METHOD FOR DETERMINING A QUANTITY OF INTEREST IN A TARGET DOMAIN, APPARATUS, AND COMPUTER PROGRAM

Abstract

A method for determining a quantity of interest in a target comprises: providing simulated statistics of a deviation, caused by a boundary distortion in observations of the physical quantity; providing an approximate mathematical model of observations of the physical quantity, the approximate mathematical model defining the physical quantity to be dependent on the quantity of interest in the target domain, and on a deviation a boundary distortion causes in the observations, said deviation being determined to behave in accordance with the simulated statistics; receiving measured values of the physical quantity; determining an observation difference between the measured values of the physical quantity and corresponding observations according to the approximate mathematical model, and adjusting the approximate mathematical model to reduce the observation difference; and determining an estimate of the quantity of interest in the target domain on the basis of the adjusted approximate mathematical model.

Claims

1. A method for determining a quantity of interest which is one of permittivity and electrical conductivity of material(s) present in a target domain comprising a cross-sectional area or a volume within or of a process pipe, container, or vessel and having a boundary surface, by means of measurements of a physical quantity dependent on the quantity of interest of material(s) present in the target domain, the target domain possibly comprising a boundary distortion such as a boundary layer of a first material on the boundary surface and/or wear of the boundary surface, the boundary surface and/or the possible boundary layer thereon limiting an inner zone within the target domain, the inner zone comprising at least one second material, the method comprising, performed at least partially automatically by means of suitable computing and/or data processing means: providing simulated statistics of a deviation, caused by an effective boundary distortion defined relative to a predetermined reference boundary distortion, in observations of the physical quantity; providing an approximate mathematical model of observations of the physical quantity, the approximate mathematical model defining the physical quantity to be dependent on the quantity of interest in the target domain with the reference boundary distortion, and on a deviation an effective boundary distortion causes in the observations, said deviation being determined to behave in accordance with the simulated statistics; receiving measured values of the physical quantity; determining an observation difference between the measured values of the physical quantity and corresponding observations according to the approximate mathematical model, and adjusting the approximate mathematical model to reduce the observation difference, thereby providing an adjusted approximate mathematical model; and determining an estimate of the quantity of interest of material(s) present in the target domain on the basis of the adjusted approximate mathematical model.

2. A method as defined in claim 1, the simulated statistics of a deviation being determined by: providing a simulative mathematical model of observations of the physical quantity, the simulative mathematical model defining the physical quantity to be dependent on the quantity of interest in the target domain; generating, by means of the simulative mathematical model, simulated observations of the physical quantity for a plurality of various modeled quantity of interest conditions in the inner zone of the target domain, one observation with the reference boundary distortion and another with a modeled effective boundary distortion for each type of modeled quantity of interest conditions, using various modeled effective boundary distortions; and determining, on the basis of the simulated observations, simulated statistics of a deviation an effective boundary distortion causes in the simulated observations.

3. A method as defined in claim 2, further comprising: determining, on the basis of the adjusted approximate mathematical model, an estimate of a deviation caused by the possible effective boundary distortion in the measured values of the physical quantity; and determining, on the basis of said estimate, the simulated statistics of a deviation an effective boundary distortion causes in the simulated observations, and the modeled effective boundary distortions used in generating the simulated observations of the physical quantity, an estimate of an effective boundary distortion present in the target domain.

4. A method as defined in claim 1, wherein the quantity of interest is an electrical quantity, such as permittivity.

5. A method as defined in claim 1,wherein the boundary surface limits the target domain.

6. A method as defined in claim 1, wherein the boundary surface lies within the interior of the target domain.

7. A method as defined in claim 1, comprising performing measurements of the physical quantity dependent on the quantity of interest.

8. An apparatus for determining a quantity of interest which is one of permittivity and electrical conductivity of material(s) present in a target domain comprising a cross-sectional area or a volume within or of a process pipe, container, or vessel and having a boundary surface, by means of measurements of a physical quantity dependent on the quantity of interest of material(s) present in the target domain, the target domain possibly comprising a boundary distortion such as a boundary layer of a first material on the boundary surface and/or wear of the boundary surface, the boundary surface and/or the possible boundary layer thereon limiting an inner zone within the target domain, the inner zone comprising at least one second material, the apparatus comprising a computing system comprising: means for providing simulated statistics of a deviation, caused by an effective boundary distortion defined relative to a predetermined reference boundary distortion, in observations of the physical quantity; means for providing an approximate mathematical model of observations of the physical quantity, the approximate mathematical model defining the physical quantity to be dependent on the quantity of interest in the target domain with the reference boundary distortion, and on a deviation an effective boundary distortion causes in the observations, said deviation being determined to behave in accordance with the simulated statistics; means for receiving measured values of the physical quantity; means for determining an observation difference between the measured values of the physical quantity and corresponding observations according to the approximate mathematical model, and adjusting the approximate mathematical model to reduce the observation difference, thereby providing an adjusted approximate mathematical model; and means for determining the quantity of interest of material(s) anestimate of present in the target domain on the basis of the adjusted approximate mathematical model.

