Before we start with the next session, before we start with the next session, I want to announce an informal developer's meeting. We want to do that later, like this. Okay,

fine. So I would like to announce an informal developer's meeting after the last session. So this should be at 6.30 p.m., and this is, as I said, informal. That means people interested to come may stay here for a little while and discuss some issues.

One issue is how we can collect the modules which have been developed and which are going to be presented here. I think there might be some possibility to make a sort of conference contribution, collecting all modules and also the documentation, of course. This can be discussed

later. And another request to me was, is there any chance, for example, to organize GRASS courses in Italy in Italian language, maybe also for other countries? And this we may discuss with people being interested later today. So after the last session,

who is interested may stay here for a while. Thank you. So welcome to this second morning session, which is dedicated to mathematical methods and techniques that is mainly to methodological issues. I have to recommend

to all the speakers to stay into the proper time, which is around 20 minutes. The first presentation is by Ludovico Biaggi, Maria Brovelli, and Marco Negrete from

Politecnico di Milano. It's presented by Ludovico Biaggi, and the title is Environmental Thematic Map and Easy Probabilistic Estimation of a Threshold Exceeded. Ladies and gentlemen, good morning to everybody. My speech will be focused on

two main topics, which we have studied and implemented within GRASS. The first topic is a new approach, a simplified new approach to the prediction of digital surface models from observation. Then second topic is quite related to the first because it focuses on the use of

digital surface models to a probabilistic classification. Okay, what is a digital surface model? A digital surface model is, generally speaking, a numerical method to store the values

of a field that is a spatially variable phenomenon like, let's say, the subsidence or the hair pollution. And in particular, within GRASS, the digital surface model is realized by a georeference, the last error matrix. It is well known that the prediction of a digital surface model belongs to the class

of the field prediction problems that is a branch of geostatistics. And it is also well known that typically we can say that the factors affecting the accuracy of a digital surface model are two. The first one is the observation accuracy, and the second one

is the consistency of the method adapted to predict the field. Without entering into details, we can distinguish two main families of prediction method. The first one family is the so-called deterministic method, and within them,

we can remember the inverse distance weighting. The common factor of deterministic methods are that they typically impose strong constraints on the field without considering the observation's behavior. And typically, they cannot perform an accurate modeling of the observation noise.

On the other hand, we have the stochastic methods. Typically, they are more complex to use than the deterministic methods, but provide, if properly used, more accurate results and can be robust with respect to outlier on presenting the observations.

The classical example of stochastic method is the well-known universal kriging. The basic assumption in universal kriging is that the observations contain three components. Two

components are the field components, that is, the deterministic component and the stochastic component. The third component within the observation is the observation noise that is always supposed to be correlated. Particularly within the field, within the actual field,

we can distinguish the deterministic component that is usually mathematically modeled as a polynomial model. In the slide, we can see the sample monodimensional. The trend is a linear trend, the red line. The stochastic component can be graphically depicted as a fluctuation

around the trend and is mathematically represented by a secondary order stationary process that has some nice mathematical properties. The properties, the basis for the universal kriging prediction are two basic. The first one is that the stationary secondary order

process can be completely described by its variogram. That is a kind of the correlation function. We have here an example, a typical example, and that the deterministic component can be expressed by a priori known polynomial model whose coefficients are unknown.

The prediction steps are two, simplifying the argument. The first step is the empirical variogram modeling. That is, I use the variogram of the observations to fit the theoretical variogram of the stochastic process. The second step is the prediction

that is performed according a minimum square error principle, and it is performed on the observations and by using the estimated variogram and by imposing the known polynomial model.

Universal kriging is a nice method. It seems quite simple to use, but as a main drawback. That is, in order to correctly estimate the theoretical variogram of the field, we need to know a priori, the deterministic component, because we can empirically estimate

the variogram if and only if we can subtract the deterministic component from the observations. It is obviously impossible if we don't know the deterministic component within the process. We have tried to propose a simplified approach. It is very simplified. I want

to remark that, but it has demonstrated to work quite well, so we are here to talk about. We have a pre-processing step before the universal kriging prediction. The pre-processing

basic hypothesis is that, at the first step, we can neglect the spatial correlation between observations. So, under this hypothesis, we can approximate the observation covariance matrix as an identity matrix. Under this hypothesis, we can, by least square method, estimate

a nice polynomial model, the coefficients for a nice polynomial model from the original observations. And, that is most important, we can evaluate the correctness of the senior polynomial model by a simple Fisher test on the estimated coefficients.

The basic idea, I don't want to bore you with the computation, the basic idea is that the implementation is made by constructing an ordered polynomial model library. Each polynomial model is estimated and is evaluated by Fisher test. And, within the

polynomial model library, the model is chosen that satisfies the Fisher test. After that, we can remove the trend from the original observations. So, we have the

empirical variable, and after that, we can finally use the original observation. We can use the universal kriging method to predict the field according to the universal

kriging theory. All that has been implemented within GRASS with a set of three comments. The first one is completely implemented by us, and it performs the operation of observation and the trending. The second and the third comments are simply the calling function to

the gstat package by which we can perform the empirical variable estimation and the universal kriging prediction. We have tested, obviously, the algorithms on some data. This is an example. It is a synthetic field simulated. We have simulated

the field was behavior can seems the subsidence behavior in urban territory. The field is composed by a deterministic and stochastic part. We are distracted from the synthetic field of sparse observations. We have added to the observation some noise,

and we have used the observation with our algorithm to predict the digital surface model. The comparison between the original field and the predicted field is quite good.

