 finish describing this study area. Now, before I come to the examples and the illustrations of this hypothesis testing approach, I quickly need to explain to you a slightly different way of coupling the different methods, and this is based on a so-called concept of variation of information. And what variation of information does, it measures the amount of information contained in one variable about another variable. And practically speaking, it says we assume there is a relationship there, but instead of saying like I showed in the previous example, it's a linear relationship or logarithmic relationship or whatever, we want to find this relationship as part of the inversion. And I'm going to skip the equations part for now, both in the interest of time and in the interest that can be quite difficult to grasp, but we'll show you a practical example how we calculate variation of information and what it means. So here we have two models. This is a vertical slice through some models. You can see depth here on the side, so 100 kilometer, 200 kilometer, 300 kilometer, and then longitude. So this is actually a slice through one of the models through this part in the Western US. And on the top there's a density model, and at the bottom there is a resistivity model. You can see the color scales here. And if you compare those models visually, you would say, okay, there is not a lot of similarity between those two models. We can't really see a lot of features that look the same, and this doesn't change on color scale or anything. Those two methods just individually see the world quite different. And to calculate this variation of information, what we would do is we plot the density anomaly and the resistivity in each of those cells that you can see here against each other in the horizontal plane, so in the parameter space resistivity density. And that's what you see here on the left-hand side. And what you can see is you get this block of points, this cloud of points, and there's no discernible relationship. And to calculate this variation of information constraint, we turn that into a histogram, which just sort of focuses those features and allows us to turn this parameter plot on the left-hand side into probability densities. And if you are interested in the details, there is quite a bit of literature on this out there. And what we can see now is that, for example, knowing that the density in a certain part of the model, the density anomaly is zero, doesn't tell us anything about the resistivity. So the resistivity associated with that density value can be anything. This is a log scale between 10 and several thousand meters. And vice versa, if we know something about the resistivity, we know very little about the density. So there's no real relationship. And this is then results in, yeah, a low mutual information or a high variation of information. But if we have something that starts to look like this, so we have some sort of noisy relationship, you can actually see it looks a bit like a linear relationship, but with some complexity. And on the right-hand side, we can see that histogram, which just focuses on the most important features, then we start to get a higher mutual information or lower variation of information. And this is what we do in the inversion. So in the inversion, we try to produce situations where the histogram is focused and we can get a good one-to-one relationship between those quantities. And I'm going to use this now in the next few examples for some constrained and jointed versions, and then sort of discuss in this context of this hypothesis testing that I was talking about. But before that, I just want to talk to you a little bit about sort of my joint and constrained inversion recipe on a sort of very high level. Of course, the details then depend a bit on the algorithm. So what I typically do and what I have done in the next examples that I'm going to show you is the first step is always run individual inversions of each data sets. And that's mostly to determine what I would call the technical parameters of the inversion. So you need to choose, those of you who have done a 3D inversion or a 2D inversion of some data, you need to choose things like the discretization. So what are your cells? How do you describe your model? The regularization and a few other things. There are quite a few parameters that need to be established. And also, you need to find out how well do I actually fit the data. I mean, statistically speaking, if you know your data uncertainties, there are sort of hard and fast criteria that you can use to say, okay, I fit my data statistically within the uncertainty. But again, those of you who have done this know that the uncertainties are quite uncertain. So very often we cannot fit our data to the level that we expect to. So the individual inversion is sort of this exploration for, okay, what kind of misfit is appropriate for this data set. And I find this as sort of a crucial step that needs to be done before you can do any joint inversion. And then what I do is I run the joint or constraint inversion and set this coupling factor, this coupling weight as high as I possibly can. So if you think back to the equation that I showed you, this kappa value, so I want to find the strongest possible connection between those two different models and see how far I can take the inversion. And that's also to avoid the fact that if the coupling is too low, what you're basically doing is you're running two individual inversions. But very often what you will see is you run the inversion and then you make some progress. You fit a bit of the data, but it doesn't fit particularly well, especially compared to the individual inversions. And then I go into a sort of stepwise sequence where I reduce the coupling and repeat this until I can fit the data. So the idea is to get the highest possible coupling between the data sets and at the same time fit the data to the same degree that the individual inversions fit the data. And then you are in a situation where you have a strongly connected model with a good data fit. And the first example I show you is a constrained magnetic inversion. So I take the seismic velocity model as fixed. So this is this surface wave based model and take the total field magnetic anomaly and make a constrained inversion with those two ingredients with this mutual information or variation of information constraints. So I say fit the magnetic data that it looks very