TECHNICAL FIELD

The present specification relates generally to moni- toring industrial processes where process materials are stored or conveyed in pipes, vessels, or contain ^¬ ers. In particular, the present specification is related to methods and apparatuses, as well as program codes to implement such methods, for monitoring, by means of determining a quantity of interest in a tar ^¬ get domain, various internal conditions in such pro ^¬ cess equipment. Said internal conditions may relate e.g. to phase interfaces, mixing, or material bounda ^¬ ries within such process equipment, and/or for moni- toring scaling or deposition formation on and/or wear of the surfaces of such equipment.

BACKGROUND

Tomographic investigation methods, such as tomographic imaging, cover various methods for investigating or monitoring a target domain on the basis of determining a quantity of interest in the target domain by means of non-invasive measurements thereof, or of a second ^¬ ary physical quantity proportional to or dependent on the quantity of interest. The quantity of interest may be determined e.g. as a distribution thereof in the target domain.

The general principle in tomographic methods is to compare the measured values with approximate values of the quantity of interest or the secondary physical quantity determined by a mathematical model defining the relationship between the measured values and the quantity of interest in the target domain, and to ad- just the model so as to reduce possible differences between the measured and modelled values. Such adjust- ment may be continued iteratively until a sufficient consistency exists between the measured values and the model. Then, the distribution of the quantity of in ^¬ terest provided by the adjusted model may be consid- ered as an estimate of the quantity of interest condi ^¬ tions in the target domain. This estimate can further be considered as an indication of material conditions in the target domain. For example, in the case of de ^¬ termining an estimate of electrical permittivity in the target domain, abrupt spatial changes in the per ^¬ mittivity may indicate boundaries between different materials or phases.

In some applications, one specific type of material conditions within the target domain to be investigated is the presence of so called scale material on the equipment surfaces in the target domain. In some ap ^¬ plications, e.g. in oil industry, corresponding phenomenon may be called just deposition. Scaling is a well-known contamination problem which may occur in many different applications in process industry. Scal ^¬ ing, often called also fouling, means generally unde- sired deposition or accumulation of material on the surfaces of pipes, vessels, or other containers used for leading or storing flowable materials.

As a result of scaling, or generally deposition, an extra layer of solid material is formed on a process equipment surface. Thereby, the free inner zone (area or volume) within the pipe or other container, open for the presence of a flowable material, is changed. This can lead to many problems. For example, changed shape of the free inner volume causes disturbances to the fluid flow. At least, the reduced cross-sectional area of the free inner volume of a process pipe in ^¬ creases the flow resistance through the pipe. In an extreme case, the pipe can be entirely clogged, there ^¬ by stopping the entire process at issue.

In order to prevent dramatic problems e.g. due to un- expected clogging of a process pipe, or to optimize the use of scale inhibitors or the cleaning cycle of the pipe, one should preferably be able to monitor the scaling situation and its development in time. From the point of view of determining the quantity of interest in the free inner zone of the target domain, scaling or other type of deposition may result in erroneous conclusions on quantity of interest conditions in the free inner zone. Therefore, the effects thereof on the measurements should be compensated.

In prior art, scaling or other types of deposits has been monitored or diagnosed e.g. with camera-based techniques, wherein a camera is installed in the pro- cess equipment to be analyzed, with acoustic (typical ^¬ ly ultrasound) methods, or by simple mechanical meth ^¬ ods in which special intelligent test objects are mounted onto process pipe walls. Recently, a solution enabling scale monitoring by means of an ECT process was disclosed in WO 2014/118425 Al .

In addition to scaling, another example of phenomena possibly disturbing the determination of the material properties in the inner zone of the target domain is an annular flow forming a layer of a material, different from the main material in the inner zone, on an inner surface of a process pipe or other process equipment. As one specific example, in oil industry, such annular flow may be formed by water. Similarly to scaling, an annular flow of a material differing from the material (s) in the inner volume shall be taken in- to account in the analysis to avoid false conclusions on the inner zone conditions.

Yet another problem occurring in various industrial process equipment for leading and/or storing flowable materials is the wear of the walls of the pipelines and vessels. Wear may occur due to chemically aggres ^¬ sive materials reacting with the surfaces of the pro ^¬ cess equipment, but it can also result from simple me- chanical abrasion caused e.g. by sand contained in a liquid flow. Similarly to scaling or annular flow, also wear changes the conditions of the target domain and may result in erroneous conclusions on quantity of interest conditions in the free inner zone thereof. Therefore, also the effect of possible wear in the measurements should thus be able to be taken into ac ^¬ count. Wear may also occur simultaneously with scaling or annular flow. Therefore, it would be highly useful to have an efficient solution capable of on-line moni- toring of all those phenomena.

On the other hand, it would be advantageous in some applications if both the properties of scale or other deposition material or an annular flow, and the inter- nal material conditions in the inner volume could be determined reliably in a single process.

SUMMARY

Some aspects relating to determining a quantity of in- terest in a target domain are specified by claims 1, 8, and 15.

In one aspect, a method may be implemented for deter ^¬ mining a quantity of interest in a target domain hav- ing a boundary surface, by means of measurements of a physical quantity dependent on the quantity of inter ^¬ est in the target domain. The target domain may possi- bly comprise 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 deposit thereon limiting an inner zone within the target domain, the inner zone comprising at least one second material which may be different from the first material.

The boundary layer may be formed e.g. as a deposit of a solid or solid-like first material. Hereinafter, such deposition is generally called "scaling" or "scale material". The boundary layer may also com ^¬ prise, for example, an annular flow of a first material different from the second material in the inner zone.

