VAUHKONEN, Marko (Tiilisaarentie 4 B 4, Kuopio, FI-70100, FI)
REUNANEN, Juha (Kustaankatu 9 B 28, Helsinki, FI-00500, FI)
LEHIKOINEN, Anssi (Hauenkoukku 18 A 3, Kuopio, FI-70700, FI)
KAIPIO, Jari (507 East Coast Road, Murrays BayNorth Shore, 0630, NZ)
VAUHKONEN, Marko (Tiilisaarentie 4 B 4, Kuopio, FI-70100, FI)
REUNANEN, Juha (Kustaankatu 9 B 28, Helsinki, FI-00500, FI)
LEHIKOINEN, Anssi (Hauenkoukku 18 A 3, Kuopio, FI-70700, FI)
| CLAIMS 1. A method for detecting the locations of a material boundary or boundaries included in a volume comprising at least one material, wherein at least part of the volume is liquid material and wherein a set of probes further comprising at least one probe is used, the method comprising the steps of: introducing a set of probes comprising at least one pair of electrodes in a volume under examination; feeding current or voltage via at least one pair of electrodes, and measuring current or voltage from at least one pair of electrodes; and using a measurement geometry representing the arrangement in the calculation, and selecting a calcula- tion algorithm needed in the calculation; c h a r a c t e r i z e d in that the method further comprises the steps of: disposing at least three electrodes of the set of probes in the volume under examination in an assembly substantially differing from a straight line, all electrodes of the set of probes being located only in the volume of the liquid material; calculating the electrical conductivity distribution in the volume under examination on the basis of the measurement results; and concluding, on the basis of the electrical conductivity distribution and electrode location information, the location of at least one material boundary in the volume under examination. 2. The method according to claim 1, c h a r a c t e r i z e d in that the method further comprises the step of: carrying out the feeding and measuring step using electric impedance tomography (EIT) or electric resis- tivity tomography (ERT) . 3. The method according to claim 2, c h a r a c t e r i z e d in that the method further comprises the step of: disposing the electrodes on at least one probe of the set of probes such that the vectors between electrodes span a three-dimensional subspace. 4. The method according to claim 2, c h a r a c t e r i z e d in that the method further comprises the step of: disposing the electrodes on at least two separate probes, the electrodes on each probe being substantially disposed on a straight line. 5. The method according to claim 2, c h a r a c t e r i z e d in that the method further comprises the step of: using a standard inverse problem lD-o-method in the calculation to solve the planar profiles of the boundaries . 6. The method according to claim 2, c h a r - a c t e r i z e d in. that the method further comprises the step of: determining a function of the measured voltage or current values for desired numerical values representing the boundary by methods of machine learning. 7. The method according to claim 6, c h a r a c t e r i z e d in that the method further comprises the step of: using the results obtained from simulation as training material in the methods of machine learning. 8. The method according to claim 6, c h a r a c t e r i z e d in that the employed method of machine learning is MLP-network (Multi-Layer Perceptron) . 9. The method according to claim 8, c h a r a c t e r i z e d in that the MLP-network is trained by using the Levenberg-Marquardt algorithm. 10. The method according to claim 2, c h a r a c t e r i z e d in that the method further comprises the step of: monitoring the fouling of the electrodes by esti- mating the contact impedances between each electrode and the surrounding material. 11. A system for detecting the locations of a material boundary or boundaries included in a volume comprising at least one material, wherein at least part of the volume is liquid material, and wherein a set of probes further comprising at least one probe is used, the system comprising: a set of probes (40, 52, 52a, 52b) comprising at least one pair of electrodes (41, 52) disposed in a volume under examination; feeding means (53) for feeding current or voltage via at least one pair of electrodes, and measuring means (53) for measuring voltage or current from at least one pair of electrodes; and a processor (53) using a measurement geometry representing the arrangement in the calculation and a selected calculation algorithm needed in the calculation; c h a r a c t e r i z e d in that the system further comprises: at least three electrodes of the set of probes (40, 52, 52a, 62b) disposed in the volume under examination in an assembly substantially differing from a straight line, all electrodes of the set of probes being located only in the volume of the liquid material; said processor (53) for calculating the electrical conductivity distribution in the volume under examination on the basis of the measurement results; and said processor (53) for concluding the location of at least one possible material boundary in the volume under examination on the basis of the electrical con- ductivity distribution and electrode location information. 12. The system according to claim 11, c h a r a c t e r i z e d in that the system further com- prises: current feeding and measuring means (53) arranged to carry out the feeding and measuring steps using electric impedance tomography (EIT) or electric resis¬ tivity tomography (ERT) . 13. The system according to claim 12, c h a r a c t e r i z e d in that the system further comprises: electrodes (41i, 52i) disposed on at least one probe (40, 52, 52a, 52b) of the set of probes such that the vectors between electrodes (41i, 52i) span a three-dimensional subspace. 14. The system according to claim 12, c h a r a c t e r i z e d in that the system further comprises : electrodes (52ia,b) disposed on at least two separate probes (52a, 52b), the electrodes on each probe being substantially disposed on a straight line. 15. The system according to claim 12, c h a r a c t e r i z e d in that the system further com- prises: said processor (53) arranged to use a standard inverse problem lD-o-method in the calculation to solve the planar profiles of the boundaries. 16. The system according to claim 12, c h a r a c t e r i z e d in that the system further comprises : said processor (53) arranged to determine a function of the measured voltage or current values for desired numerical values representing the boundary by methods of machine learning. 17. The system according to claim 16, c h a r a c t e r i z e d in that the system further comprises : said processor (53) arranged to use the results obtained from simulation as training material in the methods of machine learning. 18. The system according to claim 16, c h a r a c t e r i z e d in that the employed method of machine learning is MLP-network (Multi-Layer Percep- tron) . 19. The system according to claim 18, c h a r a c t e r i z e d in that the MLP-network is trained by using the Levenberg-Marquardt algorithm. 20. The system according to claim 12, c h a r a c t e r i z e d in that the system further comprises : said measuring means (53) for monitoring the fouling of the electrodes by estimating the contact impedances between each electrode (41i, 52i) and the sur- rounding material. 