Two Step Estimation of TID Parameters

The MSTID model presented in Equation 1.8 is characterized by three parameters:

{λ, θ, ϕ}. As the reconstruction error depends on the parameters through trigono-metric functions, and a division, this error is more sensitive to some parameters than others. Specifically, the impact of small deviations of the parameters{λ, θ}on the fi-nal performance is much higher than the case of ϕ. The method is extremely precise in the estimation of propagation azimuth and wavelength, but the velocity estima-tion is less accurate, possibly because it is computed from differences of ϕ at each snapshot. Taking this into consideration, the estimation strategy will have two steps, in the first step, the parameters{λ, θ}are estimated using a dictionary fine-grained in the range of these parameters and rough in{ϕ}. In the second step, the estimation is done with λ and θ fixed to the values of the first step, and the parameter ϕ will be used to construct a new dictionary with a fine grained range of values.

The first approach for solving the problem (1.14) was based on coordinate descent, as explained in detail in Hastie, Tibshirani, and Wainwright (2015). For our imple-mentation, we have used a method for solving the problem which yields equivalent solutions but is computationally faster. This method is the LARS, see for instance Efron et al. (2004) and Hastie, Tibshirani, and Wainwright (2015). The computa-tional complexity of the LARS algorithm is equivalent to the ordinary least squares estimation which has a complexity of the order O p²N
, where p is the number of elements of the dictionary, and N the number of pierce points.

The effect of splitting the estimation into two steps also has an important impact on the computational requirements of the problem. If the granularity (i.e., incre-ments) on the values of each parameter were of n, the size of the dictionary would be O(n³). By dividing the estimation into two phases, the dictionary of the struc-ture in Equation 1.19 gives a requirement of O(n²)for the first phase. The resulting nonzero-weight elements of the dictionary of the detected MSTID in the first phase M (where M  N) reduces the total number of operations on the second phase to O(Mn). The range of values n needs to be adapted to the problem at hand, which depends on the margin of phases, frequencies, and azimuths of the MSTIDs that we want to detect.

1.6.5.1 Step I: Estimation of{λ, θ}and the Number of TIDs

For the estimation of {λ, θ}, we can simplify the dictionary by means of simple trigonometry, the parameter ϕ of the elements T_i(x, y)of the dictionary (See Equa-tion 1.8) is absorbed by the amplitude of the wave. Thus the element T_i(x, y), for the snapshot t is:

α_iT_i(x, y) = β_icos 2π

λ_i (x·_{cos θi}+y·_{sin θi})



+γ_isin 2π

λ_i (x·_{cos θi}+y·_{sin θi})

 (1.16) which can be rearranged as two new dictionary elements as,

Tβi(x, y) = cos 2π

λ_i (x cos θ_i+y sin θ_i)



(1.17) T_γi(x, y) = sin 2π

λ_i (x cos θ_i+y sin θ_i)



(1.18) where β_i and γ_i are the amplitudes of the pair orthogonal sinusoidal basis, which satisfy the relationship of β_i = α_icos(ϕ_i)and γ_i = −α_isin(ϕ_i). Note the values of β_iand γ_i are used only for deciding the set of elements of the dictionary T_i(x, y)to be used in the following step. And we decided to work with an augmented dictio-nary, with elements of the form shown in equations 1.17 and 1.18, and estimate the amplitudes β_iand γ_i by means of the LASSO algorithm.

Although the phase ϕ_i can be obtained from β_iand γ_i, by their ratio and the use of the arctan function, this method is extremely unreliable, which justifies postponing the estimation of ϕ_i for step II. The estimation of ϕ is done in step II keeping the fixed values λ and θ estimated in step I.