The target domain may lie in any kind of process envi ^¬ ronment where a boundary layer effect, such as scaling or annular flow on and/or wear of process equipment surfaces may exist. These kinds of industrial process ^¬ es exist e.g. in oil production, refining, and transport, other oil based industries, energy produc ^¬ tion, pulp industry, and food industry, without limit ^¬ ing the scope of this specification to these examples only.

The investigation method may be implemented as a tomo ^¬ graphic process, in which the conditions in the target domain, e.g. the distribution of the quantity of in- terest, is reconstructed. The quantity of interest fi ^¬ nally determined on the basis of the adjusted model may be represented as images, typically as two- dimensional cross-sectional images of the target do ^¬ main. Then, the method may fall within the field of tomographic imaging. The basic principles of tomo ^¬ graphic imaging, including e.g. electrical impedance tomography EIT and electrical capacitance tomography ECT as well as ultrasound tomography, and its use in various applications, are well known for those skilled in the art. In the case of tomographic imaging, vari ^¬ ous image reconstruction algorithms known in the art may be used. On the other hand, the method is not nec ^¬ essarily pure "imaging" comprising such image reconstruction. In some applications, it may be sufficient to determine just one or more characteristic parame ^¬ ters indicating or representing the quantity of inter- est conditions in the target domain.

The quantity of interest may be an electrical quanti ^¬ ty, such as permittivity (which may be a real or a complex valued quantity) or electrical conductivity of the material (s) present in the target domain, in which case the method may lie generally within the field of electrical tomography. However, the quantity of interest is not limited to electrical quantities only but may also be, for example, a quantity indicating opti- cal scattering and/or absorption properties of the material (s) in the target domain, or an acoustic quanti ^¬ ty such as speed or attenuation of sound in those target domain material (s) . It may also be a general mate ^¬ rial property such as density. Generally, the method may be applied in investigating any quantity of inter ^¬ est which can be measured, directly or indirectly, preferably non-invasively, i.e. from the outside of the target domain. For example, in the case of density as the quantity of interest, nucleonic devices may be used to make the measurements by means of gamma, x- ray, or neutron beams, wherein the primary physical quantity to be measured may be e.g. the intensity of the radiation. Similarly, tomographic measurement set ^¬ ups utilizing radio frequency radiation, namely radio tomographic imaging, can also be used in certain situ ^¬ ations for determining attenuation properties of the material in the target domain. Then, the primary phys- ical quantity may be the attenuation/intensity of a radio frequency electromagnetic field.

By said indirect measuring is meant that the physical quantity to be actually measured may be a quantity which is just dependent on, or proportional to, the primary quantity of interest. Naturally, if the quan ^¬ tity of interest is directly measurable, the physical quantity may be the actual quantity of interest it- self. Thus, generally, the physical quantity of inter ^¬ est may be the quantity of interest itself, or it may be just dependent on the latter.

In the case of said example of permittivity as the quantity of interest, the method may generally be based on principles known in electrical capacitance tomography (ECT) . In ECT, the permittivity in the target domain may be determined. This may be implemented by finally reconstructing an image of the permittivity distribution in the target domain. Permittivity, and in particular changes thereof may provide information on the internal material properties and distributions within the target domain. A typical example of utili ^¬ zation of ECT is imaging a multi-phase flow in an in- dustrial process, wherein an image showing the areas or volumes of different phases within the material flow is generated. One example of this kind of method and different practical issues involved therein is discussed in US 7496450 B2. Recently, the inventors have found it being possible to use ECT also e.g. for monitoring scaling (fouling) of undesired deposit on, as well as possible wear of, process equipment surfac ^¬ es in various industrial processes. By "target domain" is meant in this specification generally a two-dimensional area or a three-dimensional volume, the inner conditions of which are to be inves- tigated. For example, the target domain may comprise a cross-sectional area or a volume within or of a pro ^¬ cess pipe, container, or vessel. The "boundary surface" may be any structural surface in the target domain. For example, it may be an inner surface of a process pipe, container, or vessel, or a measuring sensor integrated as part of such process equipment. It may also be an outer surface of a meas- uring probe or other instrument inserted into the in ^¬ terior of such process equipment. Location "in" the target domain means that the boundary surface belongs to the two-dimensional or three-dimensional target do ^¬ main. In other words, this definition covers both sit- uations where the boundary surface lies within the in ^¬ terior of the two or three-dimensional target domain, and situations where the boundary surface coincides with the target domain boundary, thus defining the boundary of the target domain.

The "boundary distortion", such as a boundary layer of a first material on the boundary surface or wear of the boundary surface, generally refers to a deflection or divergence from the normal, desired, initial, or any predetermined reference conditions in the target domain. A boundary distortion thus changes the properties of the boundary surface, and/or the target domain properties close to the boundary surface, which bound ^¬ ary surface may be e.g. a process equipment inner sur- face. One example of the first case is wear of the boundary surface, whereas the target domain properties close to the boundary surface may be changed e.g. due to a deposit or an annular flow of a first material on the boundary surface. Such boundary distortion often affects the tomographic analysis process in a way which may result erroneous conclusions on the quantity of interest in the target domain if the boundary dis- tortion is not appropriately taken into account in the analysis .

The deposit of a first material may be e.g. scale ma- terial deposited on the boundary surface such as a process pipe inner surface. It may be solid material, but also some non-solid deposit. For example, in some processes, deposit material may aggregate on various surfaces, which material is not truly solid, but has however high viscosity so that the deposit behaves substantially like one formed of a solid material.