21. A computer program for detecting the locations of a material boundary or boundaries included in a volume comprising at least one material, wherein at least part of the volume is liquid material, where- in a set of probes further comprising at least one probe is used, and wherein the set of probes comprising at least one pair of electrodes is disposed in a volume under examination, the computer program comprising program code which, when run on a data pro- cessing device, is arranged to control the following steps : feeding current or voltage via at least one pair of electrodes, and measuring current or voltage from at least one pair of electrodes; and using a measurement geometry representing the arrangement in the calculation, and selecting a calculation algorithm needed in the calculation; c h a r a c t e r i z e d in that at least three electrodes of the set of probes are disposed in the volume under examination in an assembly substantially differing from a straight line, all electrodes of the set of probes being located only in the volume of the liquid material; and wherein the computer program is further arranged to control the following steps: calculating the electrical conductivity distribution in the volume under examination on the basis of the measurement results; and concluding, on the basis of the electrical conductivity distribution and electrode location information, the location of at least one material boundary in the volume under examination. |
FIELD OF THE INVENTION
The invention is applied in different industrial processes involving precipitation, sedimentation, flow or storage of different materials in pipe- works or containers or, more generally, processes wherein amounts of different materials must be measured in order to secure correct operation of the process .
BACKGROUND OF THE INVENTION
In many industrial processes materials in different states flow or are stored in containers, pipeworks or similar, in which case it is necessary in the process to know for example amounts, flow rates, mixture composition and similar information of different materials. Such processes typically develop boundaries between different materials, wherein the boundaries can be defined such that the densities of the materials on different sides of the boundary are different. In practice, a boundary has a specific transition area where the physical properties of the material are altered. One example of boundaries is a mixture of oil and water, where oil, being a lighter material, forms a layer on the surface of water, and a distinct boundary is found between these materials. Another example can be to examine different soil layers and sedimentation of materials in the earth, where . interesting boundaries in the breaking of ore include for example boundaries between the rock material including precious metals and other rock material.
In many cases it is necessary to know the amount of material in a container or area to be examined. These situations occur in particular in the ore preparation process and sewage disposal process. A particular application in the preparation of ore is the thickening machine of the process. When the materials are distinctly separated and there is a distinct boundary between them the separation or thickening of the materials can take place.
The location of a boundary examined for example in the direction of height of the container to be examined is particularly interesting information for a number of processes. In general, these boundaries lo- cated for example in a container or in the earth are practically impossible to observe by visual inspection. On this account, more developed methods to detect boundaries are needed. There may be many different boundaries which may exist between different states of materials; however, the boundaries between two liquids or a liquid and a solid are particularly discussed below.
Boundaries between different materials precipitated by layers or the height of a fluid level have been measured in the prior art for example by acoustic and optical methods and methods based on gravity (pressure measurements) and electrical meas ¬ urements. These methods have been described in [1] J. Vergouw, CO. Gomez, J. A. Finch: "Estimating true lev- el in a thickener using a conductivity probe", Minerals Engineering, 17:87-88, 2004; [2] O-P Tossavainen, M. Vauhkonen, V. Kolehmainen: "A three-dimensional shape estimation approach for tracking of phase interfaces in sedimentation processes using electrical im- pedance tomography", Measurement Science and Technology, 18:1413-1424, 2007 and [3] M. Maldonado, A. Des- biens, R. del Villar: "An update on the estimation of the froth depth using conductivity measurements", 2008. One known method of measuring boundaries by acoustic waves in based on reflection from a discontinuity point. A sound wave transmitted to a material under examination is reflected from the boundary as an electromagnetic wave would reflect from the boundary of an impedance variation. By calculating the propagation times of the reflected wave it is possible to calculate the distance of the boundary from the trans ¬ mitter and further the desired height of the boundary in the y-dimension.
EIT (Electrical impedance tomography) , in turn, is a method in which electrodes can be mounted on the surface of an object to be examined. The basic principle of the method is that a set of electrodes is mounted on the surface of the study object and fed with minute alternating current, whereafter the potential differences, i.e. voltages, between the elec- trodes are measured. Typically, the voltage measure ¬ ment is made from the same electrodes as , the current feed. EIST (Electrical impedance spectroscopy tomography) , in turn, means that a number of different frequencies are used in the measurement, i.e. the meas- urements are typically made over a desired continuous frequency band. From the measured potential differences with a number of different electrode intervals it can be concluded that the electrical conductivity or permittivity of the object to be measured varies as a function of location, provided that the object in question is not completely homogenous. In practice, the conductivities are calculated by various mathematical methods in which suitable calculating models can be utilized. Such a calculation relates to the field of inversion calculation. Finally, for example a sectional view of the level of the measured object on which the electrodes have been disposed is obtained from the electrical conductivities as a function of location .
In the prior art, the height of the boundary between a solid material precipitated on the bottom of a container under examination and a liquid on top of it has been measured by introducing a measurement sensor disposed at one end of an arm directly in the precipitate. This' in conjoined with fouling of the sen- sor, which considerably affects the measuring accuracy and performance of the sensor. In addition, the lifetime of the sensor becomes in this case, at the worst, very short.