The dictionary D_I for step I has the following structure:

D_I = [D_β, Dγ] = [T_β1, T_β2, ..., T_βN, T_γ1, T_γ2, ..., TγN] (1.19) where T_βi and T_γi denotes the elements of the dictionary defined in Equation 1.17 and Equation 1.18 with the corresponding parameters θ_βi, λ_βiand θ_γi, λ_γi. We denote the vectors of the weights associated with each estimated parameter as: β_t and γ_t. Note that we did not introduce a constraint in the elements of the vectors βtand γt

so that they should be non-null simultaneously; we decided to take as candidates of MSTID the union of indices i found in both vectors, to filter the values in step II. That is, some of the candidates to be detected MSTIDs, will be discarded in the second step. The map V(x, y)at epoch t can be expressed as follows,

V(x, y) =D_I· [_{ˆβt, ˆ}γ_t]^T = D_β· _ˆβt+D_γ·γˆ_t (1.20)

where the vector of coefficients ˆβ_t and ˆγ_t are estimated by means of the LASSO algorithm. The elements of the dictionary DIassociated with the coefficients ˆβtand

ˆ

γ_tdifferent from zero, determine the wavelengths ˆλ_tand directions ˆθ_tof each MSTID candidate and are the input for the next step.

The parameter ϕtcan can be approximately computed by,

ˆ

ϕt= −arctanγˆt

ˆβt

(1.21)

Nevertheless, we have found that this estimate was extremely unreliable and in-consistent between different snap shots. Note that the solution from LASSO is the approximated decomposition of detrended VTEC map V(x, y, t)with the deviation of D· [ˆβ_t, ˆγ_t]. The method we described above in the text gives a reliable estimation for the parameters of ˆθtand ˆλtfrom the improved dictionary, but a weak estimation for ϕ_t. In order to improve the reliability of our estimation, we use another LASSO step in order to have a better estimate of ϕ_t.

1.6.5.2 Step II: Estimation of ϕ

In step I at epoch t, we detect a set of M possible MSTIDs candidates. We define the parameters of the i^th detected MSTID at epoch t as the pair(ˆλ_i, ˆθ_i), which remains fixed in step II, where we create a new smaller dictionary for estimating the ϕ_i associ-ated with each pair(ˆλ_i, ˆθ_i), from a new variable that is more suitable. We will denote anal-ysis showed that the expression 1.23 has a lower sensitivity to the estimation error of ϕ_j than 1.22. Hence, the dictionary is parametrized by B_(ˆλ

iˆθi)j.

The structure for the elements j associated with the pair(ˆλ_i, ˆθ_i)of the dictionary of step II, will be as follows:

T_(ˆλ to a given ϕ, and the structure of the new dictionary, which we will denote as D_{I I} is created by concatenation of the dictionaries specific for each MSTID associated with each pair(ˆλ_i, ˆθ_i)detected in step I. Therefore, the dictionary associated with a given MSTID D_ˆλiˆθi, will consist of a set of waveforms that characterize the MSTID estimated for fixed values of ˆθ_i and ˆλ_i and the range of values B_(ˆλ

iˆθi)j related to the

phase ϕ_j of the waveform.

D_{I I} = [D_ˆλ

1ˆθ1, D_ˆλ2ˆθ2, ..., D_ˆλMˆθM] = [T_ˆλ

1ˆθ11, T_ˆλ1ˆθ12, ..., T_ˆλMˆθMN] _(1.24) Thus, if the range of possible values of B_(ˆλ

iˆθi)j is N, the size of the dictionary D_{I I} is M×N (M is the number of candidate MSTIDs from step I).

Note that dividing the estimation into two steps, in addition to solving the estima-tion uncertainty in ϕ_j, reduces the computational needs. That is, the combined size of the two dictionaries is 2N²+MN, which is much smaller than the size N³needed for simultaneously computing all the parameters. For instance, if the resolution of θ, λ and B are 2^◦, 2 km and 0.1 km respectively, assuming M≤10 in each snapshot, the size of the dictionary is N³ ≈ 10⁸versus 2N²+MN ≈10⁶. The value of N was variable and depended on the resolution of each variable. Typically the value of N was around 500.