Examples of scale/deposition material in oil industry comprise bitumen, wax, paraffin, and asphaltene, and various scaling materials covered by a common term "mineral scaling". The latter comprises e.g. calcium carbonate and calcium sulphate based compounds, the latter comprising e.g. gypsum. In energy production, scaling can occur e.g. due to the deposition of con- taminants contained in water in surfaces of boilers. Water contaminants that can form boiler deposits in ^¬ clude e.g. calcium, magnesium, iron, aluminum, barium sulphate, and silica. The scale is typically formed as salts of these materials.

In oil industry, an annular flow as another example of a boundary layer of a first material may be formed e.g. of water. It is to be noted that the boundary layer of a first material is not limited to the above scale/deposition and annular flow examples. The method is applicable to any process environment where a boundary layer of a first material differing from a second material in the inner zone may exist. By wear is meant in this specification generally a change of the actual location of the boundary surface, at least locally, from the initial or an earlier loca ^¬ tion due to removal of the boundary surface material. This kind of wear may be result of an erosion or abra ^¬ sion process where such boundary surface material is mechanically and/or chemically eroded.

Also the inner zone, which is at least partially lim- ited by the boundary surface and/or the possible boundary layer thereon, may be a two dimensional area or a three dimensional volume.

The second material present in the inner zone of the target domain may be some flowable material, i.e. ma ^¬ terial which can flow in a process pipe or in a vessel or container. Such material can be in a liquid form but can also contain solid or gaseous substance ( s ) , and can be also free of any truly liquid substance. There may be just one second material present in the inner zone, having uniform quantity of interest condi ^¬ tions throughout the the inner zone. On the other hand, there may be two or more materials and/or dif ^¬ ferent material phases present in the inner zone of the target domain. In general, there may be any mix ^¬ ture of gaseous, liquid, and solid substance (s) flow ^¬ ing or lying in stationary state in the inner zone. For example, in the field of oil refining and transport, various flow conditions within oil pipes are known, where gaseous and liquid components are distributed differently. As known for a person skilled in the art, some of such situations are known as bub ^¬ ble, plug, slug, annular, mist, stratified, and wavy flow conditions.

In principle, the first and the second materials are not limited to any specific materials or material properties. However, particular advantages may be achieved by the method in a case where the quantity of interest of at least one first material of possible boundary layer, and that of at least one second mate- rial differ from each other.

The "possible" existence of the boundary distortion means that the method is not limited to situations on ^¬ ly where there is some boundary distortion. One pur- pose of the method is to specifically determine wheth ^¬ er any boundary distortion exists in the target domain .

The method comprises:

- providing simulated statistics of a devia ^¬ tion, 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 approxi ^¬ mate 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 which an effective boundary distor- tion 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 approxi ^¬ mate mathematical model, and adjusting the approximate mathematical model to reduce the observation differ- ence, thereby providing an adjusted approximate mathe ^¬ matical model; and - determining an estimate of the quantity of interest in the target domain on the basis of the ad ^¬ justed approximate mathematical model. By "effective boundary distortion" is meant in this specification a boundary distortion defined relative to the reference boundary distortion. Thus, also at ^¬ tributes such as "actual" or "present" could be used instead of the word "effective" to refer to such boundary distortion. Consequently, the effect of the effective boundary distortion in the observations is also defined relative to, or in comparison to, a situ ^¬ ation with the predetermined reference boundary dis ^¬ tortion .

The reference boundary distortion may refer to a situ ^¬ ation where there is actually no boundary distortion, i.e. to a situation with the boundary surface being in its initial state. The reference boundary distortion may also be defined as any type of boundary distortion causing the boundary surface conditions to differ from the initial or an earlier situation. For example, the reference boundary distortion can be defined on the basis of scale and/or wear conditions already deter- mined. It may also be defined on the basis of knowledge about typical or expected boundary distor ^¬ tion conditions in the process environment at issue.

The "simulated statistics" refer to statistical infor- mation of simulated effect of an effective boundary distortion in the observations of the physical quanti ^¬ ty. This simulated statistics of a deviation is pref ^¬ erably generated according to probable real boundary distortion conditions in the actual target domain. In principle, such statistics may generated by any means. Some preferred examples, where statistic is generated by simulating a number of various boundary distortion and inner zone samples, are described in more detail later in this specification. Statistics may comprise e.g. information about the mean value and covariance of the deviation due to variations in boundary distor- tion parameters, such as location and thickness and internal material distribution of a boundary layer on the boundary surface, and/or a changed location and/or shape of the boundary surface itself. By said deviation caused by a boundary distortion is meant simply an effect which a boundary distortion has on observations, achievable by means of measurements, of the physical quantity. "Providing" means here that the method itself does not necessarily comprise determining or generating the simulated statistics, but that such simulated statis ^¬ tics may be generated separately and be just received as a step of the method. This allows, for example, an embodiment where the simulated statistics are stored electronically in an apparatus configured to perform the steps of the method. On the other hand, it is also possible to generate or determine simulated statistics in the method, possibly using one single apparatus or system serving also for determining the simulated statistics.

The approximate mathematical model defining the rela ^¬ tionship between the quantity of interest and observa- tions of the physical quantity, which observations may be received via measurements, thus provides means for estimating the quantity of interest in the target do ^¬ main. The model is "approximate" in the sense that no effective boundary distortion in the target domain is directly modelled by it. Instead, the quantity of in ^¬ terest is modelled in the target domain according to the reference boundary distortion. The effect of an effective boundary distortion is modelled via the de ^¬ viation which an effective boundary distortion causes in the observations. In the case of no effective boundary distortion, i.e. with a boundary distortion identical to the reference boundary distortion, the deviation is naturally zero.

By restricting said behavior of the deviation in accordance with the simulated statistics, it is possible to limit the degrees of freedom of the deviation to be estimated. In practice, the deviation may be estimated by setting the simulated statistics as the prior mod ^¬ el. Thus, it is presumed that, before one has any in ^¬ formation on the target, the deviation obeys the simu- lated statistics. As a result, the estimate possesses realistic features.

