 So let's continue with our lectures for repeating and orientation. We have been working on the background of probabilistic modeling yesterday. All the day with lectures, with the exercises, maximum likelihood method. We have been introducing what it is about. The video of information analysis. We know now how we can calculate the structure reliability. We know that we can do it for a component, but also for systems. We know the system reliability characteristics. And that goes to redundancy versus known redundancy. So it's either a serial system where we don't have redundancy, and if we have redundancy, then it's a parallel system. And we can add to this logic in system modeling. So series versus parallel. We can add a mechanical behavior. And then we are in the Denny's system modeling, which goes for redundant systems and accounts for the virtual or the ductile behavior in case of failure. So we have covered this part in the last two lectures. And we are already... If you would zoom in, I can do it later again. Here we have the exposure. This here is our constituent damage states, our component damage states, damage or failure states. So this is covered with the lecture one of today. And how to calculate the probabilities of failure with form and with Monte Carlo simulation. Basically, and going a step further, so that we can also model the consequences of system failure or damage events. We have the system modeling down here. So that's for orientation. And that's what we have covered in the last two lectures. Now we look at how we can collect information and how we can model them so that they can be used in our models. I think now this is also the point of time where I think we... Now we had good ground and we have been using, but of course especially rearranging lecture material which was already there, which we already teach. Now we are touching aspects which we have not thought. Thought. We have been thinking about it, but we have not thought it. Yeah. And this will be basically starting now and will continue for most parts until tomorrow afternoon or evening. Or night. No, we have dinners. Okay, so what will we do? We will look at the NDET measurement performance modeling. That's basically inspections. So we have inspection system and how I quantify the performance. We will have a look then to damage detection system performance. So here the situation is you don't have an inspection system and you don't have a device like this one. What's that by the way? Is that microphones? Yeah. That's microphones. So now you have a measurement system attached to a structure. So that's something different. How do we model this? And we also look at the uncertainties which are associated to measurements. And here I'm providing some depth knowledge on the measurement uncertainties. So when we are able to model the holistic performance of measurements, inspections, damage detection information and also continuous measurements, SHM outcomes, I call it SHM outcomes here. When we are able to model this, we can update the structure reliability. So this will be the focus of the second part of the lecture. So we have introduced this here and now. What sources of uncertainties do we have when we conduct a measurement? So that means it's the very basic approach which an inspection is a measurement. The damage detection system provides measurement information and also continuous monitoring with the sender is also a measurement. So what uncertainties are associated to that it is the measurement technology and the process. So that means if you measure at a structure, for instance the strains with a strain gauge, then your measuring device is measuring a voltage difference and it is amplified. So behind this resistance change and the resistance change of the strain gauge is actually providing information about the physical strain on the structure. So the principle basically was developed by Lord Kelvin in the 18th century where they just invented something like or they knew about the voltage and the current and they wanted to make it measurable by physical quantities they knew and it was a strain for instance. So then this principle was discovered and strain gauges were one of the first sensors which were developed and which were applied. I think the first applications even was in aviation industry and then it was also employed for structures, for structure measurements and this is also a very important aspect. Structure, health monitoring has its roots in structure engineering but also in aviation. But let's come back to the measurement technology. So it is the four strains, strain gauges, the technology is strain gauges which we apply to a structure and the process is then voltage measurement amplification and then with the k-factor we can conclude the strains but our models are usually using the stresses as the input so we need to convert it to the stresses. So this is also an important step and there is, all the way there is all, if you really do it there is quite some pitfalls so you can easily measure with one strain gauge one-dimensional stress state but there is only under certain conditions one-dimensional stress state or strain state there is this personal ratio right in mechanics. So this is something to be aware of for designing SHM systems and to interpret the measurements. Okay, measurement, technology and process. So then we really have somehow stresses and also here we again have, if you go with this scheme here we have a desktop model of our measurement process and in the real world there will be model uncertainties also associated to these models. And then of course it's human errors. So again for the example of the strain gauge it's positioning of the strain gauge. So in practice who has glued strain gauges on the structure? Karl, super. You also? Okay, so it's a very tiny piece of sensor and it is to be glued on metal so that it becomes one that is completely bound or bonded. And as it is so small you can easily misalign it and this will have influence on again the strains you are measuring and interpretation of the data so you make errors here. And then of course it's the data analysis and all kinds of influences. It can be, I once had a case where we installed a monitoring system and then the SHM technicians they claimed we have temperature compensated it. So there's also an issue about temperature compensation which can be done in different ways but they claimed it is temperature compensated. But when I looked at the data I saw a 24 hours curve like this so obviously there was temperature influence. So what did happen? So it was temperature compensated until shortly before the sensor and then there was a box and there was a cabling from the box to the sensor and this piece just this size and the other cables were 30, 46 meters long they were temperature compensated but this small piece not so I had a full dependency