In the prior art, sensors have been used with the electrodes disposed in a single dimension, i.e. using a straight arm with electrodes at one or both ends. For example minute electric current has been fed via such a pair of electrodes, and the potential difference, i.e. the voltage, has been measured between these electrodes. By placing the sensor arm in different locations for example in a fluid container and examining the voltage variation between the different locations it has been possible to obtain information of possible boundaries between two different materi- als. However, this has required numerous reproducible measurements, and fouling or even breakdown of the sensors is an essential problem.
More specifically, the boundary has been measured in the prior art as follows. Consider a con- tainer with solid precipitated material on the bottom in a layer of a specific height, and for example water on top of the precipitate. The boundary between the precipitate and water is assumed to be distinctly defined, i.e. it can be assumed that under examination in the height direction a leap in the properties of the material occurs at one point (a single coordinate value in the y-dimension) . Consider an arrangement where a pair of electrodes is disposed at each end of a straight pipe. The pipe is placed vertically in the container such that the lower pair of electrodes is entirely within the precipitate and the upper one is entirely within the water. It is further assumed herein that the situation is not dynamic, i.e. the bounda ¬ ry between the liquid and the precipitate remains stably immoveable.
According to the principle of impedance tomography, minute electric current is fed to the electrodes, and the voltage is measured on these electrodes. In other words, the voltage is measured be ¬ tween two adjacent electrodes. From the measurement results it is easy to calculate the characteristic resistivity of the material surrounding each pair of electrodes by formulae:
From these values the electrical conductivities oi and σ 2 of each material can be calculated. If the electrical conductivities o and σ 2 differ from each other, it can be concluded that the boundary of the materials is within the region between the two pairs of electrodes. By performing a new measurement by raising or lowering the pipe in the vertical direc ¬ tion a new estimate for the height is obtained after the above-mentioned calculations.
The main problem of the prior art has been fouling of the sensors and the resulting loss of operational reliability, as well as slowness in determining the boundary as a result of reproduction of the measurements. Furthermore, the sensors can typically be used in the working order for a moment, but their lifetime is not long. From the fouling it follows that the sensors or measurement electrodes must be cleaned relatively often, which means from the process point of view implementation of an automatic cleaning system for the system or, alternatively, maintenance person- β nel must manually clean or service the apparatus on a regular basis, which further results in pauses in the use of the apparatus. SUMMARY OF THE INVENTION
The present invention discloses a method for detecting the locations of a material boundary or boundaries included in a volume comprising at least one material, wherein at least part of the volume is liquid material and wherein a, set of probes further comprising at least one probe is used, in which method:
a set of probes comprising at least one pair of electrodes is disposed in a volume under examina- tion;
current or voltage is fed via at least one pair of electrodes, and current or voltage is measured from at least one pair of electrodes; and
a measurement geometry representing the ar- rangement is used in the calculation, and a calculation algorithm needed in the calculation is selected.
The characteristic features of the method include :
disposing at least three electrodes of the set of probes in the volume under examination in an assembly substantially differing from a straight line, all electrodes of the set of probes being located only in the volume of the liquid material;
calculating an electrical conductivity dis- tribution in the volume under examination on the basis of the measurement results; and
concluding, on the basis of the electrical conductivity distribution and the electrode location information, the location of at least one material boundary in the volume under examination. In one embodiment of the present invention the feeding and measuring step is carried out using electric impedance tomography (EIT) or electric resistivity tomography (ERT) .
In one embodiment of the present invention the electrodes are located on at least one probe of the set of probes such that the vectors between the electrodes span a three-dimensional subspace.
In one embodiment of the present invention the electrodes are located on at least two separate probes, the electrodes on each probe being substantially disposed on a straight line.
In one embodiment of the present invention a standard inverse problem lD-o-method is used in the calculation to solve the planar profiles of the boundaries .
In one embodiment of the present invention a function of the measured voltage or current values for the desired numerical . values representing the boundary is determined by methods of machine learning.
In one embodiment of the present invention the results obtained from simulation are used as training material for the methods of machine learning.
In one embodiment of the present invention the employed method of machine learning is MLP-network (Multi-Layer Perceptron) .
In one embodiment of the present invention the MLP-network is trained by using the Levenberg- Marquardt-algorithm.
In one embodiment of the present invention fouling of the electrodes is monitored by estimating the contact impedances between each electrode and the surrounding material.
According to a second aspect of the present invention the inventive idea also comprises a system for detecting the locations of a material boundary or boundaries included in a volume comprising at least one material, wherein at least part of the volume is liquid material and wherein a set of probes further comprising at least one probe is used, the system com- prising:
a set of probes comprising at least one pair of electrodes disposed in a volume under examination;
feeding means for feeding current or voltage via at least one pair of electrodes and measuring means for measuring voltage or current from at least one pair of electrodes; and
a processor using in the calculation a measurement geometry representing the arrangement and a selected calculation algorithm needed in the calcula- tion.
As characteristic features of the system it further comprises:
at least three electrodes of the set of probes disposed in the volume under examination in an assembly substantially differing from a straight line, all electrodes of the set of probes being disposed only in the volume of the liquid material;
said processor for calculating an electrical conductivity distribution in the volume under examina- tion on the basis of the measurement results; and
said processor for concluding the location of at least one possible material boundary in the volume under examination on the basis of the electrical conductivity distribution and the electrode location in- formation.
In one embodiment of the present invention the current feeding and measuring means are arranged to carry out the feeding and measuring steps using electric impedance tomography (EIT) or electric resis- tivity tomography (ERT) . In one embodiment of the present invention the electrodes are disposed on at least one probe of the set of probes such that the vectors between the electrodes span a three-dimensional subspace.
In one embodiment of the present invention the electrodes are disposed on at least two separate probes, the electrodes on each probe being substan ¬ tially disposed on a straight line.
