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51 pages, 6461 KiB  
Article
The Quadrature Method: A Novel Dipole Localisation Algorithm for Artificial Lateral Lines Compared to State of the Art
by Daniël M. Bot, Ben J. Wolf and Sietse M. van Netten
Sensors 2021, 21(13), 4558; https://doi.org/10.3390/s21134558 - 2 Jul 2021
Cited by 3 | Viewed by 2573
Abstract
The lateral line organ of fish has inspired engineers to develop flow sensor arrays—dubbed artificial lateral lines (ALLs)—capable of detecting near-field hydrodynamic events for obstacle avoidance and object detection. In this paper, we present a comprehensive review and comparison of ten localisation algorithms [...] Read more.
The lateral line organ of fish has inspired engineers to develop flow sensor arrays—dubbed artificial lateral lines (ALLs)—capable of detecting near-field hydrodynamic events for obstacle avoidance and object detection. In this paper, we present a comprehensive review and comparison of ten localisation algorithms for ALLs. Differences in the studied domain, sensor sensitivity axes, and available data prevent a fair comparison between these algorithms from their original works. We compare them with our novel quadrature method (QM), which is based on a geometric property specific to 2D-sensitive ALLs. We show how the area in which each algorithm can accurately determine the position and orientation of a simulated dipole source is affected by (1) the amount of training and optimisation data, and (2) the sensitivity axes of the sensors. Overall, we find that each algorithm benefits from 2D-sensitive sensors, with alternating sensitivity axes as the second-best configuration. From the machine learning approaches, an MLP required an impractically large training set to approach the optimisation-based algorithms’ performance. Regardless of the data set size, QM performs best with both a large area for accurate predictions and a small tail of large errors. Full article
(This article belongs to the Section Sensing and Imaging)
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Figure 1

Figure 1
<p>Normalised continuous velocity profiles for five movement directions (indicated) of a source at <span class="html-italic">d</span> = 1 m from the sensor array’s centre: (<b>a</b>) <span class="html-italic">v<sub>x</sub></span> and (<b>b</b>) <span class="html-italic">v<sub>y</sub></span>. The sensors are located along the <span class="html-italic">x</span> axis and <span class="html-italic">b</span> is the source sphere is <span class="html-italic">x</span> position in m.</p>
Full article ">Figure 2
<p>A schematic view of the simulated environment. The source sphere (green) has a radius of 1 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math> and is shown to scale. A possible movement direction is shown by the arrow (not in scale). The sensor locations are shown in blue. Parallel <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> and perpendicular <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> velocity components are indicated at the right-most sensor (not to scale). The area in which the source sphere is positioned is offset by 25 mm from the array location, ensuring a minimal distance of 15 mm between the source’s edge and closest sensor’s centre.</p>
Full article ">Figure 3
<p>The signal to noise ratio (SNR) of both velocity components measured by the fifth sensor (<math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>2.86</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>). The top row shows contours of the median SNR in cells of <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math>. The bottom row shows the median SNR’s polar contours in cells of <math display="inline"><semantics> <mrow> <mn>0.02</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <mi>rad</mi> </semantics></math> × 2 cm for source states with an <span class="html-italic">x</span>-coordinate between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>7.14</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>12.86</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, indicating how the movement direction of a dipole <math display="inline"><semantics> <mi>φ</mi> </semantics></math> influences the SNR. Simulated potential flow measurements (Equation (<a href="#FD1-sensors-21-04558" class="html-disp-formula">1</a>)) and the same measurements with additive Gaussian distributed noise values (<math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>) were used to compute the SNR. Specifically, the SNR was computed as the frequency power ratio between the noisy measurements and noise floor at the source frequency (<math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>45</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi>Hz</mi> </semantics></math>). The frequency power was computed by a discrete Fourier transform (DFT) using a Hamming window.</p>
Full article ">Figure 4
<p>Graphical illustration of the movement direction estimation from the measured velocity and the source’s position <b><span class="html-italic">p</span></b> = 〈<span class="html-italic">b</span>, <span class="html-italic">d</span>〉 (<b>a</b>) A view of <span class="html-italic">ψ<sub>e</sub></span>(<span class="html-italic">ρ</span>) and <span class="html-italic">ψ<sub>o</sub></span>(<span class="html-italic">ρ</span>) along the sensor array. (<b>b</b>) The values of the wavelets can be interpreted as vectors (<math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mi>ψ</mi> <mo>→</mo> </mover> </mrow> <mi>e</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mi>ψ</mi> <mo>→</mo> </mover> </mrow> <mi>o</mi> </msub> </mrow> </semantics></math>) in a 3D <span class="html-italic">ψ</span><sub><span class="html-italic">e</span></sub>–<span class="html-italic">ψ</span><sub><span class="html-italic">e</span></sub>–<span class="html-italic">ρ</span> space. Their vector combination <math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mi>ψ</mi> <mo>→</mo> </mover> </mrow> <mrow> <mi>e</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mi>ψ</mi> <mo>→</mo> </mover> </mrow> <mrow> <mi>e</mi> </mrow> </msub> </mrow> </semantics></math> + <math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mi>ψ</mi> <mo>→</mo> </mover> </mrow> <mrow> <mi>o</mi> </mrow> </msub> </mrow> </semantics></math> is a fixed 3D wavelet structure that can be constructed solely from the source’s previously determined position <span class="html-italic"><b>p</b></span>. This vector <math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mi>ψ</mi> <mo>→</mo> </mover> </mrow> <mrow> <mi>e</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> has a magnitude <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ψ</mi> </mrow> <mrow> <mi>e</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <msqrt> <mrow> <msubsup> <mi>Ψ</mi> <mi>e</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>Ψ</mi> <mi>o</mi> <mn>2</mn> </msubsup> </mrow> </msqrt> </mrow> </semantics></math> and