on the temperature and then again strain gauge we should be aware of, so a strain gauge sensor can be like your thumb nail this size. So we glued on a structure but the structure is really big so the second challenge or often the challenge is where is the sensor? Yeah, we went in the structure and we glued it over there and there's an approximate somehow sketch handmade, yeah it's there. But if you measure such a, so locally you have all effects, all kind of effects so forget about beam theory it does not work it is very local stress measurements. So if you have a normal beam and you, so that's an IB it's standardized, if you do a design you have your beam theory, mechanical theory you can easily do it and then you measure, you have a strain gauge somewhere here and it measures the longitude in a direction, maybe you have a strain gauge also here and they will deliver different strains and stresses. Why is that? Yes, super. So you need a shell model to reproduce the exact mechanical behavior so it's the effective width, so it's basically the stiffness of the back which influences the strains here, here will be lower strains, its stresses will be higher. Also bending for example. Local bending, yeah. Okay, so basically if you install a measurement system you need to know exactly, very exactly with the precision of a tenth of a millimeter where the strain gauge is. Okay, and then there's another challenge with, at least to my experience with measurement technicians. What is the precision of the sensor? We don't know what, it's very precise. Okay, and this is what we are going to address. Usually it's to have an estimate, it's rather easy to do a full assessment, maybe not, but I have to have an idea. It's rather easy. Okay, so these are the uncertainties and strain gauges in some detail already. Sebastian, can I have another answer? Yes, please. I think that when you put a sensor, in this case you are talking about a metallic structure and you glue a strain gauge. The stress of conflict which is, for example, you invent strain gauges on the causes. There is also that by the fact that you are trying to measure a specific part of the structure you know that you need to prepare the surface to go to the sensor so you have this turbine, the stress along this area. So you have this turbine, the real stress profile of the structure. It's like, it's from physics, when you try to measure something you are preserving the field of measurement. So you are not measuring, even if everything was really perfect. So you have the rebar and then you prepare the surface and your rebar is now different. And the sensor is here, then you have a different area, right? Yeah, that's another answer. Yeah, yeah. Just for discussion. Sure, but it goes to, we need excellent SHM engineering. This is what we need, because they can do it, of course, but they need to provide information on what the thickness is. And then your answer is gone. So yeah, I mean the most important point is SHM engineering. This is what we are relying on and we need to have an eye on that it is done properly. And this will give the boundaries for our probabilistic models. So ND, E, NDT, non-destructive evaluation or non-destructive testing, performance modeling. So this is quite a field which is very known since a few decades. Again, the origins of and still the largest applications are for civil structures, for monitoring inspections but also for aviation structures. And they are basically these models have been developed in these communities over the last decades. And they are especially in aviation, they are highly standardized. Probably they are over standardized so that some people cannot forget the meaning of what is behind. So the basic ingredients here are that we have a probability of indication given a damage size. So that means if you have such a device and you have such a tube connection, the device has the size of the microphones and you put it. Basically you can imagine that you are kind of scanning the belt with this device. So that's an inspection technology called ACFM, alternating current field measurement. It could be something else, it could be also an current. So this is how to imagine where, how an inspection is done. Who has done an inspection with a measurement device? Or any? I have also done it. Together? Yes, the right. Yes, we did ACFM measurements. Yes, okay. So and if there was a defect the machine or the system will make a beep. So that's an indication. So then there is also a probability of false alarm. That means there was no defect but our system makes a beep. And we already see here as you have the probability of indication for no damage or for a damage. Then we knew we know the complementary events. So we know the probability of no indication given a damage and the probability of no indication given no damage. So it's conditioned for abilities and the complementary events are also always associated to the events which are here. At the beginning. What are these complementary events? Yeah, okay. Could you repeat the question? Yeah, an example for the complementary events. So that's basically the event of no indication that you have no damage. So that's a proper functioning system. So if you make with the system the test and you know that there's no damage and you give it to someone who doesn't know whether there's damage or not and he puts the system to that weight and he gets no indication and the test is repeated a few times and then you have the probability of no indication given there was no damage. And the other way around, so you have a damaged specimen and then someone comes with the inspection technology and it doesn't make a beep. Then again, repeated a few times, this will give you the probability of no indication given there's a damage. But the clue here is with the complementary events you just need to do either this or this and you can just calculate by one minus the probabilities of the other event or the complementary event. So I will come back to thank you for your questions. I will also come back later to this. What I just described is called the round robin test. Okay, and of course we have simple environmental conditions for an inspection here but inspections, especially in the offshore oil and gas industry they need to be done underwater and then you see that this guy probably cannot see this much. Okay, so this is already the very basic NDE and NDTV measurement performance modeling. This is all we need. Now we are finding out how we can calculate this. Yes, different means. One mean I just described. Okay, but one step before that is that our damage size is basically unknown. The damage size in a structure when we model the damage process will be a random variable. So we basically need to know the probability of indication for all possible damage sizes and that's a probability of indication curve. It looks like this. So