9. An apparatus as defined in claim 8, the simulated statistics of a deviation being determined by: providing a simulative mathematical model of observations of the physical quantity, the simulative mathematical model defining the physical quantity to be dependent on the quantity of interest in the target domain; generating, by means of the simulative mathematical model, simulated observations of the physical quantity for a plurality of various modeled quantity of interest conditions in the inner zone of the target domain, one observation with the reference boundary distortion and another with a modeled effective boundary distortion for each type of modeled quantity of interest conditions, using various modeled effective boundary distortions; and determining, on the basis of the simulated observations, simulated statistics of a deviation an effective boundary distortion causes in the simulated observations.

10. An apparatus as defined in claim 9, wherein the apparatus further comprises: means for determining, on the basis of the adjusted approximate mathematical model, an estimate of a deviation caused by the possible effective boundary distortion in the measured values of the physical quantity; and means for determining, on the basis of said estimate, the simulated statistics of a deviation an effective boundary distortion causes in the simulated observations, and the modeled effective boundary distortions used in generating the simulated observations of the physical quantity, an estimate of an effective boundary distortion present in the target domain.

11. An apparatus as defined in claim 8, wherein the quantity of interest is an electrical quantity, such as permittivity.

12. An apparatus as defined in claim 8, wherein the boundary surface limits the target domain.

13. An apparatus as defined in claim 8, wherein the boundary surface lies within the interior of the target domain.

14. An apparatus as defined in claim 8, comprising a measurement system configured to carry out measurements of the physical quantity dependent on the quantity of interest.

15. A computer program comprising program code which, when executed by a processor, causes the processor to perform the method according to claim 1.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0096] In the following, various embodiments are described with reference to the accompanying drawings, wherein:

[0097] FIG. 1 is a flow chart illustration of an investigation method;

[0098] FIG. 2 shows a schematic cross-sectional view of a measurement setup for performing capacitance or impedance measurements;

[0099] FIG. 3 shows a schematic cross-sectional view of another measurement setup for performing capacitance or impedance measurements;

[0100] FIG. 4 shows a schematic view of a measurement probe for performing capacitance or impedance measurements;

[0101] FIG. 5 shows a schematic view of another measurement probe for performing capacitance or impedance measurements; and

[0102] FIG. 6 shows an apparatus for determining a boundary layer and/or wear in process equipment.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0103] The process illustrated in the flow chart of FIG. 1 may be used to determine a quantity of interest in a target domain having a boundary surface, by means of measurements of a physical quantity dependent on the quantity of interest in the target domain. The target domain may comprise a boundary distortion such as a deposit or annular flow of a first material on the boundary surface and/or wear of the boundary surface, the boundary surface and/or the possible deposit or annular flow thereon limiting a free inner zone within the target domain, the free inner zone comprising at least one second material. The possible deposit and annular flow may be called generally a “boundary layer” of a first material.

[0104] The quantity of interest may be any quantity which is observable by means of measurements of a physical quantity dependent on the quantity of interest. One example of the quantity of interest is electrical permittivity. The target domain may lie e.g. within an industrial process equipment for storing and/or transporting various process materials.

[0105] The process starts by providing simulated statistics of a deviation caused by an effective boundary distortion in observations of the physical quantity. By the effective boundary distortion is meant a boundary distortion defined relative to, i.e. in comparison to, a predetermined reference boundary distortion. In this sense, given a “non-zero” reference boundary distortion, a boundary distortion identical to the reference boundary distortion means a “zero” effective boundary distortion.

[0106] In the example of FIG. 1, a simplified situation is assumed where the predetermined reference boundary distortion means actually no boundary distortion present in the target domain. Then, the effective boundary distortion, which generally is defined relative to the predetermined reference boundary distortion, may be a boundary distortion of any type. From this on, the effective boundary distortion is thus called simply a “boundary distortion”.