Correspondingly to providing the simulated statistics, said "receiving" measured values, i.e. measurement re- suits, of the physical quantity may mean just receiv ^¬ ing results, preferably in the form of electronic da ^¬ ta, of ready performed measurements. In other words, the method itself does not necessarily comprise per ^¬ forming the actual measurements, but the measured val- ues of the physical quantity may be generated sepa ^¬ rately and just received as a step of the method. This allows, for example, an embodiment where the results of the measurements performed at a measuring site are sent electronically to an analysis site where the ac- tual analysis and quantity of interest determination is carried out. On the other hand, it is also possible to perform the measurements and analysis steps inte ^¬ grally, e.g. using a single apparatus comprising both a measurement system for performing the measurements, and some appropriate computing system for performing the other operations of the method. The measurements can be performed according to the principles as such well known in the field of various measurement technologies and tomographic investigation methods. For example, in the case of real or complex valued permittivity as the quantity of interest, the measured results may be current signals resulting in response to various voltage excitation signals.

The measured values and corresponding observations de- termined according to the approximate mathematical model are compared to determine an observation differ ^¬ ence therebetween. The observation difference is thus a measure of the correspondence between the observa ^¬ tions determined by the approximate mathematical model and the actual measured values, the latter being af ^¬ fected by the real conditions in the target domain.

As specified above, the approximate mathematical model may be adjusted so that the observation difference is reduced. This may be performed by adjusting various parameters of the model related to the quantity of in ^¬ terest and the deviation. Said adjusting may be continued iteratively until the observation difference reduces below a predetermined level.

As a result of such adjusting, an adjusted approximate model is provided. In the adjusted model, the term "adjusted" just refers to the fact that the parameters of the model have been adjusted in comparison to the initial ones. So, the general nature of model does not need to be changed.

By means of the adjusted approximate mathematical mod ^¬ el, an estimate of the quantity of interest conditions in the target domain may thus be determined. For the estimated quantity of interest, the effect of possible effective boundary distortion on the measurements has been compensated via the separate term of the model modelling the deviation which an effective boundary distortion causes in the observations. In practice, comparing the measured values and corre ^¬ sponding observations according to the approximate mathematical model, and changing the parameters of the approximate mathematical model is generally known as an inverse problem or inverse calculation. Solving an inversion problem is typically based on rather complex computational algorithms performed at least partly au ^¬ tomatically by means of suitable computation programs installed in a suitable processor. Several different algorithms suitable for the present invention are known in the art. Some of them are discussed in more detail in the detailed description section of this specification .

In one embodiment, the simulated statistics of a devi- ation are determined by:

- providing a simulative mathematical model of observations of the physical quantity, the simula ^¬ tive 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 modelled quantity of interest conditions in the inner zone of the target domain, one observation with the reference boundary distortion and another with a modelled effective boundary distortion for each type of modelled quantity of interest conditions, using various mod ^¬ elled 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 .

The determination of the simulated statistics may be performed in advance, in which case the simulated sta ^¬ tistics may be just received, preferably in some ap ^¬ propriate electronic form, in the method. On the other hand, such determination may be part of the actual method for determining the quantity of interest.

Similarly to the approximate mathematical model, the simulative mathematical model of observations defines the relationship between the quantity of interest in the target domain and observations of the physical quantity, which observations may be received via meas ^¬ urements. Differently from the approximate mathemati ^¬ cal model, the simulative mathematical model prefera ^¬ bly determines the quantity of interest in a way tak ^¬ ing into account also an effective boundary distortion possibly present in the target domain. In this sense, in comparison to the approximate mathematical model, the simulative mathematical model does not model an approximation of the quantity of interest only but a complete representation thereof.

In general, a "mathematical model" refers in this specification to a numerical representation of the relationships between the physical material properties of the two or three-dimensional target domain, affect- ing the quantity of interest, and observations of the physical quantity dependent on the quantity of inter ^¬ est.

In generating the simulated observations for each type of modelled quantity of interest conditions in the in ^¬ ner zone both with and without an effective boundary distortion, one single modelled quantity of interest conditions for the entire target domain may be used as a basis for both situations. For the situation with a modelled effective boundary distortion, e.g. a scale deposit on the boundary surface differing from possi- ble scale according to the reference boundary distortion, the quantity of interest conditions in the area (or volume) of the boundary distortion may then be replaced with local conditions according to that mod ^¬ elled effective boundary distortion. The other way round, with a given modelled quantity of interest con ^¬ ditions for the inner zone outside the modelled effec ^¬ tive boundary distortion, the modelled quantity of in ^¬ terest conditions may be complemented for the situa ^¬ tion without a modelled effective boundary distortion by simply modelling the quantity of interest condi ^¬ tions for the zone outside said actual inner zone al ^¬ so. This may be accomplished, for example, by extrapo ^¬ lating the modelled quantity of interest conditions, on the basis of that in the inner zone, so as to cover the entire target domain.

When simulated observations are generated for one type of modelled quantity of interest conditions in the in ^¬ ner zone both with and without an effective boundary distortion, it is possible to determine a deviation which the boundary distortion causes in the simulated observations for the modelled quantity of interest conditions at issue. The plurality of various modelled quantity of interest conditions in the inner zone of the target domain may be selected according to probable conditions expected in the application at issue. Also some stochastic al ^¬ gorithms may be used to generate random or quasi- random quantity of interest conditions within some predetermined ranges. By simulating the observations corresponding the actual measured values of the physi- cal quantity both for situations with no effective boundary distortion and with an effective boundary distortion, the effect of the effective boundary dis ^¬ tortion on the observations may be simulated.