for low damage size you have a low probability of indication of 0.05 for instance here and then for a very high damage size you may reach one. But even if the system makes reliability or meaning with the probability of almost one beat there can be some operational errors and the operational errors are here. So that's basically human errors. So it should only be for very large damage sizes. So where's the probability of force allowed? Here in this diagram. Where is it? At this point. Yeah, you see there's a piece of curve here. So that's the probability of force allowed. Okay, but now we are coming to the point where we should think of how this curve can be determined. It can be, this is what we just talked about, determined by ground-robin tests. So it's interlaboratory tests. It's even standardized in some guidelines or standards. You need to have 10 laboratories across Europe doing the same test and then they are after the same result and then there is a statistical modeling and then you have accounted for quite a few uncertainties. Basically of all the types we have been discussing. So when we go back to our uncertainties with the ground-robin test, so it is 10 laboratories, it is 10 different inspector teams or it can also be different inspector teams and then there's also research about how good the inspector teams can be and how well they were educated. So there can be edits at all steps, some complexity. So it's 10 inspector teams or at different universities having the same measurement device and trying to find for here these damage sizes, they will get specimens and they take the inspection technology and the specimen and they document what they have found. But they don't know, of course, which specimen is damaged in which way. So this procedure covers the uncertainties associated to the measurement technology and the measurement process and also the human errors and then we have this curve. But there's also other means of establishing this curve and this is by simulations. So we could model the measurement process or we could also document the signals associated to each damaged size and then we are coming with these expressions. And here we see a damaged state. So there's one realization of A and that's my signal by my inspection technology and there's a distribution of this signal. And here I have a threshold. This threshold is basically to distinguish the damage from the undamaged state. So I need to define a threshold. And then I can integrate this signal in the damaged state to find probability of indication that is here, the area under this curve and the area under this piece of curve here is the probability of no indication given that there was a switch. What is your signal? That's basically any measurement output. Yes, but the threshold is a limit on the stream or is a limit on the width of the crack? It's a technology internal quantity. It's not a threshold associated to the structure. It's just the inspection technology. We are not at the structure. So just to distinguish, you have a signal and this signal has a distribution. So if you do, we have for the ACFM inspections, there is basically a certain shape of a curve and if you recognize a certain shape of a curve, it's the measurement of the magnetic field. So there's a magnetic field induced and then it's reflected back and if there was a crack in between then it will be different. So there's a signal associated to the state to A and this has a distribution. And now you need to define the threshold in your associated to your measurement technology, inspection technology. So just to talk my language, you define a damage feature, something that is related to damage and that's measured. So you have already defined something that's related to damage, which is your signal. Yes. It's not just a measurement, something related to the technology. No, no, no. Only technology. We are doing only technology, only measurement technology. It's in the measurement. Yes, but how can you relate to the damage of the structure? Yeah, well, any, so if you are after a damage in the structure, you can take any technology which somehow provides a changing value which is dependent on the damage size. So it can be any technology. Maybe I will understand that. But, yeah, okay, since you are after, okay, we are here not in a situation that we are on the structure but we need to have the, this curve we need to have before we go to the structure. But my doubt is just that you need a model between the damage on the structure and what you measure. Otherwise, how can you relate to the damage? This will be at the end of this lecture. Okay, so this is maybe I will understand. Okay. First question. You have got this signal from your measurement device, do you hear it? The signal which comes from your measurement device. Yes, the signal comes from the measurement device directly. And each time it will be different? For each damage size it will be different. So, sorry, just to understand. To obtain that curve, for example, you have a damage, you know the size of the damage. You use the instrument and then you make the distribution of the signal coming back from the instrument. It can be voltage or everything. Yes. Then in the instrument you know that there is a threshold above which there is detection or not. Yes. And you put the threshold there and it account for the probability of overcoming or not. Yes. Yes, exactly. Super. The point is that in the undamaged state or in the reference state here, you also have a signal distribution. And this is basically where you calculate from the probability of no indication if there was no damage and the probability of indication if there was no damage. And so far, and especially in the aviation industry, they are after reducing this probability. So this should be very, very small. Because if they take, if they do an inspection and they get an indication, a beep, all is a damage. And then they start to repair or to replace the component, but there was no damage. And this is really obstructing operation and, yes, and this can be really costly. So usually these MBE and MET technologies, they are optimized for giving a low probability of indication and there was no damage because this is really, can be really costly. But they are wrong. We will know better. Also information which in precise can have value. They are overlooking, they are over standardizing. They are wrong. Okay. You mean there is a lot of uncertainty about them? No, no, it's standardized and they have to. This has to be lower than 5 percent with 95 percent confidence. So this is what they are after and they are designing their technologies just to arrive at that reliability. That makes a certain sense, but it's not an overall conclusion because also, so you only reliability wise optimize the threshold. The background is threshold optimization, but you should optimize it for your