Variational Methods

ThreeDVar and FourDVar minimise a variational cost function via BFGS rather than applying a Kalman update. They share the same loop interface as the ensemble filters but differ in when and how observations are assimilated.

Cost Function

Both methods minimise

\[J(x) = \frac{1}{2}(x - x_b)^\top B^{-1}(x - x_b) + \frac{1}{2}(Hx - y)^\top R^{-1}(Hx - y)\]

where $x_b$ is the background state and $B$ is the background error covariance.

• 3D-Var solves this independently at each assimilation step, using the most recent analysis as $x_b$.
• 4D-Var extends the window to $T$ steps and optimises the initial state $x_0$ globally, accumulating observations across the window.

Setup

2D kinematic model — position and velocity, observe only position:

using Opal
using LinearAlgebra
using Random

Random.seed!(42)

n = 2
m = 1
Δt = 0.1
nsteps = 50

F = [1.0 Δt;0.0 1.0]
H = [1.0 0.0]

transition = Model(F)
observation = Model(H)

σ_b = 1.0
σ_r = 0.5
B = σ_b^2 * Matrix(I(n))
R = σ_r^2 * Matrix(I(m))

x0 = [2.0,0.5]
prior = build_prior(copy(x0))

Generate observations:

true_states = [F^k * x0 for k in 1:nsteps]
noisy_obs = map(x -> H * x .+ σ_r * randn(m),true_states)
obs = stack(noisy_obs)  # m × nsteps

3D-Var

At each step t, 3D-Var solves the cost function with the single observation at t and the analysis from t-1 as background:

tdv = ThreeDVar(transition,observation,prior;B,R)
results_3d = loop(tdv,obs)
visualise(true_states,results_3d)

4D-Var

4D-Var optimises $x_0$ over the full observation window. The gradient is computed by back-propagating through the transition model via automatic differentiation:

fdv = FourDVar(transition,observation,prior;B,R)
results_4d = loop(fdv,obs)
visualise(true_states,results_4d)

For nonlinear transition models, the tangent-linear model is derived automatically from the Julia function via AD — no hand-coded adjoint is required.

Comparison