Said using various modelled effective boundary distor ^¬ tions means that at least two, preferably more, dif ^¬ ferent modelled effective boundary distortions are used in generating the simulated observations of the physical quantity. For example, it is possible to use a unique modelled effective boundary distortion for each type of modelled quantity of interest conditions in the inner zone so that the same modelled effective boundary distortion is used for one type of modelled quantity of interest conditions only. Alternatively, one single modelled effective boundary distortion may be used for several different modelled quantity of in ^¬ terest conditions in the inner zone. On the basis of a plurality of simulated observations for various types of modelled quantity of interest conditions and various boundary distortions, the sta ^¬ tistics of a deviation which an effective boundary distortion causes in the simulated observations may be finally determined.

In addition to the estimate of the quantity of inter ^¬ est in the target domain, it is further possible to determine, on the basis of the adjusted approximate mathematical model, also an estimate of a deviation caused by the possible effective boundary distortion in the measured values of the physical quantity. On the basis of this estimate, the simulated statistics of a deviation an effective boundary distortions caus- es in the simulated observations, and the modelled ef ^¬ fective boundary distortions used in generating the simulated observations of the physical quantity, an estimate of an effective boundary distortion really present in the target domain may be then determined.

Thus, it is possible to first determine an estimate of a deviation which the effective boundary distortion, if any, caused in the measured observations of the physical quantity. On the other hand, the simulated statistics of a deviation which an effective boundary distortions causes in the simulated observations, and the known modelled effective boundary distortions used in generating said simulated observations of the phys ^¬ ical quantity, may be used to determine an estimate of a boundary distortion possibly present in the target domain. Said determining said last mentioned estimate may actually mean determination of the most probable effective boundary distortion having caused the estimated deviation in the measured values.

In the above, the method is described with regard to one specific instantaneous or stationary situation in the target domain only. Naturally, the method may be applied also for continuous monitoring of the target domain, wherein the situation in the target domain, in particular the quantity of interest conditions there- in, is determined continuously or dynamically, i.e. for different, consequent time instants. Then, the ap ^¬ proximate mathematical model may comprise information not only about the relationships between the observa ^¬ tions and the quantity of interest conditions as well as the deviation but also about the time-dependencies of those factors.

The boundary surface may limit the target domain, i.e. define at least a part of the boundary thereof. This approach is particularly useful for the cases where the boundary surface is formed of an electrically con ^¬ ductive material, e.g. a metal. Metallic surfaces can be used in high temperatures and pressures and, in general, in various harsh environmental conditions.

Alternatively, the boundary surface may lie within the interior of the target domain so that the target do ^¬ main extends behind the boundary surface. In this case, at least part of the body, a surface of which the boundary surface forms, is included in the target domain. This arrangement is suitable, for example, for use in the case of an electrically insulating boundary surface .

The method may also comprise performing measurements of the physical quantity dependent on the quantity of interest, thereby providing measured values of the physical quantity. So, the method may comprise also generating the measured values which are then received for the actual analysis steps of the method. Such measurements may be performed according to the princi- pies known in the field of tomographic investigation methods, in particular tomographic imaging, such as electrical tomographic imaging. Examples of such meth ^¬ ods include electrical impedance tomography and elec ^¬ trical capacitance tomography.

Various advantageous effects may be achieved by means of the method discussed above. First, because the ef ^¬ fect of possible effective boundary distortion on the observations is taken into account, a reliable esti- mate of the quantity of interest in the target domain, in particular in the inner zone thereof, may be determined. Said taking into account of the effect of pos ^¬ sible effective boundary distortion is carried out in a way allowing avoiding, or at least alleviating, at least some of the problems associated with the conven ^¬ tional approaches. Further, not only an estimate of the quantity of interest, but also an estimate of the possible effective boundary distortion itself may be determined. By using appropriate modelled effective boundary distortions in the simulations, it is possi ^¬ ble to determine estimates for both boundary layer on and wear of the boundary surface, thereby making the method very versatile. On the other hand, the general principles of the method are applicable to any type of tomographic investigation method with any target do ^¬ main type, and for any type of quantity of interest observable by means of measurements of a physical quantity dependent on the quantity of interest.

The steps of the method discussed above are preferably performed at least partially automatically by means of suitable computing and/or data processing means. Such means may comprise e.g. at least one processor and at least one memory coupled to the processor. The at least one memory may store program code instructions which, when run on the at least one processor, cause the processor to perform operations according to various steps of the method. Alternatively, or in addi ^¬ tion, at least some of those operations may be carried out, at least partially, by means of some hardware logic elements or components, such as Application- specific Integrated Circuits (ASICs) , Application- specific Standard Products (ASSPs) , or System-on-a- chip systems (SOCs) , without being limited to those examples . What is stated above about the details, ways of imple ^¬ mentation, preferred features, and advantages in the context of the method aspect apply, mutatis mutandis, also to the apparatus aspect discussed hereinafter. The same applies vice versa.

According to an apparatus aspect, an apparatus may be provided for determining 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 possibly comprising a boundary distortion such as a boundary layer of a first material on the bounda ^¬ ry 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. Such apparatus comprises 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 mathe ^¬ matical model of observations of the physical quanti ^¬ ty, 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 which an effective boundary distortion causes in the observa ^¬ tions, 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 observa ^¬ tion difference, thereby providing an adjusted approx ^¬ imate mathematical model; and

- means for determining an estimate of the quantity of interest in the target domain on the basis of the adjusted approximate mathematical model. As discussed above in the context of the method as ^¬ pect, the statistics of a deviation may be determined by :

- providing a simulative mathematical model of observations of the physical quantity, the simula ^¬ tive mathematical model defining the physical quantity to be dependent on the quantity of interest in the target domain;

- generating, by means of the mathematical model, simulated observations of the physical quantity for a plurality of various modelled quantity of inter ^¬ est conditions in the inner zone of the target domain, one observation with the reference boundary distortion and another with a modelled effective boundary distortion for each type of modelled quantity of interest conditions, using various modelled 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 .