angle <span class="html-italic">ψ<sup>’</sup></span> = atan <span class="html-italic">ψ<sub>o</sub></span>/<span class="html-italic">ψ<sub>e</sub></span>. The measured velocities—which are linear combinations of <span class="html-italic">ψ<sub>e</sub></span> and <span class="html-italic">ψ<sub>o</sub></span>—can be viewed as a 2D projection of this 3D wavelet. For instance, a projection on the <span class="html-italic">ρ–ψ<sub>e</sub></span> plane (bottom plane) yields <span class="html-italic">ψ<sub>e</sub></span> for <span class="html-italic">φ</span> = 0 rad. For a general angle <span class="html-italic">φ</span>, the measured velocity profile is a projection on a plane through the <span class="html-italic">ρ</span> axis subtending an angle <span class="html-italic">φ</span> with the <span class="html-italic">ρ–φ<sub>e</sub></span> plane. (<b>c</b>) Diagram illustrating the geometric relation between a measured <span class="html-italic">v<sub>x</sub></span>, the angles <span class="html-italic">α</span> and <span class="html-italic">φ<sup>’</sup></span> which are constrained via <span class="html-italic">Ψ<sub>env</sub></span>, and the movement orientation <span class="html-italic">φ</span>. We show a slice of <math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mi>ψ</mi> <mo>→</mo> </mover> </mrow> <mrow> <mi>e</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> (green) in the <span class="html-italic">ψ<sub>e</sub></span>–<span class="html-italic">ψ<sub>o</sub></span> plane for a fixed value of <span class="html-italic">ρ</span>. The velocity value at this fixed <span class="html-italic">ρ</span> is a vector <math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mi>v</mi> <mo>→</mo> </mover> </mrow> <mi>x</mi> </msub> </mrow> </semantics></math> (black) in this space. It has a length <span class="html-italic">v<sub>x</sub></span> ∝ <span class="html-italic">ψ<sub>e</sub></span> cos <span class="html-italic">φ</span> + <span class="html-italic">ψ<sub>o</sub></span> sin <span class="html-italic">φ</span> and has angle <span class="html-italic">φ</span>. The contributions of <math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mi>ψ</mi> <mo>→</mo> </mover> </mrow> <mrow> <mi>e</mi> </mrow> </msub> </mrow> </semantics></math> (blue) and <math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mi>ψ</mi> <mo>→</mo> </mover> </mrow> <mrow> <mi>o</mi> </mrow> </msub> </mrow> </semantics></math> (orange) to <span class="html-italic">v<sub>x</sub></span> are shown in yellow and purple. The angle <span class="html-italic">φ<sup>’</sup></span> of ~ <math display="inline"><semantics> <mrow> <msub> <mrow> <mover> <mi>ψ</mi> <mo>→</mo> </mover> </mrow> <mrow> <mi>e</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> can be computed directly from an estimated source position. Given that the difference between <span class="html-italic">φ</span> and <span class="html-italic">φ<sup>’</sup></span> is <span class="html-italic">α</span> = acos <span class="html-italic">v<sub>x</sub></span>/<span class="html-italic">ψ<sub>env</sub></span>, we can compute an estimate of <span class="html-italic">φ</span> at every sensor using only measured velocity values and a position estimate.</p>
Full article ">Figure 5
<p>Total areas with a median position error <math display="inline"><semantics> <msub> <mi>E</mi> <mi mathvariant="bold-italic">p</mi> </msub> </semantics></math> (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) below 1 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>, 3 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>, 5 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>, and 9 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math> for the training and optimisation sets with a varying minimum distance between source states <math display="inline"><semantics> <msub> <mi>D</mi> <mi>s</mi> </msub> </semantics></math> (see <a href="#sec2dot3-sensors-21-04558" class="html-sec">Section 2.3</a> and <a href="#sensors-21-04558-t001" class="html-table">Table 1</a>) and the (x + y) sensor configuration at the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The median position error was computed for <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">c</mi> <msup> <mi mathvariant="normal">m</mi> <mn>2</mn> </msup> </mrow> </semantics></math> cells. Note, the bar for LSQ* is based the (x + y) condition in Analysis Method 2.</p>
Full article ">Figure 6
<p>Total areas with a median movement direction error <math display="inline"><semantics> <msub> <mi>E</mi> <mi>φ</mi> </msub> </semantics></math> (Equation (<a href="#FD5-sensors-21-04558" class="html-disp-formula">5</a>)) below <math display="inline"><semantics> <mrow> <mn>0.01</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <mi>rad</mi> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>0.03</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <mi>rad</mi> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>0.05</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <mi>rad</mi> </semantics></math> for the training and optimisation sets with a varying minimum distance between source states <math display="inline"><semantics> <msub> <mi>D</mi> <mi>s</mi> </msub> </semantics></math> (see <a href="#sec2dot3-sensors-21-04558" class="html-sec">Section 2.3</a> and <a href="#sensors-21-04558-t001" class="html-table">Table 1</a>) and the (x + y) sensor configuration at the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The median movement direction error was computed for <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">c</mi> <msup> <mi mathvariant="normal">m</mi> <mn>2</mn> </msup> </mrow> </semantics></math> cells. Note, the bar for LSQ* is based on the (x + y) condition in Analysis Method 2.</p>
Full article ">Figure 7
<p>Boxplots of the position error distributions for all dipole localisation algorithms in each condition of first analysis method. This analysis varied the minimum distance <math display="inline"><semantics> <msub> <mi>D</mi> <mi>s</mi> </msub> </semantics></math> between source states in the training and optimisation set (see <a href="#sec2dot3-sensors-21-04558" class="html-sec">Section 2.3</a> and <a href="#sensors-21-04558-t001" class="html-table">Table 1</a>) and used the (x + y) sensor configuration at the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The whiskers of the boxplots indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. LSQ* is based on the (x + y) condition in Analysis Method 2.</p>
Full article ">Figure 8