decision scenario for exactly what you are after and it's not something in between. Good. We made six slides so far. Okay. What comes next? Okay. Here this is the threshold and this is what I've been talking about. The threshold influences the probability of indication curve. Here we have only the indication if there was a damage size, so we don't have anything for zero here. Here we have a high threshold and here we have a low threshold and this is how the probability of indication curve changes. So it's below the threshold that we lifted and then there's also a higher inclination of the curve. So the threshold influences the probability of indication. It's an arbitrary measure associated just to the measurement technology. It's standardized, but maybe for some scenarios it's fine, but for others not. And what do we have here? Okay, that's another concept. You can basically get out with the entity and measurement performance modeling only out of the probability of indication given with damage and probability of indication given with no damage. So this is the four probabilities I had on the first slide. And so that's called the receiver operating characteristic. And here one curve is only associated to one damage size. And here we have a low damage size and if we have a high damage size, then we see that the probability of indication given there was no damage. So this is very important for the operation, for the integrity management of airplanes or structures. So that there's a low probability of indication given that there was no damage. And if you have high damages, it looks almost like a rectangle. So this is what they like to see. Of course, they don't like to see these lines where there is quite a significant probability of indication of termination. Okay, some examples. But I have the slides anyway, okay. Good, good. So we see here the crack length in millimeters. And this is for visual inspection of steel structures and ships. And then you have easy, moderate and hard environment. And yet this has been assessed by experiments. So for instance with the probability of 80%, you will be able to recognize a crack size of around a crack length of 150 millimeter. So that's something like this. So if you take, for instance, an inspection technology like either netting current field measurement, ACFM, then you have crack depth here ranging from or until 5 millimeter. And 80%, you can detect a damage size of 3 millimeters. But it is the crack depth. And here we have the crack length. So how is this interrelated for steel? What is the relation between the crack depth and the crack length? So this is usually like a crack in steel it looks like. It has an elliptical shape. You can reproduce this by testing and then cutting through your specimen. And then you can measure. And so this is usually L divided by 2 or 3. How is it going to be? So this means this would be a crack length of around 10 millimeters. So with visual inspection it's 150 millimeter, 80%. And with this technology you can get down to 10 millimeters you can find. And again, talking about the uncertainties associated to the measurements. This does not mean that this is bad. And this does not mean it is good because it's just more reliable. This is only if you narrow your perspective to reliability. But if you are after the value of information for instance. This can have a very high value of information because you just need the costs of the human to do the inspection. Here you need the cost of the human to do the inspection, the measurement technology and the data processing. And then of course it's the issue. Are there small cracks? Do we expect small cracks or do we expect large cracks? Where do we get that information from? Any idea? Do we find it by a monitor? No, that's something strange right? We want to measure but we need to know before. We measure what the crack sizes are. How do we get it? How can we calculate crack sizes? Yeah, some people, very experienced people, they can tell. With a photography, if you can take a photography of... Yeah, but you do a measurement. You don't want to do a measurement. We want to know before. Is it a numerical model? Yes. Very detailed one. Yeah, I'm great. Is that for analysis or something? Crack size. What mechanism is that? Thirteen. With a low history. Pardon? Yeah. Yeah. You do fatigue damage modeling. You do an SN approach but then you don't get the crack sizes. So you need a fracture mechanics model. Then you can do crack size distribution for each year in the service line. We need that information. And this is also an important point here you need to remember but I have it also in the... Ah, here it is. This was an icon or the icon project in the late 90s where in the offshore oil and gas industry all the available technologies were evaluated with wrong drawbacks. So this is the information source for getting the probability of detection curves. Probability of indication curves. More clever people than me to answer how this looks in concrete. Is there someone who knows how the cracks are looking in concrete? So we have a steel crack and there's a ratio between the depth and the length. You mean the crack in concrete? Yes. There are always cracks in concrete. Yeah. You know there's this stage one and stage two and normally you have crack concrete. Yeah. So cracking the concrete is not a problem. But for some applications we consider cracking steel and reinforced concrete. Yeah. And then it's going to have some better. Yeah. Yeah. Exactly. Okay. So now we could calculate a probability of indication curve. We assume that the signal is distributed with the mean and the standard deviation that's normally distributed and the mean and standard deviation they are dependent on the crack size and the crack size ranges from 0 to 10 meter and we also have a noise distribution. So basically we have defined these distributions here for the damage state independently of 8 and then we also have the reference state. But there is a relation between the size of the average and the signal. Yes. Yes. So this is the model. This is the, yes. This is this. This is the model of the signal. This is what we did. This is the model between the signal and the damage. Yes. Yeah. Yes. We need this. Yes. But this is not a commercial. Just want to show you some code. So this is our task one. Yes. We need a new projector. Yes. I forget that. Yes, well, what should I do? I just need to make it bigger. So this is our Matlab code and it has a lot of grammar errors, right? So because I put it in Word. So can someone come up here and explain this Matlab code? Well, we can do it together. So I need to volunteer. If there was no volunteer, I have to choose someone. Who's familiar with the Matlab? Okay, Karl. Okay. All variables. Clear the screen. This looks to be the number of samples. One million. Yeah. We're defining a vector that goes from