I think that I will be able to compute corresponding wind speeds for a location between grid points by using wind speeds hindcasted at these grid points as below via weighted mean:

((Wind_1 * Distance_1)+(Wind_2 * Distance_2)+(Wind_3 * Distance_3)+(Wind_4 * Distance_4))/(Distance_1+Distance_2+Distance_3+Distance_4)

But, I am not sure if this approach is true.

I am looking forward to comments.

Best regards

Adem

An approach we are using is using squared distance weighted interpolation:

u=(u1/r1^2+u2/r2^2+u3/r3^2+u4/r4^2)/(1/r1^2+1/r2^2+1/r3^2+1/r4^2)

where u is the sought for speed, r1, r2, ... are the distances and u1, u2, ... are the wind speeds at the four grid points.

Best regards

Takvor

for your quick response

Thanks Takvor

Best regards

Adem

Takvor approach is good and you can implement it with squares or without them, depending in the weight you want to give to each wind contribution, but you must take care when one of the r_i=0 (an simple if in your program) or if it gets too small. On the other hand, this approach does not take into account if the point where you want to interpolate is in the line between, let us say location 1 and 2. In this case you will probably want a wind speed obtained as interpolation of just u1 and u2, but with no contribution from u3 and u4, and this do not happens (although their contribution may be small).

If you are working in a rectangular mesh, i.e. your data are equally spaced in x and y directions, in a Dx, Dy grid cell spacing and you can measure distances in de x and y direction, another approach could be:

u=(sum(i=1:4) ui *wi)/(sum(i=1:4) wi)

were wi is the corresponding weight (squared of not depending if you want something linear -bilinear- or squared) that write as:

wi=(Dx-dx_i)*(Dy-dy_i)

where dx_i and dy_i are the distances measure in the x and y directions respectively.

Hope it helps

Dear Jorge,

Thank you very much for your suggestions. I am trying both approaches.

Dear Takvor and Jorge,

in your approaches, How can I compute wind directions?

I think that you can try the same interpolating approach as for velocities as far as your wind directions are between 0 and 360 degrees for example (I mean that you can not have -190º and 200º). If the 4 wind directions are not too far away to avoid pathological cases that can appear, I think you can hope it will work ok too.

Use directional statistics. For the basic equations see my paper "Probabilistic modeling of directional and linear characteristics of wind and sea states”. I think that the mean direction will be the most representative for your case. Weighted distance interpolation does not work in your case with directions.

Dear Jorge and Takvor,

Thanks for your valuable comments.

It is true interpolation does not work with directions in general, but if the 4 directions you want to interpolate can be translated to the [-90, 90] degree interval, there you can do interpolation and then come back to the original angles translating directions back. In that case it will work.

Dear Jorge, Do you have any approach for your suggestion?

In the case you have speed and direction what I would do is to split the 4 velocities in x and y components and performing the same interpolation process described above for both components and them to recover the resulting wind speed and direction from the interpolated x and y components. This is simpler and avoiding the large casuistic appearing if you want to think in terms of vector directions.

As Jorge just said do not (ever) interpolate directions as if scalars….ie. as you would a vector component or temperature or … . In your case, using the navigational convention,…1,2,3,359 in degrees will average at ~90 or almost perpendicular to any of the 4 at the corners ..or ..1,2,359 355 will be in the ~ opposite !! (~180 degrees from the simplest and convincing ~average direction) . In directions a=a+n*2*pi for any integer n so …notice that the complex number exp(i*a) is the same regardless of ‘n’ ……so doing averages of complex numbers, seems “ok” but then, why ‘normalize’ each vector? , do the interpolation of u+i*v and you won’t have the ‘circular’ ambiguity or amplitude difference messing up your interpolation. This is equivalent of interpolating component by component and then figuring out the direction of the interpolated ‘vector’.

As for methods to ‘interpolate’ there is a myriad of them (or more). Why only the four corners surrounding the point in question? (with some added inconveniences as to ‘on the spot’ or aligned (in which case which pair is the one to choose or disregard..). These kind of problems are solved (theoretically ...since you’ll have to invent or assume some ‘functions’ which define the averaging/interpolating weights) by ‘objective mapping’ or Kriging or …. (these and all  the ones I know or have heard are variations of 'least square methods')

There is also a simple formula called ‘bilinear interpolation’ which will do in your case (i.e. a somehow mid-point in a rectangle or 4-sided ….).

Now, said all that, since the vector position (x,y) where you want the ‘estimate’ is known [as are the ‘corners’ where you have vectors (u,v)] try the same ‘weights’ to see how well (x,y) is reconstructed (2D so ..a little easier ). You’ll get ‘at least an idea’ of how good/poor is the method you decided.

if you consider other uncertainties in data, perhaps just simple bi-linear interpolation will suffice.