<p>Boxplots of the movement direction error distributions for all dipole localisation algorithms in each condition of first analysis method. This analysis varied the minimum distance <math display="inline"><semantics> <msub> <mi>D</mi> <mi>s</mi> </msub> </semantics></math> between source states in the training and optimisation set (see <a href="#sec2dot3-sensors-21-04558" class="html-sec">Section 2.3</a> and <a href="#sensors-21-04558-t001" class="html-table">Table 1</a>) and used the (x + y) sensor configuration at the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The whiskers of the boxplots indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. LSQ* is based on the (x + y) condition in Analysis Method 2.</p>
Full article ">Figure 9
<p>Total area with a median position error <math display="inline"><semantics> <msub> <mi>E</mi> <mi mathvariant="bold-italic">p</mi> </msub> </semantics></math> (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) below 1 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>, 3 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>, 5 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>, and 9 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math> for varying sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> for subsequent sensors, (x) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> at all sensors, (y) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> at all sensors. This analysis method used the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The median position error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">c</mi> <msup> <mi mathvariant="normal">m</mi> <mn>2</mn> </msup> </mrow> </semantics></math> cells. Note, the bar for QM* is based on the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition in Analysis Method 1.</p>
Full article ">Figure 10
<p>Total area with a median movement direction error <math display="inline"><semantics> <msub> <mi>E</mi> <mi mathvariant="bold-italic">p</mi> </msub> </semantics></math> (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) below 1 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>, 3 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>, 5 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>, and 9 <math display="inline"><semantics> <mi mathvariant="normal">c</mi> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math> for varying sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> for subsequent sensors, (x) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> at all sensors, (y) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> at all sensors. This analysis method used the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The median movement direction error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">c</mi> <msup> <mi mathvariant="normal">m</mi> <mn>2</mn> </msup> </mrow> </semantics></math> cells. Note, the bar for QM* is based on the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition in Analysis Method 1.</p>
Full article ">Figure 11
<p>Boxplots of the position error distributions for all algorithms in the second analysis method. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> for subsequent sensors, (x) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> at all sensors, (y) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> at all sensors. The <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level were used. The whiskers of the boxplots indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. QM* is based on the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition in Analysis Method 1.</p>
Full article ">Figure 12
<p>Boxplots of the movement direction error distributions for all algorithms in the second analysis method. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> for subsequent sensors, (x) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> at all sensors, (y) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> at all sensors. The <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level were used. The whiskers of the boxplots indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. QM* is based on the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition in Analysis Method 1.</p>
Full article ">Figure 13
<p>Spatial contours of the median position error <math display="inline"><semantics> <msub> <mi>E</mi> <mi mathvariant="bold-italic">p</mi> </msub> </semantics></math> (blue) (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) and median movement direction error <math display="inline"><semantics> <msub> <mi>E</mi> <mi>φ</mi> </msub> </semantics></math> (orange) (Equation (<a href="#FD5-sensors-21-04558" class="html-disp-formula">5</a>)) of the predictors using the largest training and optimisation set (<math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>) and 2D sensitive sensors (x + y) at the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The algorithms are ordered with an increasing overall median position error. The median errors were computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">c</mi> <msup> <mi mathvariant="normal">m</mi> <mn>2</mn> </msup> </mrow> </semantics></math> cells.</p>
Full article ">Figure 14
<p>These figures indicate how the movement direction <math display="inline"><semantics> <mi>φ</mi> </semantics></math> and distance <span class="html-italic">d</span> of a source state influence the median position error <math display="inline"><semantics> <msub> <mi>E</mi> <mi>p</mi> </msub> </semantics></math> (blue) (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) and median movement direction error <math display="inline"><semantics> <msub> <mi>E</mi> <mi>φ</mi> </msub> </semantics></math> (orange) (Equation (<a href="#FD5-sensors-21-04558" class="html-disp-formula">5</a>)). The quadrature method (QM), Gauss–Newton (GN), and least square curve fit (LSQ) predictors were used with three sensor configurations: (x + y), (x), (y) at the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The median errors were computed in cells of <math display="inline"><semantics> <mrow> <mn>0.01</mn> <mi>π</mi> </mrow> </semantics></math> <math display="inline"><semantics> <mi>rad</mi> </semantics></math>× 1 cm.</p>
Full article ">Figure 15
<p>An overview of the position error <span class="html-italic">E<sub>p</sub></span> (Equation (<a href="#FD5-sensors-21-04558" class="html-disp-formula">4</a>)) of QM, GN, and MLP using simulated sensors with higher velocity equivalent noise levels. (<b>a</b>) Total areas with a median position error <span class="html-italic">E<sub>p</sub></span> below 1 cm, 3 cm, 5 cm, and 9 cm. (<b>b</b>) Boxplots of the position error distributions, whiskers indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. The values for σ = 1.0 × 10<sup>‒5</sup> m s<sup>‒1</sup> are based on the <span class="html-italic">D<sub>s</sub></span> = 0.01 condition in Analysis Method 1. The (x + y) sensor configuration was used. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the <span class="html-italic">D<sub>s</sub></span> = 0.01 condition of Analysis Method 1.</p>
Full article ">Figure A1
