Page 229 - Special Topic Session (STS) - Volume 2

STS486 Alessandro F. et al.
2. Data and notation
We consider a 40 year temporal domain = [1, … , ] with 12 hour

time step starting from 1 Jan 1978 00:00UTC and ending 31 Dec 2017. The
spatial domain is denoted by = { … , } representing 655 "good"
1,
,

stations of the IGRA network. Altitude of measurements is expressed in
pressure level 925 ≥ ℎ ≥ 50 ℎ.
Let (, , ℎ) denote the response process (e.g. temperature in Kelvin) and
let the data be denoted by:
( , , ℎ ,, ) (1)

where
1. = ( , ) ∈ are the coordinates of the − ℎ station;

2. ∈ is time at 12ℎ scale;

3. ℎ ,, = 1, … , , ≤ is altitude expressed in hPa ℎ ∈ [925, 50].
Let = () be the dimensional vector of all data at time and
,

,
station and let = ( , … , )′ be the dimensional vector of all data
′
′
,

,
,1
at time t, with , = ∑ . For the sake of simplicity, we will assume that
=1
,
altitudes are time invariant. That is , = , ℎ ,, = ℎ and dimensional
,
vector. This is the case of the so-called mandatory levels.
′
′
Eventually, let ;+ = ( , … , + )′ and let = 1: be the full dataset

vector.

3. A 4D Gaussian Process
Consider the Gaussian process (GP) with 4D continuous index, given by
(, , ℎ) = (, , ℎ) + (, , ℎ) (2)
where = (; ) ∈ ℎ ℎ, ℎ ∈ [925, 50] and let = (, , ℎ).
The data described in Equation (1) are observations at possibly non-regular
time points ∈ , and spatial points = { , … , } . Hence, using the

1

−dim vector , the above equation is written as

= +

where ( ) = and covariance function given by

′
4 | − |

′
2
′
2
((), ( )) = (− ∑ ) + ( = ) .

=1 ,
The corresponding GP parameter set is = (, , … , , , ).
2
2
1
4

Although the above setup is time invariant and defines a stationary GP, in
the sequel, we assume local stationarity with respect to both space and time.

4. Recursive (local) estimation
We consider a time interval Δ for local estimation, e.g. 30 days, and define
the Δ − periods index by = () = 1, … , . If Δ is constant, and
assuming = Δ x , we have () = ( − 0 ) = 1, … . Moreover, let
Δ
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