similar to the velocity model, which I'm showing down here. So here you can see this slices at 20 and 30 kilometers. So yellow here is low velocities and blue here is high velocities. So make the structures match or establish sort of as much of a relationship as you possibly can and at the same time fit the magnetic oscillations. And on the top here you can see the susceptibility results. So you can see here the structures in terms of magnetic susceptibility at 20 and at 30 kilometers. And yeah, this is what I would typically call or what people probably typically call an unsuccessful joint inversion because you compare our reference velocity model and our susceptibility model at 20 kilometers and there isn't much similarity here. Yes, we can see some structures, actually the strongest structures that have evolved here have this north east southwest trend and we can see nothing similar in the velocity model. If we go down to about 30 kilometers we can start to see some more similarity. For example, here is some high velocity structure that corresponds in shape to high susceptibility. But still there are quite a few structures here that are in the susceptibility model and that are not in the velocity model. And if we look at the parameter relationship for this model it becomes, if you want to know even worse initially, sort of if it's what you're after establishing a connection here. Because as part of the inversion I also want to establish one-to-one relationship. And you can see that here for significant parts of the model the inversion has not been able to do that. Yes, you can see here there's a large scatter. So knowing for example that your seismic velocity is 3.5 kilometers per second doesn't tell you anything about susceptibility. There's a large variety of susceptibilities that correspond to that. And for the other way around so knowing susceptibility also doesn't tell you anything about velocity. And here you can see the color that's sort of certain depth ranges. So you can see those reddish colors that's all in the crust. And so you can see the scatter in the crust is extreme. So we cannot really establish a relationship in the crust. But then something quite surprising happens and we can see that for deeper layers and higher velocities all of a sudden we get this nice one-to-one relationship nearly like a linear relationship between the two. But then those of you who know something about magnetics will say that's a bit weird because that's 60 to 150 kilometers. We're dealing with magnetic data way above the curie temperature. So we don't expect any strong magnetic anomalies at these depths. And that's definitely true. And also what you can see when you do some tests which I'm not going to show here is that actually the magnetic data has virtually no sensitivity to this part of the model. So this is purely what the inversion is trying to do. It's a demonstration that if you want so if it can the inversion will try to make a one-to-one relationship but this is not driven by the data. The data doesn't know anything about it. But where we have sensitivity there is no direct relationship between the structures that are sensed by the magnetic data and the structures that are in the velocity model. Now you might say okay so what's that? I mean you have one part where you have no sensitivity and there you have a relationship so that's sort of a bit useless. And then you have another part where you wanted to establish a relationship in the cross and you have sensitivity but there you don't get a relationship. But this is exactly what I wanted to sort of establish here as this hypothesis testing sort of approach. If you turn it around you could say okay what we've learned about the earth in this region is the structures sensed by the magnetics are different and they have to be different. The data sort of mandates that they are different from the structures that we have in the seismic velocity model. So for example a comparison between those models or sort of independent models doesn't make any sense. Now I haven't gone into a deep analysis of these structures but of course you can then go into these structures and see how they relate to tonic features both from the magnetics and the velocities. And you can be certain that there are differences there and you can use that to drive your analysis. So if you want so you have disproved or refute the hypothesis that there's a direct connection between magnetic susceptibility and seismic velocity in this region and I think that's a quite powerful statement that you can make and maybe even a more powerful statement than if you can establish a relationship because as I said you always are a bit in doubt whether that relationship is actually true whereas here you can say I definitely know the the structures in the magnetics and the structures in the velocity have to be quite different. So that's a test with or an example with magnetic data. Now we can do the same with gravity and the seismic velocity model and many or again those of you that have done seismic inversions you will know that there are very often people assume some sort of relationship between density and velocity. So that seems maybe a bit more like a reasonable oh sorry I had I forgot I had this but I already talked about. So yeah so here are the results of the constrained gravity inversion. So I'm doing the same thing again I'm doing an inversion of the gravity data and I keep the velocity model fixed as a constraint. Now there is one change in the previous slide you can see if you look at the color scale this is absolute velocity that I'm using here and here I'm using velocity anomaly that the results for the previous example for the magnetic don't really change when you use either velocity anomaly or absolute velocity here I haven't changed this to velocity anomaly because I'm also dealing with density anomaly so the two things seem to be quite or better related to each other. And on top here you can see the result at 30 kilometers so this is a horizontal slice at 30 kilometers through the the Snake River plain and Yellowstone is somewhere over here and you can see the density anomaly model and the velocity anomaly model and here we can see quite good correspondence even visually between those those two models you can see for example this high or positive velocity anomaly also corresponds largely to a positive density anomaly and we can also see others sort of secondary structures for example