<p>Velocity profiles of a continuous sensor array for five source movement directions <span class="html-italic">φ</span>: (<b>a</b>) <span class="html-italic">v<sub>x</sub></span> and (<b>b</b>) <span class="html-italic">v<sub>y</sub></span>. The envelopes of the velocity profiles <span class="html-italic">ψ<sub>x,env</sub></span>(<span class="html-italic">ρ</span>) and <span class="html-italic">ψ<sub>y,env</sub></span>(<span class="html-italic">ρ</span>) are shown in green.</p>
Full article ">Figure A2
<p>Quadrature profiles <math display="inline"><semantics> <msub> <mi>ψ</mi> <mrow> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> <mo>,</mo> <mi>n</mi> <mi>o</mi> <mi>r</mi> <mi>m</mi> </mrow> </msub> </semantics></math> (left panel) of a continuous sensor array for five source directions <math display="inline"><semantics> <mi>φ</mi> </semantics></math> (same as <a href="#sensors-21-04558-f0A1" class="html-fig">Figure A1</a>). Furthermore, the two constituting functions <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mrow> <mi>s</mi> <mi>y</mi> <mi>m</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>ρ</mi> <mo>,</mo> <mi>φ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (middle panel) and <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mrow> <mi>s</mi> <mi>k</mi> <mi>e</mi> <mi>w</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>ρ</mi> <mo>,</mo> <mi>φ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (right panel) are shown. These functions are, respectively, even and odd in <math display="inline"><semantics> <mi>ρ</mi> </semantics></math>. At the secondary anchor points <math display="inline"><semantics> <mrow> <mo>±</mo> <msub> <mi>ρ</mi> <mrow> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> </mrow> </msub> </mrow> </semantics></math> the function <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mrow> <mi>s</mi> <mi>y</mi> <mi>m</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>ρ</mi> <mo>,</mo> <mi>φ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> provides angle independent values which may be employed to determine the source distance <span class="html-italic">d</span> before the source direction of motion <math display="inline"><semantics> <mi>φ</mi> </semantics></math> is known.</p>
Full article ">Figure A3
<p>The width between the secondary anchor points <math display="inline"><semantics> <mrow> <mo>±</mo> <msub> <mi>ρ</mi> <mrow> <mi>a</mi> <mi>n</mi> <mi>c</mi> <mi>h</mi> </mrow> </msub> </mrow> </semantics></math> of the normalised quadrature curve <math display="inline"><semantics> <msub> <mi>ψ</mi> <mrow> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> <mo>,</mo> <mi>n</mi> <mi>o</mi> <mi>r</mi> <mi>m</mi> </mrow> </msub> </semantics></math> are almost constant <math display="inline"><semantics> <mrow> <mn>1.78</mn> <mo>±</mo> <mn>0.011</mn> </mrow> </semantics></math> with respect to the movement direction angle <math display="inline"><semantics> <mi>φ</mi> </semantics></math> and is well approximated by the analytically determined value <math display="inline"><semantics> <mrow> <mn>4</mn> <mo>/</mo> <msqrt> <mn>5</mn> </msqrt> </mrow> </semantics></math>.</p>
Full article ">Figure A4
<p>Spatial contours of the median position error (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) for each localisation algorithm in all conditions of Analysis Method 1. This analysis varied the minimum distance <math display="inline"><semantics> <msub> <mi>D</mi> <mi>s</mi> </msub> </semantics></math> between sources in the training and optimisation sets (<a href="#sensors-21-04558-t001" class="html-table">Table 1</a>) while using the (x + y) sensor configuration at the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The median position error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
Full article ">Figure A5
<p>Spatial contours of the median movement direction error (Equation (<a href="#FD5-sensors-21-04558" class="html-disp-formula">5</a>)) for each localisation algorithm in all conditions of Analysis Method 1. This analysis varied the minimum distance <math display="inline"><semantics> <msub> <mi>D</mi> <mi>s</mi> </msub> </semantics></math> between sources in the training and optimisation sets (<a href="#sensors-21-04558-t001" class="html-table">Table 1</a>) while using the (x + y) sensor configuration at the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The median movement direction error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
Full article ">Figure A6
<p>Polar contours of the median position error (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) for each localisation algorithm in all conditions of Analysis Method 1. These figures indicate how the movement direction <math display="inline"><semantics> <mi>φ</mi> </semantics></math> and distance <span class="html-italic">d</span> of a source influence the error. This analysis varied the minimum distance <math display="inline"><semantics> <msub> <mi>D</mi> <mi>s</mi> </msub> </semantics></math> between sources in the training and optimisation sets (<a href="#sensors-21-04558-t001" class="html-table">Table 1</a>) while using the (x + y) sensor configuration at the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The median position error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
Full article ">Figure A7
<p>Polar contours of the median movement direction error (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) for each localisation algorithm in all conditions of Analysis Method 1. These figures indicate how the movement direction <math display="inline"><semantics> <mi>φ</mi> </semantics></math> and distance <span class="html-italic">d</span> of a source influence the error. This analysis varied the minimum distance <math display="inline"><semantics> <msub> <mi>D</mi> <mi>s</mi> </msub> </semantics></math> between sources in the training and optimisation sets (<a href="#sensors-21-04558-t001" class="html-table">Table 1</a>) while using the (x + y) sensor configuration at the <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level. The median movement direction error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
Full article ">Figure A8
<p>Spatial contours of the median position error (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) for each localisation algorithm in all conditions of Analysis Method 2. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> for subsequent sensors, (x) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> at all sensors, (y) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> at all sensors. The <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level were used. The median position error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
Full article ">Figure A9