[0107] The simulated statistics may comprise e.g. information about the mean value and covariance of some boundary distortion parameters, such as location and thickness and internal material distribution of a scale deposit on the boundary surface, or changed location and/or shape of the boundary surface itself due to wear thereof. The simulated statistics may be determined beforehand or during the process, as one step thereof. The simulated statistics is preferably determined by providing a simulative mathematical model of observations of the physical quantity, the simulative mathematical model defining the physical quantity to be dependent on the quantity of interest in the target domain; generating, by means of the simulative mathematical model, simulated observations of the physical quantity for a plurality of various modeled quantity of interest conditions in the free inner zone of the target domain, one observation without any boundary distortion and another with a modeled boundary distortion for each type of modeled quantity of interest conditions, using various modeled boundary distortions; and finally determining, on the basis of the simulated observations, simulated statistics of a deviation which a boundary distortion causes in the simulated observations.

[0108] Next, an approximate mathematical model of observations is provided, the model defining the physical quantity to be dependent on the quantity of interest in the target domain without any boundary distortion, and on a deviation which a boundary distortion causes in the observations of the physical quantity. The model is configured so that the deviation is determined to behave in accordance with, i.e. similarly to, the simulated statistics of a deviation.

[0109] Measured values of the physical quantity are then received, and an observation difference between the measured values and corresponding observations according to the approximate model is determined.

[0110] The mathematical model is then adjusted so that the observation difference is reduced. This may be iteratively continued until the observation difference goes below a predetermined limit. As a result of the adjustment, an adjusted mathematical model is provided. The model itself remains the same, “adjusted” just refers to the fact that the parameters of the model has been adjusted in comparison to the initial ones.

[0111] An estimate of the quantity of interest in the target domain may then be determined on the basis of the adjusted mathematical model, actually on the basis of an quantity of interest term included therein. In this step, the estimate of the quantity of interest may cover the entire target domain, including the area or volume of possible boundary distortion. Naturally, in such area or volume, the estimate may be possibly not accurate. However, by means of the process of FIG. 1, it is possible to determine an estimate which is, at least for the free inner zone, close to the real quantity of interest conditions. The estimate of the quantity of interest may be determined, for example, in the form of a reconstructed image(s) of the quantity of interest distribution in the target domain.

[0112] The method may also comprise determination of an estimate of the boundary distortion possibly present in the target domain. This may be carried out by first determining, on the basis of the adjusted approximate mathematical model, actually a deviation term included therein, an estimate of a deviation caused by the possible boundary distortion in the measured values. This estimate can further be used to determine, on the basis of the simulated statistics of a deviation, and on the basis of the modeled boundary distortions used in generating the simulated observations, an estimate of the boundary distortion possibly present in the target domain.

[0113] Thus, both an estimate of the quantity of interest in the free inner zone and an estimate of a boundary distortion, comprising e.g. scale deposit and/or wear of the boundary surface, may be reliably determined.

[0114] The order of the method steps is not limited to that shown in FIG. 1. The order of the steps may deviate from that of FIG. 1 whenever appropriate. For example, the step of receiving the measured values of the quantity of interest may be performed at any stage before determining the observation difference.

[0115] As stated above, the process illustrated in FIG. 1 and described is thus also suitable for situations where the predetermined reference boundary distortion is non-zero, i.e. represents some specific predetermined boundary distortion conditions. Then, what is explained in the above description about “boundary distortion” actually concerns an effective boundary distortion defined in relation to such reference boundary distortion.

[0116] In the above, the method was discussed at a conceptual level. In the following, one generic example of a tomographic imaging method is discussed by using another, more mathematical point of view. The example discussed below relates to an electrical capacitance tomography method. It is to be noted, however, that the principles of the method apply to a non-imaging tomographic analysis also, and to methods utilizing measurements of some other physical quantity than capacitance.

[0117] In tomographic image reconstruction, it is necessary to have a model that describes the relation between the quantity of interest conditions in the target domain and measurement data, i.e. measured values of a physical quantity dependent on the quantity of interest in the target domain. Typically, the goal is to find a distribution for the quantity of interest so that the data predicted by the model is in close agreement with actual measurement data. In this procedure it is often necessary to incorporate some qualitative or quantitative information on the target to the problem formulation to find a unique solution. A typical choice used in the prior art is to assume that the distribution of the quantity of interest to be estimated is a spatially smooth function.

