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## Appendix A

The sensitivity of the three-dimensional interferometric measurements depends on the angle of incidence, *θ*, and the angle of the ground swath track, *ψ*, for the ascending- and descending-orbit data which are combined.

The ground swath track angle, *ψ*, measured clockwise relative to north, depends on the target latitude, *φ*, the orbit inclination, *i*, the angle, *y*′ = *y/R*
_{e}, between the ground swath track and the nadir track, where *y* is the ground swath track distance (positive to the right), the Earth radius, *R*
_{e}, and the Earth rotation, *ω*
_{E}. If Earth rotation is ignored, the Earth assumed spherical, and the orbit circular, the track angle, *ψ*
_{s}, can be calculated from

where “+”applies to ascending and “−” to descending orbit arcs. Note, *ψ* is in the interval −90° to +90°, so it will change sign at the northern- and southernmost points of the orbit. The Earth rotation contributes an additional latitude-dependent term, with a numerical value of approximately 3.5° at the Equator, decreasing gradually to zero at the northern-/southernmost points of the orbit. The resulting track angle, *ψ*, can be approximated by

where *v*
_{E} = *ω*
_{E}
*R*
_{E} denotes the Earth rotation velocity at the Equator, *v*
_{S} is the satellite ground velocity, and “−” and “+” apply to ascending and descending orbit arcs, respectively. The accuracy of Equation (A1) is sufficient for error analysis, but not for actual three-dimensional velocity decomposition, which should be based on the actual state vectors.

## Appendix B

In this appendix an expression for the error in the vertical velocity caused by uncertainties in *Fh* in the FD term of Equation (1c) is derived.

Because of the correlation of some of the error sources, it is convenient first to consider the variance of the sum of *E* values along lines parallel to the filter boundaries. According to the discussion of the *Fh* errors in section 5, we may write the sum of *E*_{m,j}
, given by Equation (22), along one of the filter boundary lines as

where the relevant values for *p* are {− (*m* + 1), −*m, m, m* + 1}.The bottom undulation term is described by the constants *c*, *d*, *ω* = 2*π/L*, and Δ*
*_{n}
and an unknown phase, *φ*_{p}
, assumed equal for adjacent *p* values and uncorrelated from one side of the filter to the other. The ice-thickness noise, , is assumed independent from gridpoint to gridpoint. The ice-thickness bias, , is assumed equal for adjacent filter boundary lines but uncorrelated from one side to the other. The *F* bias, *β*_{F}
, is assumed constant over the entire filter.

First the cosine error term is considered. Replacing *F*_{p},_{j}
, *h*_{p,j}
and with their mean values along the boundary line *F*_{p}
_{,0}, *h*_{p}
_{,0} and approximating the summation with an integral, and introducing the filter length relative to the wavelength of the basal undulation, *L*′ = 2*m*Δ*
*_{n}/L, yields

where the worst-case value of 1 for sin(*x*)/*x* is assumed. With an unknown phase of the basal undulation the variance of the term is .

For the remaining error terms it is also acceptable to replace *F*_{p,j}
, *h*_{p,j}
and with their mean values. Taking the correlation into account, the total variance, of a ∑*E*_{p,j}
term becomes

where and are the rms of the noise and bias for an individual gridpoint, respectively, and the rms of the *F* bias. In order to simplify the notation in Equation (B3), we have omitted the subscript 0 in all quantities.

Analogous expressions can be derived for the variance of the other ∑*E*_{p,j}
terms in Equation (18). The contribution to the variance, ,of the smoothed FD terms from all four ∑*E*_{p,j}
terms is found by combining the variances with due consideration of the degree of correlation between the different sub-contributions. For calculation of the variance contribution from the bias term, *β*_{F}
, which has a correlation coefficient of 1 within the entire filter, we use the approximation *F*
_{+} = *F*_{m}
= *F*_{m}
_{+1} and *F*
_{−} = *F*
_{−m
} = *F*
_{−m−1}, and similar approximations for *h*_{p}
and . For the ice-thickness bias, *β*_{h}
, and the ice-thickness noise, which are uncorrelated from one side of the filter to the other, we further approximate *F*_{p}
, *h*_{p}
and by the central values *F*
_{0}, *h*
_{0} and . We find

The variance of the sum of the four *N* terms of Equation (18) is found similarly with Δ*
*_{x}
= Δ*
*_{e}
= Δ*
*_{n}
and the total *Fh* contribution to the variance becomes

In our case, where we assume a constant *F* within the entire area, the last two terms can be modified by multiplying with in both the numerator and the denominator. Using the constant *F*
_{0} approximation, the east term becomes

where is the flux divergence (i.e. the submergence/emergence velocity) averaged from one end of the filter to the other. Likewise for the north term.