Multistep variational data assimilation: important issues and a spectral approach

Figures & data

Fig. 1 Uniformly distributed coarse-resolution innovations (c-inno) plotted by blue+signs, and high-resolution innovations (h-inno) plotted by purple x signs. The solid black curve plots the benchmark analysis increment Δa. The dashed red curve plots the analysis increment Δa ₂₀ from SE with 20 iterations. The solid green curve plots the analysis increment Δa _I-20 from the first step of two-step experiment (TEA, TEB or TEC) with 20 iterations. The dashed blue, dotted cyan and dot-dashed grey curves plot the analysis increments Δa _A20 from TEA, Δa _B20 from TEB and Δa _C20 from TEC, respectively, with 20 iterations.

Fig. 2 C _b(x) plotted by red curve and C _a(x) plotted by dotted blue curve.

Fig. 3 Discrete background error power spectrum S(k _i ) (that forms the diagonal matrix S) plotted by red+signs, and discrete analysis error power spectrum S _a(k _i ) (that forms the diagonal part of S _a) plotted by green×signs, where k _i =iΔk and Δk≡2π/D=5.7×10⁻⁵ m⁻¹. The solid red (or green) curve plots S(k) [or S _a(k)], that is, the continuous counterpart of S(k _i ) [or S _a(k _i )] in the limit of D=NΔx→∞ for fixed Δx.

Fig. 4 (a) Full-matrix structure of S _a. (b) Magnified structure of S _a. The coloured contours plot the element value in m²s⁻².

Fig. 5 (a) Structure of benchmark A plotted by colour contours every 0.5 m²s⁻². (b) Deviation of approximately calculated A from benchmark A plotted by solid (or dashed) contours at 0.01 (or −0.01) m²s⁻². The deviation in (b) reaches the maximum (or minimum) of 0.01156 (−0.01145) m²s⁻² at the point marked by the +(or −) sign on the diagonal, and this diagonal point corresponds to a grid point that is collocated with a coarse-resolution innovation (or located in the middle between two adjacent coarse-resolution innovations).

Fig. 6 Analysis error e ₂₀ (or e ₁₀₀) from SE with 20 (or 100) iterations plotted by dashed (or solid) red curve, analysis error e _A20 (or e _A100) from TEA with 20 (or 100) iterations plotted by dashed (or solid) blue curve, and analysis error e _B20 (or e _C20) from TEB (or TEC) with 20 iterations plotted by dot-dashed cyan (or grey) curve.

Table 1. Entire-domain averaged RMS errors (in ms⁻¹) for the analysis increments obtained from SE, Step-I, TEA, TEB and TEC applied to the first set of innovations with periodic extension and consecutively increased n, where n is the number of iterations and Step-I stands for the first-step analysis (in TEA, TEB or TEC). All the RMS errors are evaluated with respect to the benchmark analysis increment Δa except for those in the third row where the RMS errors are evaluated versus Δa _I – the benchmark for the first-step analysis itself, obtained directly from eq. (1a) for the coarse-resolution innovations only

Experiment	n=20	n=50	n=100	Final
SE	0.883	0.286	0.090	0.019 at n=309
Step-I	0.713	0.588	0.596	0.599 at n=178
Step-I vs. Δa I	0.321	0.081	0.018	0.007 at n=178
TEA	0.316	0.084	0.017	0.008 at n=141
TEB	0.806	0.417	0.530	0.568 at n=178
TEC	0.661	0.402	0.420	0.428 at n=121

Fig. 7 As in Fig. 5 but for the non-uniform coarse-resolution innovations in the second set. The contours in (a) are plotted every 1 m²s⁻². The contours in (b) are plotted at ±0.2, ±0.5 and −1 m²s⁻². The deviation in (b) reaches the maximum value of 0.639 m²s⁻² (or minimum value of −1.131 m²s⁻²) at the diagonal point marked by the +(or −) sign.

Fig. 8 As in Fig. 2 but for the second set of innovations. The dotted green (or purple) curve plots C _a+(x) [or C _a− (x)] – the correlation structure intercepted from the benchmark A in Fig. 7a across the point marked by the +(or −) sign in Fig. 7b.

Fig. 9 As in Fig. 1 but for the second set of innovations.

Fig. 10 As in Fig. 6 but for the second set of innovations.