<p>Spatial contours of the median movement direction error (Equation (<a href="#FD5-sensors-21-04558" class="html-disp-formula">5</a>)) for each localisation algorithm in all conditions of Analysis Method 2. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> for subsequent sensors, (x) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> at all sensors, (y) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> at all sensors. The <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level were used. The median movement direction error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
Full article ">Figure A10
<p>Polar contours of the median position error (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) for each localisation algorithm in all conditions of Analysis Method 2. These figures indicate how the movement direction <math display="inline"><semantics> <mi>φ</mi> </semantics></math> and distance <span class="html-italic">d</span> of a source influence the error. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> for subsequent sensors, (x) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> at all sensors, (y) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> at all sensors. The <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level were used. The median position error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
Full article ">Figure A11
<p>Polar contours of the median movement direction error (Equation (<a href="#FD5-sensors-21-04558" class="html-disp-formula">5</a>)) for each localisation algorithm in all conditions of Analysis Method 2. These figures indicate how the movement direction <math display="inline"><semantics> <mi>φ</mi> </semantics></math> and distance <span class="html-italic">d</span> of a source influence the error. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> for subsequent sensors, (x) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> at all sensors, (y) measured only <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> at all sensors. The <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> noise level were used. The median movement direction error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
Full article ">Figure A12
<p>An overview of the movement direction error <span class="html-italic">E<sub>ϖ</sub></span> Equation (5) of QM, GN, and MLP using simulated sensors with higher velocity equivalent noise levels. (<b>a</b>) Total areas with a median movement direction error <span class="html-italic">E<sub>φ</sub></span> below 1 cm, 3 cm, 5 cm, and 9 cm. (<b>b</b>) Boxplots of the movement direction error distributions, whiskers indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. The values for <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> are based on the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition in Analysis Method 1. The (x + y) sensor configuration was used. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition of Analysis Method 1.</p>
Full article ">Figure A13
<p>Spatial contours of the median position error (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) for QM, GN, and MLP using simulated sensors with higher velocity equivalent noise levels. The <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and (x + y) sensor configuration were used. The values for <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> are based on the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition in Analysis Method 1. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition of Analysis Method 1. The median position error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
Full article ">Figure A14
<p>Spatial contours of the median movement direction error (Equation (<a href="#FD5-sensors-21-04558" class="html-disp-formula">5</a>)) for QM, GN, and MLP using simulated sensors with higher velocity equivalent noise levels. The <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and (x + y) sensor configuration were used. The values for <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> are based on the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition in Analysis Method 1. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition of Analysis Method 1. The median movement direction error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
Full article ">Figure A15
<p>Polar contours of the median position error (Equation (<a href="#FD4-sensors-21-04558" class="html-disp-formula">4</a>)) for QM, GN, and MLP using simulated sensors with higher velocity equivalent noise levels. These figures indicate how the movement direction <math display="inline"><semantics> <mi>φ</mi> </semantics></math> and distance <span class="html-italic">d</span> of a source influence the error. The <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and (x + y) sensor configuration were used. The values for <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> are based on the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition in Analysis Method 1. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition of Analysis Method 1. The median position error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
Full article ">Figure A16
<p>Polar contours of the median movement direction error (Equation (<a href="#FD5-sensors-21-04558" class="html-disp-formula">5</a>)) for each localisation algorithm in all conditions of Analysis Method 2. These figures indicate how the movement direction <math display="inline"><semantics> <mi>φ</mi> </semantics></math> and distance <span class="html-italic">d</span> of a source influence the error. The <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> training and optimisation set and (x + y) sensor configuration were used. The values for <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1.0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math> are based on the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition in Analysis Method 1. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> condition of Analysis Method 1. The median movement direction error was computed in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <msup> <mi>cm</mi> <mn>2</mn> </msup> </semantics></math> cells. The sensors were equidistantly placed between <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>.</p>
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33 pages, 4927 KiB  
Article
A Unified Model for Plasticity in Ferritic, Martensitic and Dual-Phase Steels
by Shuntaro Matsuyama and Enrique I. Galindo-Nava
Metals 2020, 10(6), 764; https://doi.org/10.3390/met10060764 - 8 Jun 2020
Cited by 6 | Viewed by 3586
Abstract
Unified equations for the relationships among dislocation density, carbon content and grain size in ferritic, martensitic and dual-phase steels are presented. Advanced high-strength steels have been developed to meet targets of improved strength and formability in the automotive industry, where combined properties are [...] Read more.