[0118] One example of tomographic image reconstruction for investigating a target domain was disclosed in WO 2014/118425 A1. In the proposed approach, the general principle was first to find an estimate for the distribution of the quantity of interest, and use that estimate for making inferences on the scale-liquid interfaces by means of suitable image processing tools.

[0119] In addition to the location of such interface, it may be often desirable to get information also on the quantity of interest conditions in the inner zone limited by a boundary surface. However, there may exist some complications with this, when using the conventional approach, especially for the following reasons:

[0120] a) If the properties of the scale or other boundary layer material are close to the material in inner zone (e.g. oil and paraffin with similar permittivity in ECT measurements), it may be difficult to detect the scale-liquid interface and hence the result may be erroneous.

[0121] b) If the quantity of interest contrast between the boundary layer material and inner zone is large, it may be challenging to make inferences on the internal conditions of the inner zone.

[0122] c) In general, in an approach based on determination of the distribution of the quantity of interest on the basis of image reconstruction algorithms, it is often necessary to make the spatial smoothness assumption concerning the distribution of the quantity of interest. Abrupt changes of the quantity of interest, e.g. between a deposit material on the boundary surface and the inner zone may then result in erroneous conclusions on the location of the interface between the inner zone and the scale deposit. Similar difficulties may arise also e.g. in a situation with an annular flow of e.g. water present in a pipe for transporting oil, the annular flow limiting an inner zone where the oil may flow.

[0123] In the following example, a situation is assumed where the target domain to be investigated comprises both flowable material (which may comprise solid substance(s)) and solid-like deposition, such as scale deposit, or a liquid or gaseous material different from said flowable material, on the boundary surface, e.g. an inner surface of a process pipe or a sensor element arranged to form a part of such pipe. The deposition and the liquid or gaseous material on the boundary surface may be called generally a boundary layer.

[0124] In the following, possible ways of implementing the method are described in more detail. The distribution of the quantity of interest, e.g. a permittivity distribution, in the inner zone of the target domain is described with quantity ε, and the quantity of interest properties in the region of the boundary layer material and possible wear of the sensor (i.e. the boundary distortion) are described with parameter vector γ. Then the dependence of the physical quantity y on the target domain can be described with model

y=f(ε,γ)+e   (1)

[0125] where e is measurement noise. In the Bayesian framework the posterior density, i.e. the joint density of ε and γ given the observations y, is of the form

p(ε,γ|y)∝p(y|ε,γ)p(ε,γ)   (2)

[0126] where the likelihood density p(ε,γ|y) is defined by the observation model and p(ε,γ) is an appropriately chosen prior density. Estimates for ε and γ can be determined from the posterior density, and the most commonly used estimates are the conditional mean (CM) and the maximum a posteriori (MAP) estimate. Unfortunately, the computation of these estimates is not always straightforward. Markov Chain Monte Carlo (MCMC) methods typically needed for the determination of CM estimates are usually very time-consuming, which may be a major problem is some applications. Effective approaches to find MAP estimates usually require evaluations of various derivatives of the target functional, i.e. the posterior density. Depending on the parametric model for the boundary layer region, some derivatives may be difficult to be evaluated.

[0127] The challenge in determining the estimates is to make some simplifying assumptions about certain terms. A common choice is to assume Gaussian zero-mean observation noise, i.e. e˜N(0,Γ.sub.e). Then the determination of the MAP estimates is equivalent to the minimization problem

[00001]$\begin{array}{cc}{\left(\mathrm{.Math.},\gamma \right)}_{\mathrm{MAP}}=\underset{\mathrm{.Math.},\gamma }{\arg }.Math.\min .Math.\left\{{.Math.L\left(y-f\left(\mathrm{.Math.},\gamma \right)\right).Math.}^2+J\left(\mathrm{.Math.},\gamma \right)\right\}& \left(3\right)\end{array}$

[0128] where the weight matrix L satisfies L.sup.TL=Γ.sub.e.sup.−1, and J(ε,γ) is a “side constraint” determined by the selected prior density. Unfortunately, to solve this minimization problem effectively it is again necessary to evaluate some derivatives of the target functional, which may be a complicated issue.