Ready determined simulated statistics of a deviation an effective boundary distortion causes in the simu ^¬ lated observations may be just stored in the appa ^¬ ratus, in a proper memory element. On the other hand, it is also possible that the apparatus comprises means for providing the simulative mathematical model of ob- servations as specified above, and means for generat ^¬ ing, as specified above, the simulated observations of the physical quantity, and means for finally determin ^¬ ing, as specified above, the simulated statistics of a deviation which an effective boundary distortion caus- es in the simulated observations.

The computing system may further comprise: - means for determining, on the basis of the adjusted approximate mathematical model, an estimate of a deviation caused by the possible effective bound ^¬ ary 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 modelled effective boundary dis- tortions used in generating the simulated observations of the physical quantity, an estimate of an effective boundary distortion present in the target domain.

Any of said means for performing those various opera- tions specified above may be combined so that more than one operation is carried out by the same means. It is even possible that all those operations are car ^¬ ried out by the same means, e.g. by single data pro ^¬ cessing apparatus.

Any means for performing any of the above operations may comprise one or more computer or other computing and/or data processing components, units, devices, or apparatuses. In addition to actual computing and/or data processing means, the means for performing said operations may naturally also comprise any appropriate data or signal communication and connecting means, as well as memory or storage means for storing generated and/or received data.

Computing and/or data processing means serving as means for performing one or more of the above opera ^¬ tions may comprise, for example, at least one memory and at least one processor coupled with the at least one memory, wherein the at least one memory may comprise program code instructions which, when executed by the at least one processor, cause the apparatus to perform the operation (s) at issue. In addition to, or instead of, a combination of a processor, a memory, and program code instructions executable by the pro ^¬ cessor, means for performing one or more operations may comprise some hardware logic components, elements, or units, such as those examples mentioned above in the context of the method aspect.

From another point of view, a computing system com- prising "means for" performing a particular operation means actually that the computing system is "configured to" perform said operation.

The quantity of interest may be an electrical quanti- ty, such as permittivity. Alternatively, it may be any other quantity of interest, indications of the proper ^¬ ties of which may be received by measurements of a physical quantity dependent on the quantity of inter ^¬ est.

As discussed above in the context of the method as ^¬ pect, the boundary surface may limit the target do ^¬ main. Alternatively, the boundary surface may lie within the interior of the target domain.

In the above, the apparatus is defined as comprising the computational means only. According to the appa ^¬ ratus aspect, also a complete tomographic investiga ^¬ tion system may also be implemented, wherein the appa- ratus comprises, in addition to the computing system, also a measurement system configured to carry out measurements of the physical quantity of interest de ^¬ pendent on the quantity of interest. Thus, in this ap ^¬ proach, the apparatus also comprises means for per- forming the measurements of the physical quantity. Such means, i.e. the measurement system, may be in ^¬ cluded in the means for receiving the measured values, or it may be a separate system configured to just per ^¬ form the measurements, which may then be received by said receiving means. The measurement system may be implemented according to the principles and means known in the field of tomo ^¬ graphic investigation methods, in particular tomographic imaging, such as electrical tomographic imag ^¬ ing, e.g. electrical impedance tomography or electri- cal capacitance tomography.

According to yet another aspect, a computer program may be provided comprising program code which, when executed by a processor, causes the processor to per- form the method according to any of the embodiments discussed above in the context of the method aspect.

BRIEF DESCRIPTION OF THE DRAWINGS

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

Figure 1 is a flow chart illustration of an investiga ^¬ tion method;

Figure 2 shows a schematic cross-sectional view of a measurement setup for performing capacitance or imped ^¬ ance measurements;

Figure 3 shows a schematic cross-sectional view of an ^¬ ther measurement setup for performing capacitance or impedance measurements;

Figure 4 shows a schematic view of a measurement probe for performing capacitance or impedance measurements; Figure 5 shows a schematic view of another measurement probe for performing capacitance or impedance measure ^¬ ments; and

Figure 6 shows an apparatus for determining a boundary layer and/or wear in process equipment. DETAILED DESCRIPTION OF THE EMBODIMENTS

The process illustrated in the flow chart of Figure 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 lay- er" of a first material.

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 per ^¬ mittivity. The target domain may lie e.g. within an industrial process equipment for storing and/or trans ^¬ porting various process materials. 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 distor ^¬ tion, a boundary distortion identical to the reference boundary distortion means a "zero" effective boundary distortion .

In the example of Figure 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 bounda ^¬ ry 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 simp ^¬ ly a "boundary distortion".

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 observa ^¬ tions of the physical quantity, the simulative mathe- matical model defining the physical quantity to be de ^¬ pendent on the quantity of interest in the target do ^¬ main; generating, by means of the simulative mathemat ^¬ ical model, simulated observations of the physical quantity for a plurality of various modelled quantity of interest conditions in the free inner zone of the target domain, one observation without any boundary distortion and another with a modelled boundary distortion for each type of modelled quantity of interest conditions, using various modelled boundary distor- tions; and finally determining, on the basis of the simulated observations, simulated statistics of a de ^¬ viation which a boundary distortion causes in the simulated observations. Next, an approximate mathematical model of observa ^¬ tions 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 mod ^¬ el is configured so that the deviation is determined to behave in accordance with, i.e. similarly to, the simulated statistics of a deviation.