Unified equations for the relationships among dislocation density, carbon content and grain size in ferritic, martensitic and dual-phase steels are presented. Advanced high-strength steels have been developed to meet targets of improved strength and formability in the automotive industry, where combined properties are achieved by tailoring complex microstructures. Specifically, in dual-phase (DP) steels, martensite with high strength and poor ductility reinforces steel, whereas ferrite with high ductility and low strength maintains steel’s formability. To further optimise DP steel’s performance, detailed understanding is required of how carbon content and initial microstructure affect deformation and damage in multi-phase alloys. Therefore, we derive modified versions of the Kocks–Mecking model describing the evolution of the dislocation density. The coefficient controlling dislocation generation is obtained by estimating the strain increments produced by dislocations pinning at other dislocations, solute atoms and grain boundaries; such increments are obtained by comparing the energy required to form dislocation dipoles, Cottrell atmospheres and pile-ups at grain boundaries, respectively, against the energy required for a dislocation to form and glide. Further analysis is made on how thermal activation affects the efficiency of different obstacles to pin dislocations to obtain the dislocation recovery rate. The results are validated against ferritic, martensitic and dual-phase steels showing good accuracy. The outputs are then employed to suggest optimal carbon and grain size combinations in ferrite and martensite to achieve highest uniform elongation in single- and dual-phase steels. The models are also combined with finite-element simulations to understand the effect of microstructure and composition on plastic localisation at the ferrite/martensite interface to design microstructures in dual-phase steels for improved ductility. Full article
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>(<b>a</b>) Multi layered structure and interior of martensite. Schematic images of (<b>b</b>) dislocation density increase by pinning obstacles and (<b>c</b>) increase in glide distance at a constant dislocation density.</p>
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<p>Schematic illustration of dislocation interaction with carbon atoms.</p>
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<p>Schematic illustration of the process of a gliding dislocation to overcome a dislocation forest: (<b>a</b>) intersection of forest dislocations; (<b>b</b>) glide with jogs; (<b>c</b>) pinch-off of dipoles; and (<b>d</b>) glide without jogs.</p>
Full article ">Figure 4
<p>Schematic representation of dislocation pile-up at a grain boundary.</p>
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<p>Schematic illustration of the volume and the area of atoms involved in an interaction with: (<b>a</b>) dislocation; and (<b>b</b>) grain boundary.</p>
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<p>Schematic illustration of mesh and morphology used in FEM for DP5.</p>
Full article ">Figure 7
<p>Model-Experiment plot for: (<b>a</b>) <span class="html-italic">K</span>; (<b>b</b>) <span class="html-italic">f</span>; and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>Effectiveness of <math display="inline"><semantics> <msub> <mi>X</mi> <mi>C</mi> </msub> </semantics></math> on: (<b>a</b>) <span class="html-italic">K</span>; and (<b>b</b>) <span class="html-italic">f</span>.</p>
Full article ">Figure 9
<p>Effectiveness of <math display="inline"><semantics> <msub> <mi>D</mi> <mi>g</mi> </msub> </semantics></math> on: (<b>a</b>) <span class="html-italic">K</span>; and (<b>b</b>) <span class="html-italic">f</span>.</p>
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<p>Model-Experiment plot of the yield strength.</p>
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<p>Results of stress–strain curve prediction (Martensitic steels) for several values of: (<b>a</b>) <math display="inline"><semantics> <msub> <mi>D</mi> <mi>g</mi> </msub> </semantics></math>; and (<b>b</b>) <math display="inline"><semantics> <msub> <mi>X</mi> <mi>c</mi> </msub> </semantics></math>.</p>
Full article ">Figure 12
<p>Stress–strain curve prediction for various <math display="inline"><semantics> <msub> <mi>D</mi> <mi>g</mi> </msub> </semantics></math> of Ferritic steels.</p>
Full article ">Figure 13
<p>Stress–strain curve predictions for DP steels which martensite volume fraction <math display="inline"><semantics> <msub> <mi>V</mi> <mi>M</mi> </msub> </semantics></math> is: (<b>a</b>) less than 40%; (<b>b</b>) 40% to 50%; and (<b>c</b>) larger than 50%, respectively.</p>
Full article ">Figure 14
<p>Optical micrographs of (<b>a</b>) DP5, reproduced from [<a href="#B63-metals-10-00764" class="html-bibr">63</a>], with permission from Elsevier, 1992, and (<b>b</b>) DP2, reproduced from [<a href="#B64-metals-10-00764" class="html-bibr">64</a>], with permission from John Wiley and Sons, 2016, respectively, where the areas indicated by red broken lines were used in FEM simulations.</p>
Full article ">Figure 15
<p>Comparison of stress–strain curve prediction between iso-work and FEM in DP5 and DP2.</p>
Full article ">Figure 16