[0129] To avoid the complications in the estimation of ε and γ, it is possible to formulate the problem in a different way. The idea then is to fix the parameter γ and rewrite the observation model as

y=f(ε,γ.sub.0)+v+e   (4)

[0130] where γ.sub.0 is a fixed representation for the reference boundary distortion and v is the error due to the fixed value γ.sub.0. Eq. (4) is called the approximate mathematical model. The parameter γ.sub.0 can be chosen e.g. so that it does not affect the observations y but the target distribution is fully defined by any arbitrarily selected ε. The term v is the deviation due to the fixed boundary distortion γ.sub.0 and it is naturally unknown since it depends on the actual target. From eqs (1) and (4) it can be seen that the deviation is

v=f(ε,γ)−f(ε,γ.sub.0)   (5).

[0131] With this relation it is possible to obtain information on the statistical properties of v. By generating a sufficient set of representative values of (ε,γ) and by evaluating the deviation v (eq. (5)) for each sample, it is possible to get approximate statistics of the deviation v. Once the statistics of v is approximated, it can be utilized in defining a prior density for v as v is considered as a quantity to be estimated. The posterior density is

p(ε,v|y)∝p(y|ε,v)p(ε,v)   (6)

[0132] where the likelihood is defined by the approximate mathematical model (4). The posterior density (6) can be understood to define the observation difference. As an example, if e is Gaussian with zero-mean and we assume that v is Gaussian, i.e. v˜N(η.sub.v,Γ.sub.v), the MAP estimate can be obtained as

[00002]$\begin{array}{cc}{\left(\mathrm{.Math.},v\right)}_{\mathrm{MAP}}=\underset{\mathrm{.Math.},v}{\mathrm{argmin}}.Math.\left\{{.Math.L.Math.\left(y-f\left(\mathrm{.Math.},{\gamma }_0\right)-v\right).Math.}^2+J\left(\mathrm{.Math.},v\right)\right\}& \left(7\right)\end{array}$

[0133] where the regularizing constraint contains the prior models for ε and v. It is also possible to write a parametric model for v when the deviation is v=v(α) where α is the parametric representation of the deviation. Then the MAP estimate is

[00003]$\begin{array}{cc}{\left(\mathrm{.Math.},\alpha \right)}_{\mathrm{MAP}}=\underset{\mathrm{.Math.},v}{\mathrm{argmin}}.Math.\left\{{.Math.L.Math.\left(y-f\left(\mathrm{.Math.},{\gamma }_0\right)-v\left(\alpha \right)\right).Math.}^2+J\left(\mathrm{.Math.},\alpha \right)\right\}& \left(8\right)\end{array}$

[0134] As an example of parametrization, consider a case where v is Gaussian, i.e. v˜N(η.sub.v,Γ.sub.v). Then the deviation can be written as

v=η.sub.v+Wα  (9)

[0135] where the columns of the matrix W are the eigenvectors of the covariance matrix Γ.sub.v. Furthermore, the number of parameters can be decreased by dividing the last term in eq. (9) into two parts as

Wα=W.sub.1α.sub.1+W.sub.2α.sub.2=v′+v″  (10)

[0136] where the columns of W.sub.1=[w.sub.1, . . . , w.sub.p] are the eigenvectors corresponding to p appropriately chosen (typically largest) eigenvalues, and the rest of the eigenvectors are the columns of W.sub.2=[w.sub.p+1, . . . , w.sub.m]. The approximate mathematical model can now be written as (see eqs (4), (9) and (10))

y=f(ε,γ.sub.0)+W.sub.1α.sub.1+η.sub.v+v″+e   (11)

[0137] where v″ is considered as additional measurement noise. In this case the MAP estimate is

[00004]$\begin{array}{cc}{\left(\mathrm{.Math.},{\alpha }_1\right)}_{\mathrm{MAP}}=\underset{\mathrm{.Math.},{\alpha }_1}{\mathrm{argmin}}.Math.\left\{{.Math.\tilde{L}\left(y-f\left(\mathrm{.Math.},{\gamma }_0\right)-W_1.Math.{\alpha }_1-{\eta }_v\right).Math.}^2+J\left(\mathrm{.Math.},{\alpha }_1\right)\right\}.& \left(12\right)\end{array}$

