Deriving the standard deviation of a resultant vector

[selane]

I am working with some wind speed data and need to work out the standard deviation of the magnitude of the horizontal wind. Unfortunately the only information I have is the standard deviation for the streamwise and cross-stream components of the wind (which are orthogonal). Is it possible to derive the standard deviation of the magnitude of the horizontal wind from these?

Any help greatly appreciated.

 

[Stephen Tashi]

For non-meteorologists like me, you need to define what "streamwise" and "cross-streamwise" mean and confirm that "horizontal" mean the component of the wind vector projected on the surface of the earth.

For example, does "streamwise" have something to do with the predominant direction of the wind? - perhaps it's average direction over several hours? Or does it refer to the jet stream? Is a vector that points "streamwise" a two dimensional vector or a 3 dimensional vector?

 

[selane]

For non-meteorologists like me, you need to define what "streamwise" and "cross-streamwise" mean and confirm that "horizontal" mean the component of the wind vector projected on the surface of the earth.

For example, does "streamwise" have something to do with the predominant direction of the wind? - perhaps it's average direction over several hours? Or does it refer to the jet stream? Is a vector that points "streamwise" a two dimensional vector or a 3 dimensional vector?

Sorry - forgot to exit my little meteorology bubble before posting...

My data is averaged into 30 minute blocks, so the streamwise direction is the average direction of the wind over those 30 minutes. The cross-streamwise component is orthogonal to the streamwise component, and both are parallel to the ground. You could think of it as the x and y directions in a Cartesian coordinate system if you prefer.

If x is the wind in the streamwise direction and y is the wind in the cross-streamwise direction, I would usually work out the magnitude of the horizontal wind as h=sqrt(x^2 + y^2). I have the standard deviation of x and y over each 30 minute period, and I need the standard deviation of h.

I hope that makes things clearer.

 

[Stephen Tashi]

I think the answer to your question is that you need to know the joint distribution of x and y in order to have a chance of calculating the standard deviation of h.

If we take x as the streamwise direction, since it is defined as an average, it wouldn't be surprising if the deviations from it, given by y, are statistically independent of x. If you can assume that the distribution of x and y are independent and that the distribution of each is one that is completely determined by its mean and standard deviation ( for example a normal distribution) then you can compute the standard deviation of h numerically by using a simulation - or perhaps someone has worked out a formula for it.

A question that frequently arises in posts to the forum is: if x and y are independent normal random variables with different means and variances, what is the distribution of h = sqrt(x^2 + y^2). Unfortunately h will not be a normally distributed random variable and I don't know any formula for its standard deviation. (In the case that x and y have the same standard deviation and zero means, we get a Rayleigh distribution and a formula is known.) However, it shouldn't be hard to write a program to compute the standard deviation numerically. There might be some papers written on this topic. We could search the web, if this fits your problem.

It's not yet completely clear to me what data you have and what the random variables are. For example, let's say that the x direction is the stream direction. For each 30 minute interval, do you have one data vector (x,y) representing the 30 minute average x velocity and the 30 minute average y velocity? Or does a block of data consist of measurements taken over shorter intervals of time?

It's tricky to talk about the standard deviation of a variable that varies continuously in time because the standard deviation of a series of measurements of such a variable can depend on the time interval between the measurements. You say that you have the standard deviation of the x and y components. This is relative to some specific time interval, correct?

 

[selane]

It's not yet completely clear to me what data you have and what the random variables are. For example, let's say that the x direction is the stream direction. For each 30 minute interval, do you have one data vector (x,y) representing the 30 minute average x velocity and the 30 minute average y velocity? Or does a block of data consist of measurements taken over shorter intervals of time?

It's tricky to talk about the standard deviation of a variable that varies continuously in time because the standard deviation of a series of measurements of such a variable can depend on the time interval between the measurements. You say that you have the standard deviation of the x and y components. This is relative to some specific time interval, correct?

Unfortunately I only have the standard deviations of x and y over each 30 minute time period. I don't have any velocity data for x or y. I do have the mean of h over the 30 minutes, but don't have the standard deviation of h or any of the raw wind speed data (which would have a sampling frequency of 20 Hz).

 

[Stephen Tashi]

Suppose the raw 20 mhz data is in the p-q cartesian coordinate system as N vectors $(p_1,q_1), (p_2,q_2)...$ How do they compute the streamwise direction s from that?

One conjecture is

$$p_{av} = \frac{\sum p_i}{N}$$
$$q_{av} = \frac{\sum q_i}{N}$$
$$r = \sqrt{( p_{av}^2 + q_{av}^2)}$$
$$s = ( \frac{p_{av}}{r},\frac{q_{av}}{r} )$$

Another conjecture is
$$p_{av} = \frac{\sum |p_i|}{N}$$
$$q_{av} = \frac{\sum |q_i|}{N}$$
$$r = \sqrt{( p_{av}^2 + q_{av}^2)}$$
$$s = ( \frac{p_{av}}{r},\frac{q_{av}}{r} )$$