<p><math display="inline"><semantics> <mrow> <mi>K</mi> <mo>/</mo> <mi>f</mi> </mrow> </semantics></math> against: (<b>a</b>) <math display="inline"><semantics> <msub> <mi>X</mi> <mi>C</mi> </msub> </semantics></math>; and (<b>b</b>) <math display="inline"><semantics> <msub> <mi>D</mi> <mi>g</mi> </msub> </semantics></math>.</p>
Full article ">Figure 17
<p><math display="inline"><semantics> <msub> <mi>ϵ</mi> <mrow> <mi>n</mi> <mi>e</mi> <mi>c</mi> <mi>k</mi> </mrow> </msub> </semantics></math> against (<b>a</b>) <math display="inline"><semantics> <msub> <mi>X</mi> <mi>C</mi> </msub> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <msub> <mi>D</mi> <mi>g</mi> </msub> </semantics></math> in ferrite and martensite.</p>
Full article ">Figure 18
<p>Distributions of plastic equivalent strain of ferrite at the start of necking in: (<b>a</b>) DP5; and (<b>b</b>) DP2.</p>
Full article ">Figure 19
<p>Ferrite volume fraction in each cross section: (<b>a</b>) from upper left to lower right in DP5; (<b>b</b>) from upper right to lower left in DP5; (<b>c</b>) from upper left to lower right in DP2; and (<b>d</b>) from upper right to lower left in DP2.</p>
Full article ">Figure 20
<p>(<b>a</b>) Schematic images of three simplified microstructures; and (<b>b</b>) ferrite volume fraction of each cross section in those microstructures.</p>
Full article ">Figure 21
<p>Distributions of plastic equivalent strain in simplified microstructures: (<b>a</b>) DP5-chess-board; (<b>b</b>) DP5-lattice; (<b>c</b>) DP5-chess-board (small grain); (<b>d</b>) DP2-chess-board; and (<b>e</b>) DP2-lattice, respectively.</p>
Full article ">Figure 22
<p>Plastic equivalent strain distribution in DP5-chess-board assuming (<b>a</b>) 3D morphology and its (<b>b</b>) respective distribution on the x-y plane. (<b>c</b>) Microstructure and load configuration used for the simulation.</p>
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22 pages, 9523 KiB  
Article
Geophysical Investigation of the Pb–Zn Deposit of Lontzen–Poppelsberg, Belgium
by Maxime Evrard, Gaël Dumont, Thomas Hermans, Michel Chouteau, Olivier Francis, Eric Pirard and Frédéric Nguyen
Minerals 2018, 8(6), 233; https://doi.org/10.3390/min8060233 - 29 May 2018
Cited by 22 | Viewed by 10282
Abstract
The drillhole information from the Lontzen–Poppelsberg site has demonstrated three orebodies and has allowed the estimation of the extension of the lodes, their dip, and the location at the ground surface. The localisation of the lodes makes them excellent targets for further exploration [...] Read more.
The drillhole information from the Lontzen–Poppelsberg site has demonstrated three orebodies and has allowed the estimation of the extension of the lodes, their dip, and the location at the ground surface. The localisation of the lodes makes them excellent targets for further exploration with geophysics. This deposit is classified as a Mississippi Valley Type (MVT) deposit. It consists mainly of Pb–Zn–Fe sulphides that display contrasting values in resistivity, chargeability, density, and magnetic susceptibility, with regards to the sedimentary host rocks. The dipole–dipole direct current (DC) resistivity and induce polarization (IP) profiles have been collected and inverted to successfully delineate the Pb–Zn mineralization and the geological structures. Short-spacing EM34 electromagnetic conductivity data were collected mainly on the top of Poppelsberg East lode and have revealed a conductive body matching with the geologically modelled mineralization. Gravity profiles have been carried out perpendicularly to the lode orientation; they show a strong structural anomaly. High resolution ground magnetic data were collected over the study area, but they showed no anomaly over the ore deposits. The geophysical inversion results are complementary to the model based on drill information, and allow us to refine the delineation of the mineralization. The identification of the geophysical signatures of this deposit permits targeting new possible mineralization in the area. Full article
(This article belongs to the Special Issue Mining and Mineral Exploration Geophysics)
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Figure 1

Figure 1
<p>Petrophysical properties of Pb–Zn sedimentary ore deposits, represented by black rectangles (modified after [<a href="#B18-minerals-08-00233" class="html-bibr">18</a>]). Geophysical properties are represented by white rectangles.</p>
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<p>Location of the Belgian Mississippi Valley Type (MVT) ore deposit (<b>top</b>) (modified after [<a href="#B23-minerals-08-00233" class="html-bibr">23</a>]) and their formation process (after [<a href="#B23-minerals-08-00233" class="html-bibr">23</a>,<a href="#B24-minerals-08-00233" class="html-bibr">24</a>,<a href="#B25-minerals-08-00233" class="html-bibr">25</a>,<a href="#B26-minerals-08-00233" class="html-bibr">26</a>,<a href="#B27-minerals-08-00233" class="html-bibr">27</a>,<a href="#B28-minerals-08-00233" class="html-bibr">28</a>,<a href="#B29-minerals-08-00233" class="html-bibr">29</a>]) (<b>bottom</b>).</p>
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<p>Geology of the Lontzen–Poppelsberg ore deposit showing projection of the drillhole and modelling of the deposit (geological map modified from Laloux et al. [<a href="#B30-minerals-08-00233" class="html-bibr">30</a>]).</p>
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<p>Typical chargeability decay curves (Time domain induce polarization (IP)). (<b>A</b>,<b>B</b>) exponential curves type are kept while non-exponential decreasing curves (<b>C</b>,<b>D</b>) type are rejected.</p>