this this structure here that is sort of expressed similarly in the density model and now if we look at the parameter relationship we get a very different picture than we had with magnetics yes on the magnetics we had this sort of large scatter in the in the shallow part and then this yeah quite artificial structure this relationship in the deep part whereas here we get a very strong linear relationship between density anomaly and and velocity anomaly and that's compatible with what people think about how this should look like within the earth but then we also get some structures that sort of are off this main relationship so we can see here that what the joint inversion would like to do under this sort of variation of information constrained is what you have just one line here but there are certain parts that are that deviate from this line now some of them this light sort of peachy color that's sort of probably very near surface scatters so where the velocity model is not particularly great and there's heterogeneity in the in the density so that's probably just yeah a fact that the velocity model is not particularly good but we can see there are deeper parts here and again in the mantle and there are structures that do not conform to this this simple relationship and again in the spirit of this hypothesis testing I would say these areas where things do not conform to our expectations where we where the joint inversion tells us okay you cannot have a simple one-to-one relationship between your parameters the data compels us to deviate from those are the most interesting parts so I'm not going to go into this because I haven't had the time to really think about sort of what it means tectonically but for example my first approach or my next step now would be to to really focus on these structures here that seem to be both mandated by the data but also deviate from from this assumption because it means that they are yeah they are features that are in this joint inversion because the data compels them to so those are if you're also the most interesting aspects of of this joint inversion um yeah so again the sort of what I just said in a in a bit of a summary slide and with a sort of bigger um figures for this relationship um so we have sort of a strong relationship here and uh some extra scatter um and I mean if you if you think about it uh this is quite suspicious or quite strange because we have a low or negative velocity anomaly and a positive density anomaly so something some material that's slower um and um more dense than the surrounding so that's quite counterintuitive but to me certainly this is one of the features that I would then investigate most as in that as a next step because it yeah it is counterintuitive and it is sort of mandated by the data to to to be in this in this model and as a final example I want to show you this a sort of a similar thing with the joint inversion so this is a slightly bigger region so we're zooming out we're going back to sort of the whole western US here and this is now joint inversion with the same kind of coupling but with magnitude lyrics um and and density and uh I'm showing you here resistivity on the left hand side and density anomaly on on the right hand side um at 30 kilometers so the same slice the same depth we've looked at but also now at 100 kilometers and yeah it's it's the same principle we can see some some very nice correspondence for example here especially at 100 kilometers you can see for example the high resistivity structure here this dark blue that corresponds to high density um and this is perfectly compatible in terms of physical properties that we know with a descending slab so this is the Juan de Fuca slab so we have a subduction zone here of the Pacific Coast the western Pacific Coast in the US um and the interesting thing here for example is that you can see this dashed line that's the the boundary of the Juan de Fuca slab from a global geodynamic model so we can see how here the joint inversion has um made a nice um relationship between density and then resistivity um and also helped us to focus on certain features and we can see other things here so here um in the in this 100 kilometers slice the high resistivity always corresponds to positive um density anomaly and low resistivity corresponds to negative density anomaly so that we have established a very good connection between between the different methods and we can image um sort of structures such as the Wyoming craton or um here the remnant of another subduction zone known as the Silesia slab curtain um and at crustal levels levels the picture looks looks fairly similar we have this low density anomaly and low resistivity that correspond to each other but if you look carefully there are deviations from this assumption again this for me this is now at or at the moment the most interesting aspect of this so we can see there's a region here where we have low resistivity and high density and this will become even more apparent when we look at the vertical slice that goes along this black line so here is a a vertical profile we can see these um low density structures so negative density anomaly red and uh high or low resistivity and yellow but for example here on in the towards the east we can see that we have a continuous conductive structure low resistivity structure but a flip in density anomaly and again this is uh violating our the assumption of our joint inversion so we are sort of if you want so refuting the hypothesis that there is a one-to-one relationship between density and resistivity that's what we're aiming for but the data tell us this is not possible we to fit the observations there has to be a change in density across this structure but the resistivity has to stay the same and the really interesting thing for me is you can see those black triangles those are the boundary of sort of the large tectonic boundaries in this region and you can see that this this change matches exactly with that tectonic boundary so again we are learning something about the earth here from this sort of violation of um the constraint or the thing that we put in the um in the joint inversion in the sense that yeah something different operates in these two uh tectonic domains and we can look at also the relationship that comes out of this of this joint inversion so I'm plotting here density against resistivity for this whole model and you can see this yeah I call it the the dinosaur this dinosaur