We have errors always less than one millimeters and a standard deviation of 0.4 millimeters. The second topic of my speech is how can I use a digital surface model

within a classification proposal? What I mean for a classification, I mean given a digital surface model, I want to identify the sales of the digital surface model which belong to a class that is defined by an interval, by a range, a minimum and the maximum value.

The classic approach in the classification is the following. Given a class C, given a generic cell of the DSM, the cell belongs to the class if and only if its predicted value is within the class range. Obviously, this approach is too simplistic if we know

the prediction accuracy because it completely neglects the errors in the predictions. So the idea is that given a digital surface model, we know that the cell accuracies can

vary from cell to cell. And moreover, we know that universal kriging provides us not only the digital surface model prediction but also a digital accuracy model that is thereafter for each digital surface model cell contains the prediction accuracy.

So we have studied a simplified approach to take into account also this information when performing a classification on a digital surface model.

To do that, we need two hypotheses. The first one is that the prediction errors are distributed according to a normal distribution with zero mean and variance equal to the prediction variance. The second hypothesis is that the prediction errors are spatially correlated. So for each cell, we have a probability distribution that is centered on the predicted

value and whose length is proportional to the standard deviation, the Gaussian distribution. Starting from these hypotheses, we can identify a method to take into account the prediction

error when classifying a cell. Some remarks on the working hypothesis. The first hypothesis is a classical hypothesis, arbitrary but reasonable. The second hypothesis is for sure wrong. It is wrong because it is well-known that prediction are spatially

correlated and so are the prediction errors. But it is a needed working hypothesis because we don't know the full covariance matrix of the predictions and also if we could know these, we cannot manage numerically at this stage. I want only to remember that our starting hypothesis

are less simplified than the classical hypothesis. That is an hypothesis not only of the correlation but also of zero error in the predictions. Okay, so we can compute

for a generic cell the probability that the cell belongs to a class simply by integrating the distribution function within the cell interval. And we also compute the probability that a cell belongs to a senior attention class that is a class going from a minimum

values to plus infinite, again by integrating the probability function. We have tested the algorithm. We have chosen as interest area the urban territory of a town. We have used as a digital

surface model the predicted model of the subsidence during a period of four years. And we have seen an attention class of eight centimeters.

At first I show you the results provided by the deterministic approach. Obviously if we choose the deterministic approach, the attention areas are those which satisfy the simple condition predicted values greater than eight centimeter and the area are

If we choose a probabilistic approach, we have a priori to define the probability level at which we want to classify a cell as belonging to the attention class because

for each cell we can compute the probability to belong to the class and in order to select the cell we have to decide what is the probability level. For example, we can choose to classify as attention cells. The cells whose subsidence is greater than eight centimeters

with a probability at least of 34 or five percent. This is our choice. When we apply a probability level of 34 percent, we obtain two areas greater than those provided by the

deterministic approach and obviously greater than the previous ones when we choose a probability level of five percent. One conclusion is not a conclusion, it is a consideration.

The probabilistic results are equal to those provided by the deterministic approach when we choose a probability level of 50 percent. And we think that this simplified probabilistic approach furnishes results that are consistent with the probabilistic information

on the digital surface model are not completely rigorous results but are less simplified than the results provided by the classical approach. Moreover, by adopting a probability level we can

choose how much risk we want in the classification. This is chosen by environmental consideration on the risk consideration. The

proposed method has been implemented as a GRASS command and will be available. Thanks. Two little remarks. First, could you please say something about the ISOLAR project

and the second remark? If I understood you right, you use a polynomial fit, but

of course you can fit every function with polynomials. But if you know from theory that there is, for example, another function, for example exponential, then I think it would be

better to use this function, not to use your approach. I understand, yes.

ISOLAR project was a European-founded project for the implementation of GIS for urban planning, taking into account environmental consideration. In my first trial of presentation, I had some slides on ISOLAR, but I cannot say on time.

And about your second remark, yes, you are right. If I know, if I a priori know what is the deterministic component within the field, because I have other information, it's better to apply the a priori information I have. But our approach is studied when I don't

know anything about the field. So I try to extract from the observations as much as I can do. In our comment, we have implemented only polynomial models, but I can implement also

other models for the deterministic component. The speech was on the theoretical idea. After that, we have to implement something, so we have implemented polynomial model, the simple mean, the linear trend, the linear surface, the quadratic, we can proceed to other

models, yes. Just a desiderata, and you implemented bounded variograms, but there are many phenomena in hydrology, and I feel that we don't know the physical reasons

are better modeled by the unbounded variograms, let's say with power laws and things like that, and Kriging could support it. Do you think to implement it?

Okay, no, I won't be clear in that. Our implementation is only related to the trend estimation by Fisher test. The whole field prediction is done by calling G-STAT program.

I don't want to nail it off. And we only implemented two simple comments to call G-STAT within graphs and to do the variogram modeling by using G-STAT. So we use the G-STAT library. More remarks or questions? Okay, thank you, LucaDico.