In one embodiment of the present invention said processor is arranged to use a standard inverse problem lD-o-method in the calculation to solve the planar profiles of the boundaries.
In one embodiment of the present invention said processor is arranged to determine a function of the measured voltage or current values for the desired numerical values representing the boundary by methods of machine learning.
In one embodiment of the present invention said processor is arranged to use the results obtained from simulation as training material for the methods of machine learning.
In one embodiment of the present invention the employed method of machine learning is MLP-network (Multi-Layer Perceptron) .
In one embodiment of the present invention the MLP-network is trained by using the Levenberg- Marquardt-algorithm.
In one embodiment of the present invention said measuring means are arranged to estimate the contact impedances between each electrode and the surrounding material to monitor the fouling of the electrodes .
According to a third aspect of the present invention the inventive idea additionally comprises a computer program further comprising program code which, when run on a data processing device, is ar- ranged to control the steps of the method described above which are executable on a processor or other computing device.
The advantage of the invention is that the electrode assembly of the measuring system need not be disposed in the region of the precipitate in a measurement volume comprising precipitate that includes solid material and a liquid. In addition, the boundary (or boundaries) can be detected by using only one measuring arrangement without the need to move the electrodes from one place to another.
LIST OF DRAWINGS
Fig. 1 presents an example of discretion, i.e. a calculation grid, for a T-probe using the Netgen software,
Fig. 2 presents three simulated conductivity profiles and the conductivity profiles estimated therefrom by the lD-a-method,
Fig. 3 presents an example of' a so-called MLP-network (Multi-Layer Perceptron) ,
Fig. 4 presents an example of a probe used in the invention, wherein the electrodes are disposed in a T-assembly,
Fig. 5a presents an example of a container which is an application of the invention, wherein material layers have been sedimented and wherein the boundaries are measured by an L-probe, and
Fig. 5b presents an application similar to Fig. 5a wherein the measuring instrument comprises two separate bar-shaped probes disposed side by side.
DETAILED DESCRIPTION OF THE INVENTION
The present invention discloses a new type of a measuring probe based on conductivity sensor measurements for observing the boundaries between two ma- terials. The method also utilizes modern data processing methods and calculation methods. The method utilizes EIT-measurements typically performed within the volume to be measured. Since it is computationally very complicated to create complete conductivity dis ¬ tribution of a volume under examination, the invention utilizes methods of machine learning. In particular, the invention allows the use of an electrode assembly which need not be disposed within the solid material in a container with solid precipitate on the bottom. In other words, the probe can be used to detect conductivity distributions also in other regions than the area between the measuring electrodes.
The basic idea of the invention comprises use of a measurement based on electric impedance tomography, wherein different conductivities between different sides of a boundary of two materials or, more generally, as the composition of the material under examination varies as a function of three-dimensional lo- cation can be utilized. It is the consequence of these different properties of the materials that boundaries may exist in the first place. In locations of the material volume where the properties of the material vary more intensively than normally a more intensive variation than normally can also be found in the electrical properties such as electrical conductivity as a function of location. In order for boundaries to be able to emerge and be observable as well, segregation, precipitation or other non-homogenization must have taken place in a mixture of several different materials. In this case, for example substances having different masses can be separated from each other such that more intensive variation than on the average in electrical conductivity, characteristic impedance or other measurable property can be observed in the boundary area of the materials. The so-called contact impedances between each electrode and the material can also be taken into account in the calculation, typically meaning additional electrical resistance due to dirt surrounding the electrode or unidealistic conditions of the contact (between the electrode and the surroundings). This further results in additional voltage drop. In this case, it can be stated that the voltage measured by a pair of electrodes is a function of the electric cur- rent fed, the conductivity distribution of the path of the electric current, and the contact impedance between the electrodes and the surrounding material. In this situation it is possible to continue the measurements as normally, even if the electrodes were a bit fouled (typically, a layer of gypsum develops on the surface during use) . On the other hand, if the amount of dirt exceeds a specific limit, this can be observed and for example a warning signal can be given in the situation. By calculating the contact impedance, the need for maintenance of the sensor assembly is also decreased, which further reduces the costs incurring from everyday use.
The distance between electrodes can be selected freely to be suitable for each application. This is also influenced by the dimensions on the mate- ■ rial volume to be measured. The number of the employed electrodes (disposed on one or more probes, in total) is typically at least three.
As the currents and voltages are known after the EIT measurement, the task is to determine the internal electrical conductivity distribution of the material volume under examination. This type of a problem is a so-called inversion problem, which is characterized by being ill-posed, i.e. a solution to the problem is not unambiguous or does not exist. The opposite of an inversion problem is a so-called forward problem, which in this embodiment would be detection of the electrode voltages on the basis of the known feed currents and the electrical conductivity distribution. In this connection a so-called complete elec- trode model can be used which is described for example in Cheng et al.: "Electrode models for electric current computed tomography", IEEE Transactions on Biomedical Engineering, 36:918-924, 1989; and Somersalo et al.: "Existence and uniqueness for electrode models for electric current computed tomography", SIAM Journal on Applied Mathematics, 52:1023-1040, 1992. In a forward problem, in turn, the so-called Finite Element Method (FEM) is used, which is described for example in Vauhkonen et al.: "Tikhonov regularization and pri- or information in electrical impedance tomography", IEEE Transactions on Biomedical Engineering, 45:486- 493, 1998.