[0138] Assuming that v″ and e are independent, the weight matrix {tilde over (L)} satisfies {tilde over (L)}.sup.T{tilde over (L)}=(Γ.sub.e+Γ.sub.v″).sup.−1, where Γ.sub.v″=Σ.sub.i=p+1.sup.mλ.sub.iw.sub.iw.sub.i.sup.T with λ.sub.i being the eigenvalue corresponding to eigenvector w.sub.i. Again, the regularizing constraint J(ε,α.sub.1) can be constructed on the basis of the statistics of α.sub.1. The estimate ε.sub.MAP represent the target without providing information on the boundary layer region so basically it describes the distribution in the free-volume region. At this point, no information is available on the boundary layer region but there is an estimate for the effect that is caused by the boundary layer region to the measured data, i.e. there is an estimate for v or for the parameters α in the parametrized case. In the early phase of the method, simulated statistics of v were determined, and these simulation results can be utilized to determine the joint density of v and γ, i.e. the density p(γ,v). Similarly, in the case of parametrized deviation, the joint density p(γ,α) or p(γ,α.sub.1) can be determined on the basis of the results obtained when the simulated statistics of v is generated. Since there is now an estimate v.sub.est for v (can be e.g. the MAP estimate given above), what is to be solved is the density of γ conditioned on v, and it is of the form

p(γ|v)=p(γ,v)/p(v)   (13)

[0139] which defines the density for γ that can be used to determine estimates and credibility intervals for the boundary distortion γ. In the general case, given an estimate v.sub.est for the error v, estimates and credibility intervals for γ can be computed from density (13). For instance, the most probable value for γ is defined as

[00005]$\begin{array}{cc}\hat{\gamma }=\underset{\gamma }{\mathrm{argmax}}.Math..Math.p\left(\gamma |v_{\mathrm{est}}\right).& \left(14\right)\end{array}$

[0140] Assuming a Gaussian joint distribution

[00006]$\begin{array}{cc}p\left(\gamma ,v\right)\propto .Math.\exp \left(-{{0.5\left[\begin{array}{c}\gamma -{\eta }_{\gamma }\\ v-{\eta }_v\end{array}\right]}^T\left[\begin{array}{cc}{\Gamma }_{\gamma }& {\Gamma }_{\gamma .Math..Math.v}\\ {\Gamma }_{v.Math..Math.\gamma }& {\Gamma }_v\end{array}\right]}^{-1}\left[\begin{array}{c}\gamma -{\eta }_{\gamma }\\ v-{\eta }_v\end{array}\right]\right),& \left(15\right)\end{array}$

[0141] it can be shown that the most probable value for the parameters of the boundary layer region is

{circumflex over (γ)}=η.sub.γ+Γ.sub.γvΓ.sub.v.sup.−1(v.sub.est−η.sub.v).   (16)

[0142] Alternatively, in the parametrized case the conditional density of γ is

p(γ|α)=p(γ,α)/p(α),   (17)

[0143] from which, given an estimate α.sub.est and assuming Gaussian joint density

[00007]$\begin{array}{cc}p\left(\gamma ,\alpha \right)\propto .Math.\exp \left(-{{0.5\left[\begin{array}{c}\gamma -{\eta }_{\gamma }\\ \alpha -{\eta }_{\alpha }\end{array}\right]}^T\left[\begin{array}{cc}{\Gamma }_{\gamma }& {\Gamma }_{\gamma .Math..Math.\alpha }\\ {\Gamma }_{\alpha .Math..Math.\gamma }& {\Gamma }_{\alpha }\end{array}\right]}^{-1}\left[\begin{array}{c}\gamma -{\eta }_{\gamma }\\ \alpha -{\eta }_{\alpha }\end{array}\right]\right),& \left(18\right)\end{array}$

[0144] the most probable value for γ is of the form

{circumflex over (γ)}=η.sub.γ+Γ.sub.γαΓ.sub.α.sup.−1(α.sub.est−η.sub.α).   (19)

[0145] The above-described approach and its modifications can be employed for the estimation of a single target of interest on the basis of a single data vector y. However, the same approach can also be used in dynamical estimation where a temporal model is constructed to describe the time evolution of the quantities to be estimated. This straightforward extension results in some extra phases that are well known in the field of recursive Bayesian estimation. The use of temporal models and Bayesian filtering approaches can be very beneficial in real-time process imaging.

[0146] FIG. 2 shows a schematic cross-sectional illustration of section of an electrically insulating pipe 1 forming a support body, on the outer surface of which eight electrodes 2 are attached for performing measurements of one or more capacitance-dependent electrical quantities in a target domain 3 comprising the inner free volume 4 of the process pipe 1 as well as the pipe wall. Thus, in the example of FIG. 2, the boundary of the target domain 3 coincides with the outer surface of the pipe 1 and the inner surfaces of the electrodes 2 thereon. Alternatively, the electrodes could lie at least partly embedded in the pipe wall.

