Hi everybody,I want to try something in the field of wind modeling, but I do not know how to get what I need:I would like to employ the Vasicek model for wind strength development over time, such that the result (r_t in Vasicek's terms) is Weibull distributed (for each t). I have no clue how to choose the distribution of the epsilons (instead of the normal distribution) in order to obtain this result. (is that even possible?)Any hints are greatly appreciated!Best,Knut

Why not use Navier Stokes?

Cuchulainn,thanks for the suggestion!The wind simulation is supposed to be only a by product and serves as an input for a stochastic simulation.Therefore,a) I want to keep things simple b) I need to simulate wind characteristics (as opposed to a forecast)Nevertheless, I am grateful for your input, thanks! If you have other ideas, please let me know.(The aim is only to depict fundamtental "stylized facts", such as time- dependency of wind speed and direction(space dependence can be ignored). Of special interest for me would be an idea to model speed and direction simultaneously; I have no idea how to do that.)

I assume you are you trying to model a single turbine or small wind farm (otherwise you would need to consider spatial effects due to the movement of squalls and weather systems across a greater spacing of turbines).The Vasicek model will miss both the daily and seasonal patterns prevalent in real wind. But maybe you don't care about that for your purposes.You could model "direction" as the sum of two orthogonal processes (the East-West wind component, and North-South wind component) but that requires a symmetric process both positive and negative values. If you want to use a mean-reverting non-negative process (e.g., Vasicek), you could model it as four processes (East, West, North, South) with East minus West and North north South creating the X and Y components of the wind vector which then combine as an aggregate strength and direction.

The question as posed doesn't make too much sense to me, but there is a related FAQ and answer that goes something like this.I have this XYZ distribution and I want to write down an SDE or some kind of process that generates it. How can I do it?One answer to the FAQ is check to see if your distribution is infinitely divisible, usually done bygoogling with the phrases "XYZ distribution" and "infinitely divisible". So I googled with XYZ=Weibull and found the answer is 'yes'.When the answer is 'yes', you can create a Levy process (and hence an SDE) for X(t), whosedistribution at T=1 will be the target distribution. Typically, the resulting Levy process is a drift + pure jump process (no Brownian component), which looks the case here.How to create the Levy process?For example, see this paper, where the authors claim that the Weibull (and many others) are a special case:http://www.emis.de/journals/PS/images/getdoc1a86.pdfSo, you can apparently just specialize their results.

Try with this Stochastic process for time series of wind speed

QuoteOriginally posted by: KnutSchnuteCuchulainn,thanks for the suggestion!The wind simulation is supposed to be only a by product and serves as an input for a stochastic simulation.Therefore,a) I want to keep things simple b) I need to simulate wind characteristics (as opposed to a forecast)Nevertheless, I am grateful for your input, thanks! If you have other ideas, please let me know.(The aim is only to depict fundamtental "stylized facts", such as time- dependency of wind speed and direction(space dependence can be ignored). Of special interest for me would be an idea to model speed and direction simultaneously; I have no idea how to do that.)So the ambient wind speed and direction is known beforehand and are probably the input to your Vasicek model? Then you simulate (solar, a plume??) one fluid in a surrounding fluid using Vasicek? Are you interested in (general) convection effects?//I remember working with plume models for steel factory emissions. The model itself was quite simple (not even an ODE) but there was lots of historical data stored on 6 boxes of Hollerith cards that I had to pick up by auto because it was in the basement of another building:-) This was the transport protocol pre TCP/IP

That is interesting.(I know NOTHING on this subject so my questions are probably basic)When people model wind:- are they interested in wind speed at an exact location? or in wind speed over a surface? (how do you cope with the surface issue?) - are they only interested in horizontal wind speed?- do people use original wind data? are wind acceleration/deceleration of interest (differenced time series) ?- how do people cope with the many perturbations 'around' that are likely to affect wind speed? (air convections, air pressure, ...)

I apologize for not being specific from the start:Background:I need a stochastic model for wind (speed and direction) as input for an advection diffusion model to model the development of an ash could after an volcanic outbreak.As the model is supposed to be as generic as possible, but I need only a rough approximation, I will ignore certain characteristics, the most restrictive one being that the area of interest is so small that I dont have to consider spacial variations. Further, I ignore:- vertical wind- surface characteristics.I do not need to look at data at this point, but allow for data input by the structure of the model (i.e. the more information I put in the model, the more I'll get out of it)- the perturbations "around" are captured by the diffusion of the PDE (advection diffusion).- I don't want to model the dependence on the hidden factors such as pressure, temperature, etc.This leaves the dynamics of the wind speed and direction to be modeled, I thought of AR processes or Ornstein-Uhlenbeck/Vasicek.There is also a connection between wind and direction, as depicted in wind roses http://en.wikipedia.org/wiki/File:Wind_ ... ot.jpgThis is probably difficult to consider (at least I do not have any idea how to do that in a consistent manner), so independent speed and direction will do just fine.I know that the distribution of wind speed over a year is Weibull distributed, and I do not know how (and if) this can be connected to a time-varying model.Additionally, I would like to consider daily (day/night) and seasonal variation in the mean wind speed (e.g. by a time-dependent mean in the Vasicek model)I am grateful for any idea for an consistent approach to this!

So, you wish to model volcanic ash plumes? The goal is to measure the dispersion of ash?

Yes! The scope of my question is only how to model wind though. I think I am all set with the Advection-Diffusion setup. (wind will serve as the advection coefficient, more or less)

QuoteOriginally posted by: KnutSchnuteYes! The scope of my question is only how to model wind though. I think I am all set with the Advection-Diffusion setup. (wind will serve as the advection coefficient, more or less)Thinking out aloud. The ash concentration PDE dC/dt = -u dC/dx + tubulent diffusion factors in (y,z) directions.where u is the transport by the mean wind value. Then you could sample u from your historical Weibull distribution.If y,z directions can be ignored you could use a pure advection (convection) plume model?

yes, that's almost it.dC/dt = -u1 dC/dx -u2 dC/dy + turbulent diffusion factors in (x and y)I want to find a model for u1 and u2

Thanks, frenchX, but this paper just tries to fit one month of wind data to AR, ARMA, ARFIMA models etc.I am less interested in the specific fit and rather aim at picturing some general characteristics.

QuoteOriginally posted by: KnutSchnuteyes, that's almost it.dC/dt = -u1 dC/dx -u2 dC/dy + turbulent diffusion factors in (x and y)I want to find a model for u1 and u2Knut,I am thinking out aloud One way to suppress spatial differences is to transform the advection part to an ODE on a curve. Then you might be able to use space-independent wind data? QuoteI am less interested in the specific fit and rather aim at picturing some general characteristics. Like lowlands (e.g. Heimay) versus uplands(Vatnajukell) wind profiles?