shaped um structure if you have focus on the those sort of the main features so we have a tail here yes and the back and the head and then the two legs and again it's I repeat this because it's a sort of a new concept and maybe not completely intuitive but the joint inversion if if it if it could would just put everything on this sort of one line we would want to have a single relationship here so the fact that we do have these legs here means that these are features that are driven by the data so we can only explain our data by having this deviation from from a one-to-one relationship um and in this case I have done a bit of analysis what this could possibly mean um and so this material here you can see it has nearly neutral buoyancy um so whatever causes the the resistivity has to have a similar density as the as the background and if you know a little bit about um electrical conductivity in the earth um the current assumption is that these um low resistivity structures are caused by some small amounts of things like graphite or sulfites or fluids so graphites and sulfites have a similar density as the background so this is some sort of um graphite or fluid sorry graphite or sulfite induced um low resistivity structure whereas fluids have a um lower density than the background so um increasing the fluid content decreases resistivity and um decreases resistivity and I've done some first order calculation and the predictions actually match um quite well I have a plot here but I'm sort of going to skip over this in the interest of time but what we what we can do is we can map these structures sort of back into the spatial domain and we can see for example where we have these high density conductors and these low density conductors and get some some really interesting patterns and this I'm not going to go into into those details um tells us something for example about the distribution of of fluids and melts in this in this region so here in the southern active basin and range province we have a lot of pervasive fluids in the crust whereas these blue regions they are probably more relate to ancient deformation zones where we have graphites and sulfites um that sorry cause the these these conductivity anomalies yeah so again I think the the the theme here or the topic here is that very often I find in in the joint inversion it's not the the aspects that that work or that conform to our expectations but the unexpected results that give us um the most interesting results and that we can use to analyze uh our data and learn something new about the earth okay now to to finish um I want to show you one last example from on a quite different scale that also sort of highlights some of the sort of issues with joint inversion and again sort of how we can make things consistent and this is now from from exploration imaging so this is um the problem of sub basalt imaging so if you're exploring for hydrocarbons you want to find oil and gas you need to look at sedimentary structures but if those sediments are below a flat basalt for example here of the faroe islands um northern europe uh then this becomes very difficult because the the the salts have very high velocity and they scatter a lot of the seismic energy so you can't image the sediments underneath and the interesting thing here and why I picked this study there are two studies that have been performed with this these data and both do um joint inversion um so it's it's very interesting to compare how there's sort of different groups approach the same problem differently so one is uh the study of Heineke et al that I was involved in with as well that uses magnitude to lurks of electromagnetic data seismics and gravity and then there's the study of panzne et al that was done independently and they use magnitude to lurks seismics and control source em so another electromagnetic methods they don't have the gravity data but they have two different electromagnetic methods and because it's an exploration context there is borehole data so this goes back to what I showed in the tutorial in the very beginning um so the this is now a plot of real um borehole data and you can see um the graphics from the the two publications so on the left hand side this is the plot of the borehole data in Heineke et al on the right hand side the same data in panzne et al one thing you can already see that panzne et al seem to have rejected this sort of red cluster now both have a color coding for depth unfortunately exactly with the opposite sense of the color scales you need to sort of switch color scales when you go back and forth you can see that otherwise the other data are quite similar I think yeah panzne did a bit of of data curation sort of rejected a few um points here but the more important thing is that both studies assumed okay we know something about the relationship between seismic velocity and resistivity so so we should use that and use that in our joint inversion approach so on the left hand side this black line that's the parameter relationship used by Heineke et al and on the right hand side that's the relationship used by panzne et al and if you go to the paper they discuss that they specifically sort of design the relationship and use sort of a combination of two different functions to capture this steep rise here of um seismic velocity sorry resistivity with seismic velocity so they made took special effort to capture this this this increase here whereas you can see in the study of Heineke et al um that this is based on on a single function that sort of just continuously um sort of rises but it doesn't capture that steep rise but there actually just goes sort of you know moderately um towards um higher velocities and then higher resistivities and yeah you can you can put this in a single plot so the the black is the Heineke et al curve the blue is the panzne et al curve and um the dots here now are again the Bohol data but instead of using so these individual points are the individual Bohol measurements and these are taken like set a few tens of centimeters or centimeters apart and here now is the same data but averaged to a scale of about a hundred meters because when we do the joint inversion we use sort of cell sizes that are on the order of yeah about a hundred meters so you can see two things first of all the the scatter is reduced quite a lot and you can also see that for for large part of the range they actually do overlap or quite quite closely and only at