Table 2. As in Table 1 but for the second set of innovations with periodic extension

Experiment	n=20	n=50	n=100	Final
SE	0.865	0.264	0.112	0.029 at n=324
Step-I	0.509	0.405	0.419	0.423 at n=135
Step-I vs. Δa I	0.250	0.064	0.022	0.008 at n=135
TEA	0.246	0.072	0.033	0.039 at n=120
TEB	0.581	0.229	0.303	0.351 at n=103
TEC	0.498	0.293	0.316	0.323 at n=118

Fig. 11 As in Fig. 7 but without periodic extension. The contours in (b) are plotted at ±0.2, ±0.5, ±1, ±3, −5 and −7 m²s⁻².

Fig. 12 As in Fig. 10 but without periodic extension.

Table 3. As in Table 2 but without periodic extension

Experiment	n=20	n=50	n=100	Final
SE	0.753	0.267	0.127	0.017 at n=324
Step-I	0.558	0.435	0.415	0.422 at n=135
Step-I vs. Δa I	0.320	0.125	0.029	0.010 at n=135
TEA	0.305	0.128	0.049	0.029 at n=136
TEB	0.602	0.263	0.350	0.354 at n=121
TEC	0.524	0.315	0.331	0.322 at n=118

Fig. 13 (a) Uniformly distributed coarse-resolution innovation points plotted by red+signs over the analysis domain, and high-resolution innovation points plotted by the green dots in the nested domain of D _x /6=20 km long and D _y /6=10 km wide. (b) As in (a) but for the second set in which the coarse-resolution innovations are not uniformly distributed. The coarse-resolution innovations in (a) are spaced every Δx _co=6 km (or Δy _co=6 km) in the x-direction (or y-direction). The domain-averaged resolution for the non-uniformly distributed coarse-resolution innovations in (b) is estimated also by Δx _co ≈ 6 km (or Δy _co ≈ 6 km) in the x-direction (or y-direction).

Fig. 14 C _a(x−x _c) plotted by the green contours for x _c=(0, 0), and benchmark correlation pattern plotted by the black contours with x _c also placed at x=(0, 0) – the centre of the analysis domain. Here, x _c denotes the correlation centre.

Fig. 15 Deviation of σ _e ² (= 3.1527 m²s⁻²) from benchmark analysis error variance plotted by contours at 0, ±0.05, −0.1, −0.2, −0.5, −1, −3 and −5 m²s⁻² over the analysis domain.

Fig. 16 Benchmark analysis increment Δa plotted by black contours, analysis increment Δa ₂₀ from SE with 20 iterations plotted by red contours, and analysis increment Δa _A20 from TEA with 20 iterations plotted by blue contours. The rectangle plotted by thin black lines shows the nested domain.

Fig. 17 Analysis error e ₂₀ from SE and analysis error e _A20 from TEA with 20 iterations plotted by red and blue contours, respectively. The rectangle plotted by thin black lines shows the nested domain.

Table 4. As in Table 3 but for the first set of innovations in the two-dimensional case as shown in Fig. 13a

Experiment	n=20	n=50	n=100	Final
SE	0.694	0.380	0.176	0.043 at n=312
Step-I	0.727	0.636	0.589	0.588 at n=219
Step-I vs. Δa I	0.393	0.194	0.094	0.030 at n=219
TEA	0.388	0.199	0.104	0.059 at n=81
TEB	1.371	0.780	0.698	0.632 at n=242
TEC	0.615	0.500	0.443	0.448 at n=121

Fig. 18 As in Fig. 14 but for the second set of innovations.

Fig. 19 As in Fig. 15 but for the second set of innovations with contours plotted every 1 m²s⁻².

Fig. 20 As in Fig. 17 for the second set of innovations.

Table 5. As in Table 4 but for the second set of innovations as shown in Fig. 13b

Experiment	n=20	n=50	n=100	Final
SE	0.653	0.341	0.177	0.064 at n=237
Step-I	0.756	0.661	0.622	0.616 at n=201
Step-I vs. Δa I	0.406	0.210	0.088	0.042 at n=201
TEA	0.416	0.220	0.130	0.111 at n=59
TEB	1.151	0.800	0.791	0.712 at n=229
TEC	0.601	0.456	0.434	0.427 at n=135