zero in increments of 0.5 to 10. So what could that be from our model? You have the slides. Yes. Good. Yep. So now we're generating normal random variables with, I believe, a mean of one. Standard deviation 0.5. And we're generating one million samples. Yeah. What is one? One. The mean. No, the second one. The vector. This is the size of the vector. Yes. Also make two lines of... So it's a... It's a matrix, but... But it's a vector. We make it a vector by having more network. Set threshold 0.5. Yeah, we fix the threshold. And then we're looking for... We're counting the number of elements of this vector that are greater than this threshold. So this is a vector and there's a one every time the element is greater than the threshold. Zero every other time. Some will count up to one. So we have the number of elements that are exceeding the threshold and divide by the number of samples. So the fraction of samples that are exceeding the threshold and then this is the fraction of samples that are less than the threshold. And... Well, you would recognize from the name, but what probabilities are this? Indication. And which state? Reference. Reference states, no damage. And then for the damaged state, we're looping through i, which is going along this vector. So we're going... I guess this was for... For i equals one, we have asyn zero. i equals two asyn 0.5. Our signal... We're now generating another set of random variables where the mean is... 0.7. Yeah, 0.7 plus... plus 0.1 times the loop. So it's 0.7 plus, I guess, 0.2 for the first loop. So 0.9 for the first loop. And then the variance or the standard deviation also decreases. So this is, I guess, the function you showed where it's a function of the damage, which is kind of indicated by i, same number of samples. But what is that? Let's have a few words to that one. Why do I start here with two? Because you have a little jump for the probability. Because... Yes, here we are after a damage but asyn starts with zero. So that's no damage. That's why there's two here. And then you look at this line and at the task definition I gave you and to tell me whether there's a mistake or not. It shouldn't be the damage size. It shouldn't be asyn or i. Very good. Very good, yeah. Good. So we will change this. We need to write asyn. All the way, am I? Yeah, all right. Okay. And then, now the same as above, we count how many samples are above the threshold. So probability and then by the numbers are probability indication and the cap number of samples below the threshold probability of no indication and then it's indexed by the i. So it's four specific sides of damage. And then we're filling in probability indication. The first entry corresponds to the first entry of this which is no damage. So we fill in the reference state probability of indication for all the other states. Now we are adding another line or we're making a matrix so that we have the matrix of the probability of indication where we have the probability of indication in one line and in the other line it's the damage side because that's what we are after. Thank you, Carl. This is threshold. How can we get that? Yeah, it's an... It's an arbitrary definition. You can fix it by thinking of the probability of false alarm should be low. But in the end, it should be a decision parameter associated to the measurement system, the inspection system and the employment scenario. And what is really important there? Will it work? No, no, no, I copied it now in Tomato. And we are here. And now this is the most exciting event of all this, right? Will it work? It looks like... Oh, but... Then you set a round number. In the pdf you set minus 0.1 but the map not called you said minus 0.1. Oh, super. Thanks. The standard deviation? Yeah, standard deviation. So it should be here and what should... 0.1. 0.5 minus 0.1. Okay. So now it works. But we need to reformulate the task. Okay, so this is our first average size here and this is the probability of false alarm. That's why it just connects so we would need a few more seconds here. And then goes up until 1. So this is how we can get out of singles the probability of indication. What is the minimum point? Where you put the threshold? The threshold is not visible here. No, yes, but there is a minimum on the record. Where you put the threshold? That one is the... No. But it's just the first data point. So we generated a crack size of I think 0.5 or 1. And for that crack size this is the probability of indication. But this... Yes, but when you are at the threshold the probability of the indication should be zero because you will... I mean, if you are below the threshold it will not indicate anything. It should not indicate anything. But I don't know... What you are getting is for each measurement so if you have a defined crack size and you put your measurement technology on it then you get a distribution of... you don't get just one signal. It will vary. Or you may say, okay, the measurement technology is very good and I put it on and then there will be just one signal which doesn't have a distribution. But if you go in other environments and other operators then they will have... they will know a slightly different signal and that's how you get your distribution. But... No, but I was just... You said before you put your instrument and then if there is a damage it says being... but it could be that it's stronger or not. Okay? If you put the threshold at 1.5 it means I put... you say your instrument if the value is below 1.5 don't say being. It should be that. It doesn't affect anything if it's below the threshold. It should work like that. So when it's below 1.5 you don't get... you always get zero. You should. There is a probability of false alarm. So the minimum that if I understood should be corresponding to the threshold then should be zero. We need to distinguish the threshold and the damage size. There may be a way of recalculating the threshold and I understand that it has been often done to define the threshold for crack size. Or if you think of a section of it then you can derive a threshold and it could be... this threshold could be derived like this but this is a different situation. Here we are not defining the threshold with the damage size. We are simply... we just want to know how the inspection system performs in dependency of the damage size. This is what we are after. We are not after what could be a critical damage size. But there will be a way of associating the threshold to damage size. Clearly. We can work it out maybe after the... Maybe with you. Okay. So I will send you an update of this task. Okay, now let's start with the main part of the lecture. Can you have a break? Yeah, let's have a coffee break. Yes. So we... we have been finishing with this one and now we... I will make the next try to increase the number of the slides per time unit, basically. But of course if