the extreme ends those those two curves overlap and if you look at the data points now in this average space um you can also see that where there is a discrepancy that's where we on this sort of and this average don't really have any any data anymore so for the most part um these um overlap but I think this is a very good sort of illustration of okay you give the same data to two different groups and they come to two different conclusions and of course this will have an impact on what you get out in your joint inversion and here is now the comparison between sort of the joint and the individual inversions for a vertical slice so here's depth this is um profile length and kilometers and along one of the seismic lines and the seismic model especially illustrates quite nicely um that um the problem of imaging below the the basalt so it it catches the top of the basalt that's this uh interface between blue and and red here extremely well but below this there's virtually nothing um on the other hand the mt in this case um doesn't have a good um expression of that top of the basalt you can see it has sort of a similar boundary but then sort of deviates quite strongly but has some structure below the basalt and then when we put these two things together and this is now an example where yeah the two methods if you also talk to each other quite well because we have this strong connection from the parameter relationship is and we get a combined model that looks actually quite different than the individual models because the seismic data provides information about the top of this basalt boundary and once that is fixed the mt then provides information about the internal structure of the basalt and what's happening um underneath and we can then go back from our joined model back to the um to the borehole data so here you can see the the blue line that's the really scattered borehole data that was plotted as these individual plots uh points in the previous plot uh and then an averaged version and then the yellow dots are the um extracted from the model at the location of this borehole and you can see how that um matches quite nicely from with the general trend of the borehole data but you can see there's a lot more heterogeneity in the borehole then we can capture with the joined inversion because as you can see here now the distance between those cells is like on the order of a couple of hundred meters and we can then also look at the comparisons of the models so um this is just for for for our study for the Heinke-Dahl study and here I'm comparing the models um that we get from from from our study and at the bottom is the the study of Pansner-Dahl and uh yeah again color scales of course people have sort of individual choices so here it's plotted in terms of resistivity here it's plotted in terms of velocity but because the two are related you can at least structurally compare those you can see there's actually quite a lot of similarity between those um those different structures but there are also a few differences and maybe the biggest one is here this brook done that's where this borehole is um and you can see for example that Pansner-Dahl have um a sort of thicker bulge here in their um in their bottom structure which is um you know what for example what oil industry um would be interested in whereas here it's it's a it's a bit a bit more flat also Pansner-Dahl have a bit more deep structure um whereas our model sort of loses detail here at the bottom uh that's of course also um partially due to the different methods that have been used but also due to the different choices um that have been made in in the joint inversion um and yeah but I think that the the the main message here is okay um we've shown that with this parameter relationship is um we can produce some reasonable models and there are two ways if you want to to look at it you can say okay as long as the parameter relationship um is sensible the models that come out of here are also sensible and I would call this the sort of explorative mode or if we go back to our sort of hypothesis testing idea you could say okay um we cannot refute the hypothesis that both either of these uh relationships um are sensible even though they are quite different both sort of models fit the data and both sort of produce sensible structures um and in this case that's probably due to the fact that actually where it matters in this velocity range of 6000 meters per second to one and a half thousand meters per second those relationships um are actually quite similar and where they diverge there there's actually no data and you can also see this um in the model see I mean the highest velocities about that we get is on the order of 5000 meters per second so somewhere around here where those two relationships are quite quite similar but I think it would also be quite interesting now to try maybe some other relationships and see if um sort of which ones of those we can sort out and again that can help us to learn a bit more uh about the earth and sort of how this relationship works yeah so this brings me to the end so joint and constraint inversions are useful tools to investigate the earth I hope I showed you a little bit of that um the how you couple how you connect the different properties that's really the crucial element of joint inversion it describes what we assume about sort of what we know about the earth and that's what distinguishes it from from individual inversions and we can approach this um joint inversion from two perspectives we can say we're just we're exploring right we know don't know very much about this the the structures in the earth so if all the ingredients that we use uh particularly that that coupling the the connection between the parameters are representative then the the resulting models are you know valuable useful representations of what's happening within the earth or we could say it's we're using a hypothesis based approach and that means we we're formulating a hypothesis that we want to refute and yeah so that's most powerful in cases where we then sort of the result of the joint inversion is that we we cannot fit the data we cannot make it make a connection for example as I showed in the with the magnetic data and I think um if you combine those two views yes you have a sort of quite two quite powerful tools to to um learn more about sort of what's happening within the earth and I think with that I'll I'll stop for now and open for questions