[0147] FIG. 2 arrangement may be used as a part of a measurement system for performing e.g. measurements of capacitance as a physical quantity dependent on permittivity representing a quantity of interest in the target domain. Naturally, when measuring capacitance, the primary physical quantity to be measured may be e.g. voltage or current. Measurement results thereby provided may be used as measured values to be received in the example methods discussed above.

[0148] The electrically insulating pipe 1 is surrounded by a cylindrical metal sheath 5, comprising flanges 6 extending radially from the sheath to the outer surface of the pipe 1. For performing the measurements, the metal sheath and the flanges thereof may be grounded (not illustrated in the drawing) to serve as a screen to isolate the system of the electrodes and the target domain from its surrounding and to prevent the electrodes from “seeing” each other directly via the exterior of the electrically insulating pipe. In the absence of such flanges, also the material(s) between the metal sheath 5 and outer surface of the pipe 1 would affect the capacitance-related measurements. In such case, the target domain should extend to the inner boundary of the metal sheath in order to take this effect into account in the calculations.

[0149] The free inner volume 4, forming a free inner zone, within the pipe 1 is filled with a process material flowing through the pipe. Scale material 7 in the form of solid deposit has been formed of the substances included in the flowing material on the pipe inner surface 8. As another change in comparison to the initial situation, the material of the electrically insulating pipe 1 has been eroded at one location of the pipe inner surface 8 so that a slight recess 9 has been formed thereon. Also the recess changes the flow conditions within the pipe 1.

[0150] The pipe inner surface 8, including the changed pipe inner surface at the location of the wear 9, forms a boundary surface which, together with the scale deposit surface, limits the free inner volume 4 within the pipe.

[0151] An alternative example of a measurement setup enabling determination of scale and wear in a process pipe is shown in FIG. 3. As an essential difference in comparison to FIG. 2, there is an electrically conductine process pipe 11 formed e.g. of some metal. In the point of view of measuring capacitance-dependent electrical quantity values, an electrically conductive pipe necessitates the electrodes 12 being in a direct contact with the free inner volume 14 forming a free inner zone inside the pipe. In this kind of situation, the target domain 13 in which the measurements are to be made is limited by the electrodes and the electrically conductive pipe inner surface 18 itself. Further, due to the electrically conductive material of the pipe 11, each electrode is electrically insulated from the pipe by means of thin electrically insulating layer located between the electrode and the pipe wall. The pipe inner surface forms here a boundary surface together with the free surfaces of the electrodes.

[0152] Also in the situation of FIG. 3, there is scale deposit 17 formed on the pipe inner surface 18, and a wearing process has eroded the pipe inner surface 18 and one of the electrodes forming a slight recess 19 thereon. Naturally, such recess could also extend to areas of more than one electrode only.

[0153] In the examples of FIG. 2 and FIG. 3, the scale deposit and the wear of the process pipe form boundary distortions changing the conditions of and close to the original boundary surfaces.

[0154] Using the measurement setups as those of FIGS. 2 and 3, capacitance measurements may be performed according to the principles known in the art. In general, the field of electrical capacitance tomography ECT, the measurements are typically carried out as follows. Voltage supply (e.g. in square-wave, sinusoidal or triangular form) is applied to one of the electrodes (an excitation electrode) while the other electrodes are grounded. Capacitances between all electrode pairs are measured (in this example, each “group” of electrodes comprises just one single electrode). The capacitance measurement is repeated so that each of the electrodes is used as an excitation electrode. Therefore as a general rule, if there are N electrodes in the measurement system, N*(N−1)/2 independent capacitance values are obtained. Capacitances depend on the permittivity distribution in the target domain. Permittivity distribution of the target domain can then be estimated based on the set of the measured capacitance values. On the basis of the permittivity distribution, behavior and/or some physical quantities of the underlying process can be investigated.

[0155] In FIGS. 2 and 3, eight and ten electrodes, respectively, have been installed on the pipes. However, these are just examples only, not limiting the applicability of the present invention to any number of electrodes suitably configured to allow measuring capacitances or other capacitance-dependent electrical quantities between the electrodes. Further, FIGS. 2 and 3 illustrate cross-sectional views of process pipes and one electrode ring only, thus referring to a two-dimensional target domain. However, it is possible to measure and monitor a three-dimensional target domain by arranging electrodes in several rings or layers along the axial direction of the process pipe.

