By Geweke J.

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In other words, Gγ (ωk1 , ωk2 ) and Gµ (ωk1 , ωk2 ) contain the columns of G(ωk1 , ωk2 ) that correspond to the indices in G and U, respectively. By introducing the following matrices: ˜γ G TGγ (ω0 , ω0 ) .. 40) 24 SPECTRAL ANALYSIS OF SIGNALS and TGµ (ω0 , ω0 ) .. 38) can then be written as min µ where ˜ α ˜ µµ + G ˜ γγ − α ˜ G 2 , ˆ 0 , ω0 )a L1 ,L2 (ω0 , ω0 ) α(ω .. 42) . 42) is easily obtained as ˜ ˜ HG ˆ = G µ µ µ −1 ˜ ˜H α G µ ˜ − Gγ γ . 44) A step-by-step summary of 2-D GAPES is as follows: Step 0: Obtain an initial estimate of {α(ω1 , ω2 ), h(ω1 , ω2 )}.

2 Two-Dimensional GAPES Let G be the set of sample indices (n1 , n2 ) for which the data samples are available, and U be the set of sample indices (n1 , n2 ) for which the data samples are missing. The set of available samples {yn1 ,n2 : (n1 , n2 ) ∈ G} is denoted by the g × 1 vector γ, whereas the set of missing samples {yn1 ,n2 : (n1 , n2 ) ∈ U} is denoted by the (N1 N2 − g ) × 1 vector µ. The problem of interest is to estimate α(ω1 , ω2 ) given γ. Assume we consider a K 1 × K 2 -point DFT grid: (ωk1 , ωk2 ) = (2πk 1 / K 1 , 2π k 2 /K 2 ), for k1 = 0, .

3(c) shows the image obtained by the application of APES to the full data with a 2-D filter of size 24 × 36. Note that the two closely located vertical lines (corresponds to the loader bucket) are well resolved by APES because of its super resolution. To simulate the gapped data, we create artificial gaps in the phase history data matrix by removing the columns 10–17 and 30–37, as illustrated in Fig. 3(d). In Fig. 3(e) we show the result of applying WFFT to the data where the missing samples are set to zero.

### Note on the Sampling Distribution for the Metrolis-Hastings Algorithm by Geweke J.

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