there are questions to something which we have been going through, please. Yes, one question is that this noise distribution that you just wrote is just like an error you introduced to the model. The noise that you wrote there. The noise? Yeah. The signal basically in the undamaged state. The signal in the undamaged state? Yeah. So that's something that's... This one. We have one model that depends on the damage size of the signal and then there's another model for the reference state. It's also the signal but just for one state, the undamaged state. So it's different from the error of the environment or natural noise? Yeah. Yeah. So in this reference state undamaged there should be... all the environmental influences should be covered. So that you have the proper signal distribution in the undamaged state. Actually, I also wanted to come back to this slide. Thank you. So when we think about promising modeling and we see here that this is obviously a probability density function. How can I calculate, for instance, this probability of indication? Just by taking basis in the distribution functions. Yes, okay, we can integrate but what would that mean if you just work with distribution functions? So it's just 95% you said, I think we should consider... Yes, okay, we can work with 95 what? You said 95% is on the other side and 5% is the other side. Yes, and that goes to which distribution? Over the least. That's probability density function but do we get the 95 on there? Okay, my integration but if you integrate the probability density that's what I'm after. What do we get? Yes, yeah. So of course, if you have the distributions we can work with some other color simulation for integration or simply with the cumulative distribution functions. This is what we need a few minutes later. So I've made a few adjustments here so this is 20 and there I entered an easier to do also in your task. Okay, there's a few literature sources we already talked about this. There is the probability of indication curves to be found. You don't have to do any integration there they are just given associated to different inspector technologies and then there are some statistical best practices of how to determine the detection curves of the European project and there's a lot of material guidelines to be found but we just most important is to have this basic modeling I've introduced this is quite generic. So, okay, now we are not in a situation where we just have a very small area which we are observing with our inspection technology but we have put sensors to do this structure and we are obtaining responses from this bridge and these responses they can be analyzed in various forms there are various methods over the last decades which were developed can do experimental model analysis for instance or you can do output only analysis schemes you can look at you can do a model analysis and also look at the higher frequencies what they are doing and are they dependent on the damage damage development this is what VCE is doing so what should we do now? we don't have we don't have our component and our inspector technology but we have the bridge and we have a model analysis and the model analysis that means the complete structure is what the behavior of the complete structure is analyzed I think this was also the intention for the development of these damage detection technologies that you with a few sensors or sensor network you can basically have a parameter which depends on the complete structure system may have a damage indicator so obviously this information is on system level and so where is the difference to the previous slides? find the difference yes now we have a damage indicator here not a single we may have a damage indicator in the reference state so the undamaged state and the damaged state this was quite easy but how what happens to our probability of indication but on the next slide and I remember that I should increase the number of slides before time unit so now we have probability of indication surface it could look like this so if we have a system of two components the simplest system we can think of it's one component is not a system but two components is a system so we have a probability of damage curve and we see here that could be component one and that could be component two and here are the single probability of indication curves associated to the damage of one component and no damage of the other component here probability of indication curve component two no damage of component one but if both components are damaged then the probability of indication will be higher and the surface here may look like this so what we need is to establish the probability of indication in the dependency of A1 and A2 associated to the two components so that's basically all the points in here but we need to think a little what should be the probability of indication at this location that's obvious both so that's the probability of loss alarm if both components are undamaged so that's the undamaged system state but what may not be so obvious is that here along the line we have one component in the reference state and the other component is damaged so this is the probability of indication here and so we also need to define the appropriate or we need to do the integration in these states basically along the line here and also along the line here with a very good basis of the NDE and NDT performance modeling we can basically also the damage detection system performance modeling so we just redefine the signal to a damage indicator and we should be aware of that the response is on the structural system level and also the indicator is on the structural system level so we have information for the structural system and one can imagine if we talk about the robin test for this bridge this will be a little complicated and costly and costly but in the iris project VCE organized and coordinated there was at least five inspector teams at one bridge and not this one but another one which was deconstructed so then it was found out that what damage sizes basically and damage scenarios could be identified so this was cutting of the of the pre-stress of the pre-stress tenants only a few tenants which could be identified and then it was a settlement of appeal and made a hole in them it was a controlled damaged damage that's the S-101 bridge and I think we can access the dataset yes we know that on the website I guess anyway they are available they are available so we have an idea of how with this test we have an idea of how how precise the damage detection can be we don't have the full curve but we have an indication but of course such systems can be so meaning the probability of indication can be for such system performed by simulation and that's actually a topic in surely an area of structures but also an aviation so the large aviation companies Airbus for instance