The inverse problem comprises examining a regularized minimization problem of the form: σ,ζ = argmin{|£/ meaj - U{a,zf + a, (σ - σ, 2 + (Z - » 2 }
(2) where σ is unknown conductivity distribution, z in- eludes effective contact impedances between the electrodes and the surrounding material, U meas denotes the voltages measured from the electrodes, U(o,z) is the voltage calculated for example by the FEM-method associated to a forward problem, L 1/2 are regularization matrices, ΩΊ,2 are regularization parameters and σ, and z, are the previous values for conductivity and contact impedance, respectively.
Estimates for conductivity distribution and contact impedances can be iteratively calculated for example by the Gauss-Newton method. In this connection, Tikhonov regularization is additionally used, which is described in the above-mentioned publication Vauhkonen .
In the method of the invention, planar pro- files of the boundaries can be estimated for example by a so-called lD-o-method. To be able to utilize EIT to estimate the boundary, the measurement geometry must first be determined for calculation. This encompasses determination of the geometry of the probe and geometry of the electrodes disposed on the surface thereof so as to be able to create a calculation grid for the element method. Often, the object (for example a thickener) including the boundary is so large that the calculation area to be discreted can be limited to the vicinity of the probe, sufficiently far from the electrodes. When this positioning of the measurement geometry is complete, the geometry can be discreted as having finite dimensions. This can be done by ready- made programs, for example Netgen. An example of such discretion, i.e. a computing grid, for a T-probe is presented in Fig. 1.
The method based on impedance tomography solve the boundaries from the current-voltage data is referred to as lD-o in this connection and complies with the same basic principles as the above-mentioned known Tikhonov-regularized solution. Estimates of the conductivity distribution and contact impedances can be calculated iteratively using the Gauss-Newton method. In iteration round "i + 1" the quantity θ + ι = [oi + i, z + i] can be expressed by the relation:
Θ Μ = Θ, + κ,δθ, (3) where κι is the step length. In addition, search direction δθ ί is defined by the formula: δθ, = [J J, + L e Y [jf (U meas - U(9 l ))+ L- (0 i - A)] (4) where J, = J CT): ,J . j is Jacobian matrix calculated for conductivity and contact impedances, L e is the regularization matrix and is the first guess for conductivity and contact impedances. The regularization matrix for quantity # = [σ,ζ] can be built in the following form where L \ and L2 are regularization matrices of conductivity distribution and contact impedances, and ofi and 0:2 represent the respective regularization parameters.
When the conductivity distribution is assumed to vary only in the direction of depth of the object, the problem is reduced to a one-dimensional inversion problem where variations in conductivity only in a single dimension are solved. The lD-o-method can briefly be described in the following way:
1. The solution to the inversion problem is calculated in a 1-dimensional discreted grid where one point corresponds to discrete conductivity at a specific depth.
The obtained solution is interpolated to a real 3D-calculation grid to■ calculate the voltage values (quantity £(#,) in equation (4)).
2. Columns of the Jacobian matrix calculated for conductivity in the 3D-calculation grid and corresponding to the above-mentioned heights are summed. Then, a new Jacobian matrix is formed, wherein the number of columns corresponds to the heights of the discreted calculation area. The terms of the Jacobian matrix for the contact impedance are added to the final Jacobian matrix. 3. As prior knowledge it is assumed that the conductivity distribution and effective contact impedances are smooth, i.e. the values of physically adjacent points and adjacent electrodes are not largely variable. Thus, the regularization matrices for conductivity and contact impedances are differential matrices. Also another approach can be utilized in the regularization .
The above is a method for determining the ab- solute values of ID-conductivity distribution. As is known, the same can be performed for variation of electrical conductivity of the object (differential function) .
In the following, determination of the param- eters of a boundary profile from conductivity distribution obtained with lD-σ is discussed.
As a result of the lD-o-method, one- dimensional conductivity distribution, o(s), as a function of depth s is obtained. The objects of inter- est in this distribution normally include conductivities at the bottom and the top, width of the conduc ¬ tivity variation region, i.e. the transition area, and depth of the boundary. Conductivity of the upper surface is expressed as o 0 , conductivity of the lower surface as θχ, depth of the boundary as D and width of the transition area as T. In the following, one way of generating the parameters D, T, o 0 and θχ from a solution obtained as an inverse problem o{s) is shown.
To calculate the depth of the boundary D the adjacent values of o(s) are divided by respective distances of s. This gives approximation for the derivative of o(s)
(6) d S k S k+\ where ke I \N = {l,2,3,...,N-l} is the index of a single point on a discreted depthwise coordinate axis. This allows to estimate D as a maximum of ' the absolute values of these values
To calculate the conductivities at the top and the bottom, oo and oi, a minimum o m i n and a maximum o max of the conductivity distribution o(s) are determined. If ε is a predetermined tolerance value and I = I \N {/V}, then the subsets of the indices of the minimum and maximum are 7 min = {k e < <7 min + ε] and ( 8 )
Using these subsets and assuming σ, < σ 0 , estimates for conductivities of the upper and the lower surface can be calculated by averaging the values of sets
min 's/ mi „ where N max and N m i n are cardinalities of sets max and Imin- When σ,≥ σ 0 , the calculation is performed inversely .
Width of the transition area Γ can be estimated using the above-given subsets I min and I max . When σ, < σ 0 and To and Ti are determined by equations T Q = ma j^, } and ( 12 )
Ά = min{5, } (13) the result is that T^T-T 0 . When σ, > σ 0 , the calculation is performed inversely.
The following discloses simulations of lD-σ estimates and calculated boundary parameters.
The simulations were performed in a geometry similar to Fig. 1. The boundary including conductivity variation is located in a cylindrical object having a diameter of 400mm and a height of 600mm. A T-type EIT measurement sensor provided with 18 band-type elec ¬ trodes, 10 of them disposed on the vertical section and the remaining 8 on the horizontal section, is introduced in the object from above. From the geometry determined above, a simulation grid Mi was created for the simulations, which was divided in 31845 discrete points. For lD-o, a calculation grid M 2 was prepared, consisting of 14562 discrete points. For lD-σ, also a depthwise ID-grid M 3 was created wherein 31 points were created at a distance of 20mm from each other.
