Finding hidden magnetic objects is difficult when their signals overlap. This study offers a way to count, locate, and distinguish several magnetic sources even when one source’s field covers another. Finding several hidden magnetic sources matters for detecting unexploded weapons, exploring mineral resources, studying spacecraft, and biological medical engineering.

The challenge is that the fields from multiple sources overlap and change in a nonlinear way with distance. So the task becomes determining how many sources are present and solving for each source’s position and magnetic strength. When a source is far enough from the measuring point, it can be treated like a tiny magnetic arrow.

Rather than separating every sound directly, the method uses a measure that falls off rapidly with distance, so it shows little mutual interference and becomes much larger near each magnetic source. This measure becomes much larger near a magnetic source than elsewhere, while signals from different sources interfere less with one another.

It also works when a source points only slightly upward or downward, and it does not care whether that direction is positive or negative. The process first uses local peaks in this measure to estimate how many sources exist. It then narrows the possible horizontal locations and combines a broad search with a faster refinement step to optimize the source parameters.

The broad search first finds an approximate solution close to the real parameter values. That approximate solution becomes the starting point for a second method, which can quickly converge to a solution meeting the required accuracy.

In noise-free conditions, both DE and DE-LM reached high-accuracy solutions, while DE-LM converged three point zero three zero one times faster than DE. Another comparison method settled on a locally optimal answer, showing poorer convergence when many source parameters had to be found.

With ten percent Gaussian noise, DE and DE-LM achieved similar parameter accuracy, with position errors of eleven and ten millimeters respectively and moment errors of zero point one four six and zero point two zero one ampere meters squared. The combined DE-LM search still converged seven point one zero five times faster than DE, preserving its speed advantage in the comparison.

Even with measurement noise, the maps place rose-red markers near the true locations of all forty magnetic sources. That means the method can both count hidden sources and estimate where they are, rather than losing them in the surrounding magnetic signal. The method still detects a smaller source when the field from a larger source completely covers it.

But both this method and the older tilt-based method have limited ability to detect sources that are very close together. The final results from the DE-LM optimization estimated the magnetic source position with a position error of twenty millimeters, as recorded in the reported results.

The recovered directions matched the planned positive and negative directions, and the magnetic strength matched the number of magnets. The method can detect some sources whose signals interfere, but stronger interference makes it difficult to estimate the correct number of sources.

Nearby magnetic sources can interfere, and as that interference increases, correctly estimating how many sources are present becomes difficult, a limitation shared by existing methods. The method is better at counting sources when their magnetic directions differ, and it can locate and classify them quickly.

That could make searches for buried objects and underground resources more dependable, though nearby sources can still confuse it.