Measured values of the physical quantity are then re ^¬ ceived, and an observation difference between the measured values and corresponding observations accord ^¬ ing to the approximate model is determined.

The mathematical model is then adjusted so that the observation difference is reduced. This may be itera- tively continued until the observation difference goes below a predetermined limit. As a result of the ad ^¬ justment, an adjusted mathematical model is provided. The model itself remains the same, "adjusted" just re ^¬ fers to the fact that the parameters of the model has been adjusted in comparison to the initial ones.

An estimate of the quantity of interest in the target domain may then be determined on the basis of the ad ^¬ justed 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 Figure 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 quanti ^¬ ty of interest distribution in the target domain. The method may also comprise determination of an esti ^¬ mate 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 pos ^¬ sible boundary distortion in the measured values. This estimate can further be used to determine, on the ba ^¬ sis of the simulated statistics of a deviation, and on the basis of the modelled boundary distortions used in generating the simulated observations, an estimate of the boundary distortion possibly present in the target domain . Thus, both an estimate of the quantity of interest in the free inner zone and an estimate of a boundary dis ^¬ tortion, comprising e.g. scale deposit and/or wear of the boundary surface, may be reliably determined. The order of the method steps is not limited to that shown in Figure 1. The order of the steps may deviate from that of Figure 1 whenever appropriate. For exam ^¬ ple, the step of receiving the measured values of the quantity of interest may be performed at any stage be- fore determining the observation difference.

As stated above, the process illustrated in Figure 1 and described is thus also suitable for situations where the predetermined reference boundary distortion is non-zero, i.e. represents some specific predeter ^¬ mined 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. 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 an ^¬ other, 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 ca- pacitance.

In tomographic image reconstruction, it is necessary to have a model that describes the relation between the quantity of interest conditions in the target do- main and measurement data, i.e. measured values of a physical quantity dependent on the quantity of inter ^¬ est 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 proce ^¬ dure it is often necessary to incorporate some quali ^¬ tative 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 es ^¬ timated is a spatially smooth function.

One example of tomographic image reconstruction for investigating a target domain was disclosed in WO 2014/118425 Al . 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. 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 : 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. 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. 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 erroneus 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.

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 .

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 y. Then the dependence of the physical quantity y on the target domain can be described with model y = f(e,Y +e (1) 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

ρ(ε,γ\γ) < ^χ ρ(γ\ε,γ)ρ(ε,γ) (2) where the likelihoood density ρ(ε,γ\γ) is defined by the observation model and p(e,y) 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. 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 _β ) . Then the determination of the MAP estimates is equivalent to the minimization problem

(^7)MAP = argmin{||L(y - (ε, ₇ ))|| ² +/(ε,/)} (3)

ε.Υ where the weight matrix L satisfies L ^T L = Γ _β ^_1 , and /(£,y) 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. To avoid the complications in the estimation of ε and y, 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{e, Y ₀ + v + e (4) where y ₀ is a fixed representation for the reference boundary distortion and v is the error due to the fixed value y ₀ . Eq. (4) is called the approximate mathematical model. The parameter y 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 y ₀ and it is naturally unknown since it dependes on the actual target. From eqs (1) and (4) it can be seen that the deviation is v = ί(ε,γ) -/(ε,/ο) (5). 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 ρ(ε, v\y) oc p(y|ε,ν)ρ(ε,ν) (6) 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. ν~Ν(τ] _ν ,Γ _ν , the MAP estimate can be obtained as

(^) _MAP = argmin{||L(y -/(ε,/ ₀ ) - ^)ll ² +Κε,ν)} (7)

ε,ν 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 a) where a is the parametric representation of the deviation. Then the MAP estimate is (ε, α) _ΜΑΡ = argmin{||L(y - /(ε, / ₀ ) -ν(α))\\ ² +]ε, )} (8) ε,ν

As an example of parametrization, consider a case where v is Gaussian, i.e. v~N(] _v ,T _v ). Then the deviation can be written as v = η _ν + Wa (9) where the columns of the matrix W are the eigenvectors of the covariance matrix Y _v . Furthermore, the number of parameters can be decreased by dividing the last term in eq. (9) into two parts as

Wa = Wa + W ₂ a ₂ = v' + v" (10) where the columns of W ₁ = ...,w _p are the eigenvectors corresponding to p appropriately chosen (typically largest) eigenvalues, and the rest of the eigenvectors are the columns of W ₂ = [w _p+1 , ... , w _m ] . The approximate mathematical model can now be written as (see eqs (4), (9) and (10)) γ = ί( _. ε,γ ₀ )+νν ₁ α ₁ + η _ν + ν" + β (11) where v" is considered as additional measurement noise. In this case the MAP estimate is

(^I)MAP = argmin {||Z(y- ( ⁶ , 7o) - - ^) || ² (12) Assuming that v" and e are independent, the weight matrix L satisfies L ^T L = (T _e + Γ _Ρ „) ^_1 , where

^v ⁼ ∑i^ _p+ i^ _t w _t w with λι being the eigenvalue corresponding to eigenvector w _t . Again, the regularizing constraint /(ε, α^) can be constructed on the basis of the statistics of a . The estimate £ _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 a 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 y, i.e. the density p(y,v). Similarly, in the case of parametrized deviation, the joint density ρ(γ,α) or ρ^γ,α^ 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 _est for v (can be e.g. the MAP estimate given above), what is to be solved is the density of y conditioned on v, and it is of the form ρ(γ\ν) = p(y,v)/p(v) (13) which defines the density for y that can be used to determine estimates and credibility intervals for the boundary distortion y. In the general case, given an estimate v _est for the error v, estimates and credibility intervals for y can be computed from density (13) . For instance, the most probable value for y is defined as

Ϋ = argmax p y\v _est ) . (14)

Y

Assuming a Gaussian joint distribution it can be shown that the most probable value for the parameters of the boundary layer region is f = η _γ + Γ _γν Γ _ν ¹ (ν _β5ΐ -η _ν ) . (16)

Alternatively, in the parametrized case the conditional density of γ is p(y\a) =p(y,a)/p(a), (17) from which, given an estimate a _est and assuming Gaussian joint density the most probable value for γ is of the form Ϋ = η _γ + Τ _γα Τ- ¹ ( _β5ί -η _<χ ). (19)

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 . How ^¬ ever, 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.

