accumulated TLM increments to test tangent linear hypothesis

[stef]

I'm a beginner in 4D-Var and I'm wondering how to get the accumulated TLM increments at observation locations/times for I4DVAR (I know it's deprecated, but I use it for learning). According to [1] (their section 4.4), this is required to test the tangent linear hypothesis. Maybe I'm misunderstanding this. They write:

Thus the important requirement is that the difference between

H_{nonlinear} (x) - H_{nonlinear} (x_b)

and

H_{linear} (x - x_b)

should be much smaller than the typical observation errors (defined by R), for all model state perturbations

x - x_b

of size and structure consistent with typical background errors, and also with the amplitude of the analysis increments

x_a - x_b.

I would like to test the latter, i.e. the case x=x_a.

I already know how to get H_{nonlinear} (x_a) - H_{nonlinear} (x_b), the question is about the H_{linear} terms.

I saw the field

Code: Select all

        double TLmodel_value(datum) ;
                TLmodel_value:long_name = "tangent linear model at observation locations" ;

in the mod.nc file, but aren't these the increments for for each individual loop, i.e.

H_{linear}(x_i - x_{i-1})

sampled at the obs locations/times. What I would need is

H_{linear}(x_a)

Is this available? And if not, how do you test the tangent linear hypothesis?

Of course I would try to modify the code myself, all that's required is to accumulate the increments - just wondering if it actually makes sense.

I'm aware of the Desroziers et at. 2005 paper [2]. I don't understand much of it, but I get the impression they are not concerned with the linearization itself, and already depart from the assumption that the linear problem makes sense?

[1] Bouttier, F., & Courtier, P. (2002). Data assimilation concepts and methods March 1999. Meteorological training course lecture series. ECMWF, 718, 59.

Link: https://www.ecmwf.int/sites/default/fil ... ethods.pdf

[2] Desroziers, G., Berre, L., Chapnik, B., & Poli, P. (2005). Diagnosis of observation, background and analysis‐error statistics in observation space. Quarterly Journal of the Royal Meteorological Society: A journal of the atmospheric sciences, applied meteorology and physical oceanography, 131(613), 3385-3396.