Note that the many formulae on the web if you google "propagation of error" apply to independent samples only, which is generally not the case for geophysical measurements and will lead to an overly- optimistic estimate (even Wikipedia, need to fix that). One source that gives the correct formulas, taking into account the possible correlation among the variables, is:
http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc552.htm However, this simply lists formulae without explanation.The basic idea is a Taylor series approximation. If A is a function of (x,y), then:
deltaA = (dA/dx)deltax + (dA/dy)deltayEstimating the standard deviation of A by squaring and averaging, leads to:
s_A = [ (s_x^2)*(dA/dx)^2 + (s_y^2)*(dA/dy)^2 + 2*r_xy*s_x*s_y*(dA/dx) *(dA/dy) ]^.5 (1)
where s_q is the standard deviation of q and r_xy is the correlation of x and y.
(1) is the general form for any function A(x,y) and can be extended to more than 2 variables in the obvious way. If the quantities are independent, then r_xy=0 and (1) reduces to the simple expressions you find on the web.
So in this case, the centered derivatives you take are operations like:w[i,j] = (-1)*( (u[i+1]-u[i-1])/(2*deltax) + (v[j+1]-v[j-1])/ (2*deltay) )
so the expression in (1) for dw/du is just:dw[i,j]/du = (-1)*( (u[i+1]-u[i-1])/(2*deltax) ), and correspondingly for dw/dv.
The standard deviation of of the average value of w at each gridpoint is therefore estimated by taking the time average of dw/du and dw/dv, finding the standard deviations of u and v and their correlation, and applying (1).
Billy K On 26Nov 2007, at 10:49 AM, Steve Guimond wrote:
Hi, Assuming the flow is Boussinesq (density constant everywhere except where coupled with gravity i.e. incompressible) you can use du/dx + dv/dy = -dw/dz To do this in Ferret, !div is dw/dz let div = -1*(u[x=@ddc] + v[y=@ddc]) By imposing boundary conditions at the top and bottom of w=0 you can integrate dw/dz to find the vertical velocity at each level. Steve ======================================================= Stephen R. Guimond Graduate Research Assistant Center for Ocean-Atmospheric Prediction Studies (COAPS) ======================================================= On Mon, 26 Nov 2007, yangxing zheng wrote:Hi ferret_users, I have gridded ocean data with variables including u,v, temp, ... Unfortunately I need vertical velocity w which is not an output of model. One way to get w is to use continuity equation. Does anyone have any idea how to do it in ferret, also how is potential error produced using continuity equation to calculate w? Thanks a lot! Yangxing_____________________________________________________________________ _______________Get easy, one-click access to your favorites. Make Yahoo! your homepage. http://www.yahoo.com/r/hs