obtaining measurements when the Earth’s field is in a direction that produces a significant non-zero longitudinal, athwartships and vertical integral of the signature is also shown.
The author thanks Mr. Venkatesu Samala, Senior Application Engineer at ICON Design Automation, Bangalore, India for the creation of the submarine Opera finite element model. He also thanks his colleagues at Cobham Technical Services Opera Software for suggestions of simplified model topologies.
 John Holmes, “Exploitation of a Ship’s Magnetic Field Signatures – Synthesis Lectures on Computational Electromagnetics”, published by Morgan & Claypole, 2006, ISBN: 9781598290745
 John Holmes, “Reduction of a Ship’s Magnetic Field Signatures – Synthesis Lectures on Computational Electromagnetics”, published by Morgan & Claypole, 2008, ISBN: 9781598292480
 Paula Y. Zivi, Amy LeDoux, Robert D. Pillsbury, Jr., “Electromagnetic Signature of a Ship Rolling in Earth’s Magnetic Field”, presented at Compumag 2001, Evian, France
 P. Neittaanmäki, M Rudnicki, A. Savini, “Inverse Problems and Optimal Design in Electricity and Magnetism”, Oxford Science Publications, 1996, ISBN: 0-19-859383-X
 Glenn Hawe, Jan Sykulski, “Scalarizing cost-effective multi-objective optimization algorithms made possible with kriging”, COMPEL Journal, vol. 27, no. 4, 2008
 G. I. Hawe, J. K. Sykulski, “An enhanced probability of improvement utility function for locating Pareto-optimal solutions”, Proceedings of COMPUMAG 2007, pp. 965 – 966
 C. P. Riley, “Solving Inverse Electromagnetic Sensing Problems using Pareto Optimization”, Sensor Letters, Vol. 11, No. 1, January 2013
 C. S. Biddlecombe, C. P. Riley, “Improvements to finite element meshing for magnetic signature simulations”, Proceedings of Marelec 2015, poster 7.