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<p>Depth of investigation (DOI) models for electrical resistivity tomography (ERT) and IP data after processing for Profiles 2 and 10. High reliability areas have DOI lower than 0.1.</p>
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<p>Resistivity results showing DOI 0.1 isolines and drillholes. The rectangles on the drillings represent mineralized area. Colour of these rectangles indicate the distance to the drillholes from the tomographic section, as follows: white rectangles: &lt;10 m; grey rectangles: between 10 m to 20 m; and dark rectangles &gt;20 m. A, B, C, D and E correspond to the anomalies which could be attributed to Pb–Zn mineralization (geological map modified from Laloux et al. [<a href="#B30-minerals-08-00233" class="html-bibr">30</a>]). Profiles 11 and 12 were combined in one single profile by juxtaposition.</p>
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<p>IP profiles (normalized chargeability) displaying drillings and mineralized area. Rectangles represent the thickness and location of the Pb–Zn mineralization. Colour of these rectangles indicate the distance of the drillings from the tomographic section, as follows: white rectangles: &lt;10 m; grey rectangles: between 10 m to 20 m; and dark rectangles &gt;20 m. The percentage on the right of each profile indicates the proportion of remaining data after data selection (see <a href="#minerals-08-00233-f004" class="html-fig">Figure 4</a>). The white dashed lines represent the DOI limit corresponding to a value of 0.1. A, B, and C corresponds to the anomalies attributed to Pb–Zn mineralization (geological map modified from Laloux et al. [<a href="#B30-minerals-08-00233" class="html-bibr">30</a>]).</p>
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<p>Electromagnetic results on the Poppelsberg East lode. (<b>A</b>) 20 m coaxial, (<b>B</b>) 40 m coaxial, (<b>C</b>) 20 m coplanar, and (<b>D</b>) 40 m coplanar. Ppe—Poppelsberg East lode; Ppw—Poppelsberg West lode.</p>
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<p>Bouguer gravity anomaly after processing of the dataset (geological map modified from Laloux et al. [<a href="#B30-minerals-08-00233" class="html-bibr">30</a>]). Units are µGal. The rectangles present on the profiles correspond to the supposed location of the Pb–Zn vein according to the 3D modelling.</p>
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<p>Bouguer anomaly map of the survey area. Gravity data were interpolated using the minimum curvature method with a cell size of 2.5 m<sup>2</sup>. Ppe—Poppelsberg East mineralization; Ppw—Poppelsberg West mineralization. Circled f2, f3, and f4 correspond to extensional faults. First 300 m of gravity Profile 4 have been removed in this map because they correspond to inconstant measures in swamp area.</p>
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<p>Conceptual model (<b>right down</b>) from gravity and ERT observation along Profile A–B oriented SSW–NNE (geological map modified from Laloux et al. [<a href="#B30-minerals-08-00233" class="html-bibr">30</a>]).</p>
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<p>2D modelling of resistivity data of Profile 2 (1) with Res2DMOD [<a href="#B54-minerals-08-00233" class="html-bibr">54</a>]. Inversion of the two parallel lodes model (2) and the horizontal lode model (3) are represented at (2′) and (3′), respectively. As seen in (2′), the only presence of two conductive lodes, as supposed in the 3D model, cannot reproduce the low resistivity anomaly (1). The presence of a horizontal conductive layer between the two vertical lodes (3) is more appropriate to explain the geophysical signature of (1).</p>
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<p>2D modelling of the IP data in the case of Profile 2 (1) with Res2DMOD [<a href="#B54-minerals-08-00233" class="html-bibr">54</a>]. Inversion of the two parallel lodes model (2) and the horizontal lode model (3) are represented at (2′) and (3′). As seen in (2′), the presence of only the two conductive lodes, as assumed in the 3D model, cannot explain the low chargeability anomaly (1). The presence of a horizontal conductive layer between the two vertical lodes (3) is more appropriate to explain the geophysical signature of (1).</p>
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<p>Redrawing of the Pb–Zn lodes at the ground surface using geophysical survey information. Areas 1–3 represent possible extensions of the ore deposit according to geophysical investigation (geological map modified from Laloux et al. [<a href="#B30-minerals-08-00233" class="html-bibr">30</a>]).</p>
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<p>On the left: Resistivity results showing DOI 0.1 isolines and drillholes. The rectangles on the drillings represent mineralized area. On the right: IP profiles (normalized chargeability) displaying drillings and mineralized area. Rectangles represent the thickness and location of the Pb–Zn mineralization. The percentage on the right of each profile indicates the proportion of remaining data after data selection (see <a href="#minerals-08-00233-f004" class="html-fig">Figure 4</a>). The white dashed lines represent the DOI limit corresponding to a value of 0.1.</p>
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