Chapter 16 · Adjoint sensitivity and optimal perturbations

Overview

You have one extra observation to place, anywhere in the domain, and it will be taken now. Where should it go?

Chapter 15 answered a nearby question — what does this forecast quantity depend on? — with the gradient $\partial J/\partial x_0$. That is not quite what an observing plan needs. The gradient says where an error would matter most; it does not say where an error is most likely to grow.

All three come out of the same propagator, and this chapter separates them.

The model

Over a finite window $\mathbf{M}(x_0,\tau)$ maps a unit sphere onto an ellipsoid, and its SVD names the axes: $v_1$ is the initial perturbation that grows most, $u_1$ is what it becomes, and $\sigma_1$ is the amplification. At operational size $\mathbf{M}$ is never formed — $\sigma_1$ and $v_1$ come from iterative methods needing only the action of $\mathbf{M}$ and $\mathbf{M}^{\top}$ on a vector, which is one tangent linear and one adjoint integration each. That is why chapter 15 came first.

Three results

Optimal growth beats the Lyapunov estimate, systematically. Averaged over 33 base points on the attractor, $\sigma_1$ exceeds $e^{\lambda_1\tau}$ by a factor of 1.6–2.6 at every window tested, and the effective rate $\ln\sigma_1/\tau$ falls from 3.6 at $\tau = 0.25$ to 1.0 at $\tau = 8$ — still above $\lambda_1 = 0.906$, and only there. The two agree asymptotically, and “asymptotically” is well beyond any useful forecast range. This is non-normality: $\mathbf{M}$ is not symmetric, so its singular vectors are not its eigenvectors, and a well-chosen direction transiently amplifies faster than the asymptotic rate of any direction. It is why operational centres perturb along singular vectors rather than randomly.

The averaging is not cosmetic. At a single base point $\sigma_1(\tau)$ is not even monotonic, and beyond about τ = 3 MTU it can fall below $e^{\lambda_1\tau}$ — because $\lambda_1$ is a long-time average and one particular stretch of trajectory may be quieter than average. The inequality is a statement about the attractor, not about any one window.

“Fastest-growing” is undefined without a norm. Singular vectors solve $\max_v \|\mathbf{M}v\|_E/\|v\|_E$, which depends on $\mathbf{E}$. Weighting one component by 25 rotates the optimal direction by tens of degrees. Operational singular vectors are computed in a total-energy norm, and that choice was argued over for years, because a norm favouring small scales produces perturbations that grow impressively and matter little. Choosing $\mathbf{E}$ is choosing what “an important error” means.

Sensitivity is not growth — but you cannot see that in Lorenz 63. Expanding the gradient in the singular basis, $\mathbf{M}^{\top}\partial J/\partial x_\tau = \sum_i \sigma_i (u_i\cdot\partial J/\partial x_\tau)\,v_i$, shows two conditions for $v_1$ to dominate: $\sigma_1 \gg \sigma_2$, and the metric must overlap $u_1$. In Lorenz 63 both hold for any natural metric ($\sigma_1/\sigma_2 \approx 34$), so the gradient lies within a few degrees of $v_1$ and the two questions look interchangeable. In Lorenz 96 at τ = 0.5, with $\sigma_1/\sigma_2 \approx 1.8$ and a local metric overlapping $u_1$ by 0.003, the angle is 88.8° — near-orthogonal.

Nor is that a short-window artefact that a longer optimisation cures. Stepping the window from 0.25 to 2 MTU gives 88.6°, 88.8°, 68.6°, 24.6°, 64.8°, 84.1°, 85.2°, 85.9°: one dip, at τ = 1, where the overlap happens to reach 0.44. $\sigma_1/\sigma_2$ stays between 1.1 and 2.9 the whole way, so the angle is set by the overlap alone — by whether the fastest-growing structure happens to land on the site being forecast. In high dimension with a local metric, near-orthogonality is generic and agreement is coincidence. The distinction is real; it is invisible in a three-variable model, which is a useful reminder about what low-order models can and cannot show.

Exercises

Analytic. Derive the singular-basis expansion of the gradient above, and state the two conditions under which $\nabla J \parallel v_1$. Then construct a metric for which the angle is exactly 90°, and explain why it is not a metric anyone would forecast.

Computational. Verify that the returned $\sigma_1$ is the amplification actually achieved in the chosen norm — $\|\mathbf{M}v_1\|_E/\|v_1\|_E$ — for a diagonal and for a full symmetric positive definite weight.

Exploratory. Predict how the Lorenz 96 gradient–singular-vector angle changes as the window grows, then step through the whole range. Reconcile what you see with the $\sigma_1/\sigma_2$ argument, and say what the result implies for designing an observing network from singular vectors alone.

Further reading

• Buizza, R. and Palmer, T. N. (1995). The singular-vector structure of the atmospheric global circulation. Journal of the Atmospheric Sciences, 52, 1434–1456.
• Palmer, T. N., Gelaro, R., Barkmeijer, J. and Buizza, R. (1998). Singular vectors, metrics, and adaptive observations. Journal of the Atmospheric Sciences, 55, 633–653 — the norm question, argued properly.
• Errico, R. M. (1997). What is an adjoint model? BAMS, 78, 2577–2591.
• Kalnay, E. (2003). Atmospheric Modeling, Data Assimilation and Predictability, §6.4.
• Palmer, T. and Hagedorn, R., eds. (2006). Predictability of Weather and Climate, ch. 5 [citation needed: pages].