For simulations, different boundary profiles were prepared on grid Mi by the formula where σ = o(s) is conductivity at depth s, o 0 and oi are conductivities of the upper and the lower side, D is depth of the boundary, T is width of the transition area, s is depth and a is a parameter representing the shape of the curve. The employed values of parameters σ 0 , oi, D and T are presented in the table below. The value of parameter a was 7. In addi- tion, 30 different current feeds were created between the electrodes. Corresponding to the boundary profiles and current feeds in question, voltages U S i m were calculated by the forward problem, expressing the potential differences of adjacent electrodes at each cur- rent feed. To the simulated voltages was added 1% of Gaussian noise as determined from each simulated value. Similar values of contact impedances used in the simulated voltages were set for "packs" of 4-5 electrodes, and the value of each pack was drawn from a uniformly distributed random interval [0,3 Qmm 2 , 1,8 Qmm 2 ] .
Corresponding to voltages U S im, boundary profiles were estimated by the above lD-o-method as a solution to the inverse problem. The calculation was performed in grids M 2 and M 3 . The parameters of the inverse problem were set as follows: the mean value of the boundary profile simulated for all grid points was set as the prior value of conductivity cr, , the mean value of simulated contact impedance values was set as the prior value of contact impedance z, , respectively, value 5 was set as a regularization parameter ai and value 0,5 was set as a≥ . The boundary parameters were calculated from the estimated boundary profile on the basis of that given above. Fig. 2 shows visualizations of the correct and estimated boundary profiles. The following table presents the correct and estimated boundary parameters, i.e. the values of boundary parameters D, T, Oi and σ 0 of the simulated conductivity profile and estimates D , T , σ, and σ 0 calculated from the conductivity profile estimated by lD-σ.
D D T f σ, σ ο
100 110. 0 32. 9108 20 .0 0. 1294 0 .1264 0 .2916 0 .1823
130 130. 0 32. 7214 40 .0 0. 0811 0 .0876 0 .5653 0 .3503
160 170. 0 47. 3858 20 .0 0. 1029 0 .1050 0 .2610 0 .2639
190 190. 0 41. 5941 20 .0 0. 0666 0 .0759 0 .5303 0 .4233 220 230.0 40.9972 40.0 0.1102 0.1134 0.4153 0.3691
250 250. 0 32. 8991 20. 0 0. 0763 0. 0866 0. 5985 0. 4957
280 290. 0 47. 0606 40. 0 0. 1154 0. 1065 0. 2313 0. 2272
310 310. 0 42. 4411 40. 0 0. 1189 0. 0850 0. 3771 0. 3337
340 350. 0 37. 0190 40. 0 0. 1248 0. 1247 0. 2427 0. 2441
370 330. 0 40. 2650 40. 0 0. 0951 0. 2963 0. 5848 0. 5318
400 390. 0 38. 0362 40. 0 0. 0584 0. 0809 0. 2019 0. 2004
Alternatively, instead of the lD-o-method, in one embodiment of the invention a function of the measured voltage or current values can be determined for desired numerical values representing the boundary by methods of machine learning.
Since the calculations involved in the lD-σ- method are demanding in the general case, its use may require a relatively large and efficient calculating unit. As one exemplifying embodiment of the invention, methods of machine learning can be used to approximate the results of the lD-o-method at an accuracy which is sufficient in practical terms. In this case, a simple calculating unit (e.g. digital signal processor with only limited ' memory available) is sufficient for online implementation of the embodiment.
For methods of machine learning, specific training material is required, which in the case of the invention is obtained from the above simulation results. The inputs, i.e. independent variables, in the learning model include simulated voltages U S i m , whereas the quantities OQ, Oi, D and T to be estimated are set as outputs, i.e. dependent variables, of the model .
The methods of machine learning aim at finding rules for determining a function of any inputs for a space (four-dimensional in this case) corresponding to the outputs such that the inputs of the training material are a sufficiently accurate function of the corresponding desired outputs of the training material. Once this function has been determined, it can be used to estimate outputs also for new recently meas- ured inputs.
In one embodiment of the invention a so- called Multi-layer Perceptron, MLP-network, is selected as the method of machine learning. This is limited to a two-layer network determining a function of the inputs Xi for outputs y± according to Fig. 3. In the figure, the nodes indicated by a circle represent so- called Rosenblatt perceptrons (Rosenblatt: "The perceptron: A probabilistic model for information storage and organization in the brain", Psychological Review, 65(6): 386-408, 1958), each calculating a specifically weighed sum from inputs corresponding to the incoming arrows and finally possibly attenuating the response by using a so-called specific sigmoid function. In this connection attenuation is used only in the first layer depicted on the left in the figure. Thus, the rules executed by MLP can be expressed as a function y=f(x) = W 2 -g(Wix+bi) + b 2 , (15) where y =[>, ...y„ is output of the model for input x=[x,...x m ] at specific network parameters, i.e. weighing matrices W\ and Wi and so-called bias vectors b\ and bi . The function g(z) is not a general multiple variable function but is applied to vector z = [z, ··· z k by elements, i.e.
where a hyperbolic tangent function tanh(z ; ) = (exp(2z ( .)-l)/(exp(2z,.) + l) is used for attenuation.