[0156] In cases where scaling, or some other type of boundary layer, or wear in a process can be assumed to be uniform (e.g. when the scale material deposits uniformly onto the walls of a pipe or a vessel walls or there is e.g. an annular flow of a uniform thickness), it is possible to reduce the computational cost of the method by taking advantage of symmetry. FIG. 4 shows a rod 31 forming a support body, and an electrode configuration wherein a plurality of ring electrodes 32 is mounted on the surface 38 of the rod-shaped support body, said surface representing here a boundary surface correspondingly to those pipe inner surfaces of FIG. 2 and FIG. 3. Potential fields generated by ring-like electrodes are cylinder-symmetric. Thus, a Finite Element Method (FEM) approximation used to model the target domain can be formulated in two dimensions (axial and radial) only, which reduces the computational complexity remarkably.

[0157] As yet another alternative, the support body can be formed as a simple plate-like body 41, as is the case in the measurement probe 40 shown in FIG. 5. The exemplary measurement probe 40 of FIG. 5 is configured to be installed through a wall of a cylindrical vessel so that the actual support body 41, having a plurality of electrodes 42 thereon, faces towards the interior of the vessel. The backside of the measurement probe comprises connectors 46 for connecting the electrodes 42 to appropriate measurement electronics. In the example of FIG. 5, there are electrodes 42 with different sizes.

[0158] The boundary surface 48 of the support body is shaped curved so as to coincide with the inner surface of the wall of the cylindrical vessel. Naturally, the boundary surface of a plate-like support body could also be planar or have some other non-planar shape than the curved one shown in FIG. 5. Also, it is to be noted that the thickness of a “plate-like” support body can vary according to the conditions of the actual application at issue.

[0159] In FIG. 5, the electrodes are arranged so that they lie at the level of the boundary surface 48 and they are curved similarly to the boundary surface. This way, it is possible to have a continuous, smooth boundary surface without any protrusions (or recesses) possibly adversely affecting the flow conditions in the vicinity of the boundary surface. In some applications, e.g. in the field of food industry, such smooth configuration may be advantageous also from hygiene aspects.

[0160] It is to be noted that permittivity as the quantity of interest and capacitance (or a current or a voltage signal in response to a voltage or current excitation, respectively) as the physical quantity to be observed in the examples of FIGS. 2 to 5 is one example only. The basic principles of the methods discussed above may be implemented in determining any quantity of interest which may be investigated by means of one or more physical quantities dependent on that quantity of interest. For example, tomographic methods may be used to determine electrical conductivity or impedance in the target domain. Some examples of possible quantities of interest which are not electrical are speed of sound, observable via measurements of time of propagation of acoustic signals transmitted to the target domain, and scattering and absorption of light, observable e.g. via measurements of transmitted portion of a light signal transmitted to propagate though a target domain.

[0161] The support body, such as the pipes 1, 11 of FIGS. 2 and 3 or the rod or the plate of FIGS. 4 and 5, respectively, do not necessarily belong to the actual process equipment to be monitored, but may be provided in the form of a separate measurement probe located in the process equipment. In order to ensure sufficiently similar behavior of the support body and the actual process equipment itself, and thus the reliability of the scale/wear determination on the basis of monitoring scale on and/or wear of the support body, the support body is preferably formed of the same material as the actual process equipment.

[0162] FIG. 6 illustrates schematically an apparatus 50 by which any of the methods as described above may be carried out. In the operational core of system, there is a computer 51, serving as a computing system, comprising an appropriate number of memory circuits and processors for providing and storing the mathematical models and performing the computational steps of the method. The apparatus further comprises a measurement electronics unit 52 and a measurement probe 53 comprising an annular support body and a plurality of electrodes. The support body and the electrodes can be configured e.g. according to those illustrated in FIG. 2 or 3. The measurement electronics unit and the measurement probe serve as a measurement system.

[0163] The measurement electronics unit is connected to the computer so that the measurement electronics unit can be controlled by the computer and that the measurement results can be sent to and received by the computer for further processing. The computer may comprise a program code, stored in a memory and configured to control the computer to carry out the steps of the method. As a result of the method performed by the apparatus, an image 54 of the target domain is generated on the basis of the reconstructed permittivity distribution within the target domain inside the annular support body of the measurement probe 53. The image shows the scale on and the wear of the support body boundary surface.

[0164] It is obvious to a person skilled in the art that with the advancement of technology, the basic idea of the invention may be implemented in various ways. The invention and its embodiments are thus not limited to the examples described above; instead they may freely vary within the scope of the claims.