they are after establishing the probability of indication by simulation so it can be handled by simulation but surely here we have a parallel system what kind of structure system could that be parallel yes and more precisely we have a parallel system this is right and can we say more specifically which type of parallel system could it be so we would know what happens in the ultimate limit state here with the cables reader pardon reader it depends on the material properties I'm not sure it's still they can be ductile unless they are very hard or treated in a way that they become rigid and actually I need to go on the slides good so we have a system and we have many components so but then if you have many components and you would like to establish a curve here so you need for each component at least 10 but there are 100 damage states but you have a system of many components so even simulation gets computationally demanding so another approach could be the direct calculation of the probability of indication there's a paper I think it's later of the EWSHM 2016 where for a specific damage detection algorithm we know the properties of the damage indicator and can directly calculate the probability of indication so we just need a sample here and here and we can directly calculate it with an analytical function okay maybe this this is time for something in general how do we describe in a value of information analysis our measurement what do we need to describe in a value of information analysis our measurement how is the measurement information characterised there are 3 main characteristics did you repeat the question what are the main characteristics of measurement information for a value of information analysis accuracy accuracy is what? accuracy accuracy it could be the precision or the uncertainty it's all the same what else? tree very good what else? they are not associated they are in the decision scenario it's the more obvious water it's more obvious it's the type of the information the damage detection information on system whether it refers to component so it's the type the type basically tells us where to put it in the probabilistic models the type of the type of the information the same the type of information no that could be the technology but the information is how is it or the type refers to the situation where how is it related to our structural performance of any given example it is the inspection information which refers to the component or it could be the damage detection information associated to the complete bridge to the structure on system level so it could be about the frequency of the crack for example you mean the type of the information it could also be a strain so this goes so if we have a strain measurement we can input it in our fatigue damage model because there we have stress ranges we still need to come from the strains to the stress ranges so this is the type then again damage detection information any damage indicator on system level or on component level so is it a problem that you measure or observe on the structure yes but in relation with the characteristics on how to associate it to our probabilistic performance models of the structure this will come later try to be more specific in terms of time what you mean what's time if you would like to add a few short sentences related with the structure maybe the rubber where did I leave the pen well as a curious one because we can really understand time is quite a bit what is the time maybe the relevance of the information the relevance is the conclusion then the first question it's the relation to the structure performance what is the type specifically the first question okay what are the main characteristics of measurement information in the context of decision analysis if you wanted to do a decision analysis with measurement information we would need to know about the accuracy, precision or uncertainty we would know about the costs for SHM investment SHM installation SHM operation and SHM maintenance and replacement and we should know how what type of information do we have and how does it relate to our structure so of course we can do also a probability of indication curve for the two component system we can just take places in the signal running we have been doing anyway and then we define for the signal now normal distributions and this is an example of a multivariate normal distribution b-variate normal distribution that's why there are two and here we have the mean values and here we have the covariance matrix what is the correlation we have no correlation why did they assume a normal model because this morning all the materials that were also always different only for the permanent program yes in the end it depends on the measurement process what comes out of one signal distribution but now there is a good approximation and it's good to handle these distributions so let's go to Matlab so our program looks very similar we have the damages of one component the other component here we have the parameters in the reference state and now we are operating here with the CDF this was the point I made just after the restart of the lecture to determine the probabilities so it's very very similar to the example we just we just had it's approximately readable so now we this is basically our probability of the probability of indication here it's the probability of force alarm that's quite high that's 0.4 and then we see that the surface or that the numbers are increasing here and they are close to one here in this area where we have significant damage on one component and the other component and here it's always the same number so that's due to the but I think it seems slightly increasing this is due to the definition of the state here where one component is damaged and the other not yes I will distribute the programs for these two tasks and you are welcome to play around with that and we have been discussing about changing some parameters please do and play around and see what will be happening try to understand why it's happening okay so this slide we already know it contains our uncertainties they are associated to the measurement technology and process then if you describe probabilistically this process and the technology then there will also be more uncertainties and we have our human errors we also know that when we do ground roman tests we can account basically for these type of uncertainties and the and then we should also know that in this field this is a domain basically of psychology and there are different models so the performance of the humans will be dependent on how they are trained and how they feel basically and how difficult the task is of the inspection so that's what we have seen in the for the visual inspection so if there is a very hard accessible environment then the probability of detection or indication will be lower so the