Determination of the weighing matrices W\ and Wi and the bias vectors b\ and bi on the basis of the training material is referred to as training the MLP network. Normally, the training method aims at changing the values of these parameters such that the inputs of the training material would be as accurate a function as possible of the corresponding desired re- sponses of the training material. In one embodiment of the invention this training step is performed by using the so-called Levenberg-Marquardt algorithm (Hagan and Menhaj : "Training feedforward networks with the Mar- quardt algorithm", IEEE Transactions on Neural Net- works, 5(6) : 989-993, 1994) .
In one embodiment of the invention the training can be facilitated by calculating from inputs U S i m a specific number of so-called principal components (e.g. Jolliffe: "Principal Component Analysis", Springer, 2. edition, 2002), which is smaller than the number of the original voltage measurements, and using thus obtained values as inputs in the network to be trained. By storing the conversion from the voltage measurements on principal components, the same func- tion can also be determined for new online-measured voltage vectors before their input to the MLP network.
In one embodiment of the invention a committee formed by several networks is used instead of a single MLP-network. This is known to reduce the vari- ance resulting from uncertainty of the training and thereby improve the generalization ability of the final estimator (e.g. Dietterich, "Ensemble methods in machine learning", in Kittler and Roli (eds.) : "Multiple Classifier Systems", LNCS 1857, 1-15, Springer, 2000) . Training the MLP-network is computationally heavy but can be performed offline before or in connection with introduction of the measuring instrument, either based on simulations or real measurements. In this case, only a very simple program that executes function (15), weighing matrices W\ and Wi , and bias vectors and hi would have to be loaded on the actual measuring instrument. In using the principal component conversion, also a program that executes the con- version and a conversion matrix are needed. In the case of a committee, additionally a program that executes the committee and the weighing matrices and bias vectors are needed as many times as there are members in the committee. As a calculation, the production of estimates on the basis of new voltage measurements is simple. In particular, the hyperbolic tangent function can be represented as a so-called lookup table, in which case the apparatus is practically not required to support other calculations except additions and multiplications as well as roundings and table designations .
The outputs of the model obtained by methods of machine learning may also include, in addition to parameters σ 0 , θχ, D and- T, the contact impedances. In this case, the model can also be used, in addition to estimating the boundary, to monitor the fouling of the electrodes .
Next, the preferred embodiments of positioning the electrode assembly in the apparatus described in the invention are discussed. If, instead of straight positioning of the electrodes as proposed by the prior art, at least three electrodes are used such that the positioning of the electrode assembly differs from a straight line, the above-mentioned prior art problems are avoided. If the electrodes are positioned for example at each end and/or angle of a T-, Z-, F-, L- or Y-type assembly, it is possible to place the assembly for example entirely in water or other liquid without immersion in solid or precipitated material and yet be able to detect the height or location of the possible or at least one material boundary located below the assembly as measured from a desired, reference. There are naturally other alternatives of positioning the electrodes; they can be for example disposed in a zigzag array or in a manner differing in another way from a straight line. The electrodes can also be positioned on several separate probes such that each probe comprises at least one electrode. In this case, however, the electrodes of an assembly formed by one or more probes differ from a straight line, and, at the best, moving the probes in detecting the boundaries is avoided. The essential idea of the method and apparatus can thus be distilled in that the set formed by the electrodes is located in at least two different dimensions, i.e. essentially differing from a straight line in a single dimension. In addition, in using a number of straight probes, the core difference in respect to the prior art is that the pairs of electrodes used for signal feeds and measurements are formed between different desired probes.
One alternative in the invention is to dispose two separate rod-shaped sensor assemblies adjacently in the vertical direction at a suitable mutual distance from each other. Both sensor assemblies with electrodes can be located entirely in water or other liquid without contacting the precipitate including solid material. In one embodiment the distance of two electrodes in a single rod-shaped assembly is approximately on the same order as the mutual horizontal distance between the two rod-shaped assemblies. In anoth- er example several electrodes can be located sequentially on a single probe in a desired number. The voltages measured by two assemblies (probes) also exhibit the effect of the composition of the materials below the assembly, which is not perceived by a single rod-shaped assembly of two or more electrodes. In the measurement, test current flows in this case also outside the area between the probes defined by straight lines, i.e. via the precipitate region in the above example. This way, it is possible to observe that the electric field extends to both sides of the boundary between the liquid and the solid material, and the voltage measurement performed by the probe (s) by means of the above described modeling and inversion calculation brings out the possible existence and locations of the boundaries.
The idea of the present invention can for example be extended by examining a volume including several sedimented materials, and several different boundaries between the different materials can be located by the method. Also in this case the. probes can be located, in the case of the above-mentioned con ¬ tainer, only in the liquid region, but in practice the thickness of each material layer encompasses a specific upper limit in case the lowest boundary is to be detected by this method as well.
In one embodiment of the invention the probe or probes can be moved in the area of the volume to be measured, and the EIT-measurement and calculation process can be repeated after relocation. This way, more accurate three-dimensional information on the location of the boundary is obtained in a situation where thicknesses of the sedimented layers vary or where the boundaries may exist in unlimited directions (other than horizontal) . Finally, the apparatus allows the generation of an accurate 3-D-map or diagram where different materials and boundaries are visible for ex- ample in different colors on the screen of the user's computer .
Fig. 4 shows one typical example of a probe used in the invention. The probe 40 of the example is T-shaped, which preferably allows the formation of an assembly that differs from a straight line, yet is quite simple. Electrodes are mounted sequentially on the vertical section (one arm) of the assembly. In this example their number is eight, but another number of the electrodes can be used as well. In this example the electrodes are disposed at regular intervals, but they can also be disposed at variable mutual distanc ¬ es. Four electrodes are disposed on the two other tips, on each branch, of the T-probe of the example. In other words, there are 16 numbered electrodes in the example indicated 41i, . .., n .