Figure 2 shows a schematic cross-sectional illustra ^¬ tion 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 meas- urements of one or more capacitance-dependent electri ^¬ cal quantities in a target domain 3 comprising the in ^¬ ner free volume 4 of the process pipe 1 as well as the pipe wall. Thus, in the example of Figure 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 elec- trodes could lie at least partly embedded in the pipe wall .

Figure 2 arrangement may be used as a part of a meas ^¬ urement system for performing e.g. measurements of ca- pacitance as a physical quantity dependent on permit ^¬ tivity representing a quantity of interest in the tar ^¬ get domain. Naturally, when measuring capacitance, the primary physical quantity to be measured may be e.g. voltage or current. Measurement results thereby pro- vided may be used as measured values to be received in the example methods discussed above.

The electrically insulating pipe 1 is surrounded by a cylindrical metal sheath 5, comprising flanges 6 ex- tending 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 elec ^¬ trodes from "seeing" each other directly via the exte ^¬ rior of the electrically insulating pipe. In the ab ^¬ sence 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 in ^¬ ner boundary of the metal sheath in order to take this effect into account in the calculations. 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 in ^¬ cluded in the flowing material on the pipe inner sur ^¬ face 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 in ^¬ ner surface 8 so that a slight recess 9 has been formed thereon. Also the recess changes the flow con ^¬ ditions within the pipe 1. 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 .

An alternative example of a measurement setup enabling determination of scale and wear in a process pipe is shown in Figure 3. As an essential difference in com ^¬ parison to Figure 2, there is an electrically conduc- tive process pipe 11 formed e.g. of some metal. In the point of view of measuring capacitance-dependent elec ^¬ trical 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 electri ^¬ cally conductive pipe inner surface 18 itself. Fur ^¬ ther, due to the electrically conductive material of the pipe 11, each electrode is electrically insulated from the pipe by means of thin electrically insulating 15 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 elec- trodes. Also in the situation of Figure 3, there is scale de ^¬ posit 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.

In the examples of Figure 2 and Figure 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.

Using the measurement setups as those of Figures 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 elec ^¬ trodes comprises just one single electrode) . The ca ^¬ pacitance measurement is repeated so that each of the electrodes is used as an excitation electrode. There ^¬ fore as a general rule, if there are N electrodes in the measurement system, N*(N-l)/2 independent capaci ^¬ tance values are obtained. Capacitances depend on the permittivity distribution in the target domain. Per- mittivity distribution of the target domain can then be estimated based on the set of the measured capaci ^¬ tance values. On the basis of the permittivity distri ^¬ bution, behaviour and/or some physical quantities of the underlying process can be investigated.

In Figures 2 and 3, eight and ten electrodes, respec ^¬ tively, have been installed on the pipes. However, these are just examples only, not limiting the ap ^¬ plicability of the present invention to any number of electrodes suitably configured to allow measuring ca ^¬ pacitances or other capacitance-dependent electrical quantities between the electrodes. Further, Figures 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 do- main by arranging electrodes in several rings or lay ^¬ ers along the axial direction of the process pipe.

In cases where scaling, or some other type of boundary layer, or wear in a process can be assumed to be uni- form (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 meth ^¬ od by taking advantage of symmetry. Figure 4 shows a rod 31 forming a support body, and an electrode con ^¬ figuration 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 Figure 2 and Figure 3. Potential fields generated by ring-like electrodes are cylinder-symmetric. Thus, a Finite Element Method (FEM) approximation used to mod ^¬ el the target domain can be formulated in two dimen ^¬ sions (axial and radial) only, which reduces the com- putational complexity remarkably.

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 Figure 5. The ex- emplary measurement probe 40 of Figure 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 ex- ample of Figure 5, there are electrodes 42 with dif ^¬ ferent sizes.

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 bound ^¬ ary surface of a plate-like support body could also be planar or have some other non-planar shape than the curved one shown in Figure 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 appli ^¬ cation at issue.

In Figure 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 applica- tions, e.g. in the field of food industry, such smooth configuration may be advantageous also from hygiene aspects .

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 Figures 2 to 5 is one example only. The basic principles of the methods discussed above may be implemented in determining any quantity of in ^¬ terest 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 quanti ^¬ ties of interest which are not electrical are speed of sound, observable via measurements of time of propaga ^¬ tion of acoustic signals transmitted to the target do ^¬ main, 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.

The support body, such as the pipes 1, 11 of figures 2 and 3 or the rod or the plate of Figures 4 and 5, re ^¬ spectively, 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 sup ^¬ port body is preferably formed of the same material as the actual process equipment. Figure 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 com ^¬ prising 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- ure 2 or 3. The measurement electronics unit and the measurement probe serve as a measurement system.

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 ap ^¬ paratus, an image 54 of the target domain is generated on the basis of the reconstructed permittivity distri ^¬ bution 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.

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 in ^¬ vention and its embodiments are thus not limited to the examples described above; instead they may freely vary within the scope of the claims.