two data analysis are different I mean this is there is a data analysis in the measurement technology and another one in human errors yeah so the data analysis they can also be done improperly you mean that the analysis may be that you consider a little bit the gladness of the signal and things like that yeah and the second is just someone that doesn't know how to analyze it yeah and now we have a closer look to the measurement technology and the process or that could be modeled so here we have our sensors and then we usually have some electronic equipment to amplify the signals and to yeah that we have an output device that could be the computer or previous times of course something else where we just see the signal for instance so and what we can do here is we can describe with the process equation the measurement process and that takes basis in basically a probabilistic model which are valid for all types of measurement systems so we could say it is a measurement system with a strain gauge and an amplifier of a certain type and then a data analysis procedure so for that situation we can derive a process equation and but we also have another source of getting to know the precision or the measurement uncertainty and its observations so you simply take the measurement system to the laboratory where you can control the conditions and then you measure and see what how the distribution of the signals look like so this is another information here this may be more specific so this is valid for all types of measurement systems meaning for the specific application you have it is valid for the type of the sensor the type of the cable the type of the amplifier and the data analysis and here it is for the specific sensor the specific amplifier the specific environment you will get the distribution of the signal and then you can update the patient updating so this is an example here the uncertainties are higher the observations provide less uncertainties so lower standard deviation and the posterior will simply be that the information is more sharp so the density here is higher and there is lower standard deviation so as you can see the second one there is also the operator in the other terms what will make these carrying out measures? a good point you are only here and when we are here we have very controlled conditions and there could be some you are right there could be some uncertainties due to the how the observations have been taken but they should be very low because there should be a very controlled environment for that type so this can be done for strain gauges in a way that we can formulate the measurements equation as the model uncertainty for the amplifier strain and this is the apparent strain and the apparent strain is caused by the temperature effects in the strain gauge so this could be a process equation and then the so that is the amplifier strain it can be described like this basically you may have seen this anyone who has done a strain gauge measurements should recognize here this Ua and Ub and the K factor and then okay that is the fall that is going to the circuit so it can be done like this there are a few more factors that is the amplification and we have the gauge factor and the gauge factor variation so that is the K factor and the gauge factor variation is this one I think and then there is a model uncertainty of the gauge factor variation the transverse sensitivity so that goes to the Poisson ratio so we are measuring in one direction but the K factor is determined with a certain type of metal and a certain type of specimen and then there is a difference between this K factor determination and your real structure so that is basically the transverse sensitivity and also the specimen Poisson ratio goes in and the Poisson ratio of the gauge calibration and there is an amplifier zero deviation that is this one so this is how process equation for a straining measurement can look like and what are the influencing factors and then there is also a temperature coefficient for the gauge factor we have here the process equation for the apparent strain that is the apparent strain process equation and there is many influencing factors but there is only a few ones which are really irrelevant and that is the gauge factor and gauge factor variation for the strain gauge and the amplifier zero deviation so that is basically this line the zero deviation is relevant for very low strains for very high strains it is the gauge factor and gauge factor variation that is relevant and basically the interesting thing here is that this model has been built with the packaging information on the strain gauge and on the specifications of the amplifier and the knowledge of this one so here is the production process of strain gauges is described and there is the meaning of the parameters which are on the package like the k-factor deviation so you have the probabilistic models in here and then it is also some books where we can take information out on the measurement process and what is that tape basis in it doesn't come from civil engineering but it comes from where you really need to know what is your measurement precision it is something else than civil engineering so where on earth do we have the largest sensor network pardon ok but let's think of the earth it is weather measurements and it is the calibration of the global weather models and they have been in the 80s they become aware of this that they had many measurement information but they didn't really know the precision and that's why they started to regulate and to derive models and to standardize models on how to derive the measurement uncertainty and that's basically the iso-guide 98 that's the uncertainty and it is basically distinguishes two types of measurement determination one is process equation and one is the observations for example if you are a string gauge what is the tire probability density on the x-axis on the x-axis it's the measurement strain what you measure what you measure the value that you measure the density density this is not related to any structure I mean it's the string gauge wherever you put it you are distribution yes so then ok and this variation depends on the position of the string gauge just imagine the temperature maybe of the variation things like that what is the difference with the second the height of the probability density this is determined based on a process equation so this is this equation or it's this is the measurement equation and this is the p here that's the process equation so then you basically go with EPM and EPM in here you have distribution of the mechanical strain also you need a model for the measurement for the model uncertainty of this process equation and then you arrive at this distribution if you observe these quantities yeah ok you cannot observe this but if you observe this one then you will get this equation