Fig. 5a in turn discloses a measurement arrangement with an example of a container 50 wherein and wherefrom a material mixture with solid material precipitated in a liquid is led. The container may represent any flow application where flowing material is guided or runs from one container to another or mi ¬ grates for example through pipes. On the bottom of the container 50 in the example there has been accumulated sedimented material formed on the basis of different masses or other properties of the components in the mixture. The first material 51a, being heaviest, has been precipitated at the bottom, and the thickness of the material layer is indicated by hi, which may natu- rally vary at different sites of the bottom of the container. As a lighter material, a second material layer 51b has been sedimented on top of the first material layer, the height of the upper level of which from the bottom of the container is indicated by h 2 (which may also vary as a function of the bottom coordinates of the container) . A measurement probe 52, which in this example is L-shaped, can now be introduced inside the container 50. Another shape that differs from a straight line is also possible for the electrode assembly. As an ex- ample, this probe has 8 + 8 electrodes 52 x - 52 n mounted on the two orthogonal branches of the probe body. In the example the L-probe can be disposed entirely above the sedimented layers such that fouling of the electrodes can be prevented. The electrode assembly can also be moved inside the container such that the location of the probe can be freely selected, where necessary, in the sediment-free area of the container.
Control of the apparatus (the parts performing the current feed and the measurement) and the re- quired calculation logic can be carried out for example by a separate or integrated controller or computer depicted in the example of the figure as a server 53. The above-described measurement functions are controlled to be executed by a computer program installed on the computer, and the processor of the computer can process the required calculation operations. On the other hand it is also possible to locate the current feed and measurement parts needed by the EIT- or ERT- measurement in a separate unit not necessarily requir- ing a separate monitor (as depicted in the figure) . In this case the obtained measurement data can be transferred to be processed by a separate processor which may exist in a computer being a separate unit.
In one embodiment of the invention it is pos- sible to use a single integrated controller that controls the feeds, measurements and performs calculations, but a monitor is not necessarily needed. The controller may control for example a horn, giving an alarm where necessary, forward the measurement data as a desired signal for use by another system, or direct- ly adjust a mechanism of the process under examination or an effective signal in the process.
In another application the feeding parts, measuring parts and processing parts can all be imple- mented as separate and independently controlled units. One alternative is to integrate two functionalities of the above-mentioned and implement the third functionality as a separate unit. If necessary, monitoring means can be coupled to a desired part and implemented for example as a monitor connected to a computer (as presented in Fig. 5a) and/or an' integrated signal display of the feeding/measuring instrument.
The necessary connections between the current feeding parts, measuring parts, calculating unit (pro- cessor) 53 and further the set of probes 52 can be provided using known wired or wireless connection means such as different cablings, or by remote use for example over an internet connection. As a result of calculation, the values hi and 12 can be produced as a function of location, or for example a more accurate schematic plan for variations in the material properties and/or boundaries can be produced as a two- dimensional cross-sectional view.
If, on the other hand, at least . four elec- trodes are used so as to locate them in substantially three dimensions, i.e. the vectors between them spanning the basis of a three-dimensional space, it is possible, without moving the assembly, to produce a truly three-dimensional representation of the measura- ble electrical properties in the surroundings of the set of electrodes and the possible boundaries. The probe of Fig. 5a could be adapted accordingly for example by adding a third branch to the angle of the letter L perpendicularly to both existing branches.
Fig. 5b, in turn, shows an application using a different set of probes. In other respects, the measurement arrangement is identical to Fig. 5a. Instead of a single probe, two probes 52a and 52b can be used side by side. The first probe 52a comprises electrodes 52 a i, 52 a 2 , ... , 52 a 8 and the second probe respec- tively electrodes 52 b i, 52 b 2 , ... , 52 b 8 . The numbers of the electrodes are indicated in the figure by indices m and n, and it is possible to use numbers wherein m≠n. The container 50 still acts as a passage volume for a mixture formed by liquids and solid materials, whereupon materials of different masses can be sedi- mented on the bottom of the container in their proper layers 51a, 51b. The task is therefore to find out hi and h 2 as above, either as pure boundary distance information from a desired reference, or as . more exten- sive three-dimensional boundary data which can be drawn as a three-dimensional representation of the content of the desired volume. As stated in connection with the previous figure, to produce a three- dimensional representation requires a set of elec- trodes wherein the vectors between the electrodes span a three-dimensional (sub) space. For example, two probes both comprising two electrodes can together span a three-dimensional subspace if the probes are not disposed in parallel to each other. The process is controlled by a computer 53 connected to both probes 52a-b. The computer can be a local controller disposed in connection to the apparatus, or for example a separate server disposed over a data link. The current feed and measurement parts can, if necessary, be lo- cated in a physically different unit than the processor or computer performing the calculation, as described in connection with Fig. 5a.
Two or more separate probes may be located at a desired distance to each other, and their positions, mutual locations or the rotational angles of a probe relative to the other probes or the container can be varied. In a preferred embodiment there are typically 3 ... 16 electrodes.
The idea of the present invention can be applied to volumes that contain many kinds of liquids, solid materials, emulsions and mixtures, which may be vessels, containers, pipeworks, caves or similar places wherein mixing or sedimentation of materials is possible and wherein the electrical properties of different materials (for example electrical conductivi- ties or capacitive properties of materials) are different. Furthermore, the invention can be applied for example in ground measurement (geological surveys). Typical exemplary embodiments include different flow and sedimentation applications where separation or segregation of materials takes place or where it is necessary to detect volumes, surface levels or the presence of a desired material in a specific industrial process.
The invention is not limited merely to the exemplifying embodiments referred to above; instead, many variations are possible within the scope of the inventive idea defined by the claims.