First inversion with the toy Gaussian model

First inversion with the toy Gaussian model#

After running a first forward run with perturbed outputs from known emissions using the Toy Gaussian Model, you can carry out an academic inversion using these “true” observations.

This tutorial uses the 4D-VAR algorithm with the M1QN3 quasi-Newton minimizer — the most commonly used variational method in atmospheric inversions. For a comparison of all algorithms available in pyCIF, see Comparing inversion algorithms with the toy Gaussian model.

Prerequisites#

You need a completed forward simulation (see First forward simulation with the toy Gaussian Model) whose output obsvect directory will serve as the observation vector for the inversion.

Step 1 — Prepare the observation vector#

Copy the obsvect directory from the workdir of your forward simulation and rename it to ref_obsvect inside the same outdir:

cp -r $outdir/fwd_long_noMCF/obsvect $outdir/ref_obsvect

The inversion YAML references this directory via its obsvect/dir_obsvect key.

Step 2 — Set up the YAML configuration#

Download the reference YAML:

Edit the following two paths to match your environment:

rootdir: &rootdir /path/to/CIF/
outdir: &outdir   /path/to/output/dir/

Everything else — the model, observation operator, and data vector — is already configured for the same 5-day Toy Gaussian setup used in the forward tutorial.

Step 3 — Run the inversion#

python -m pycif path/to/config_inversion_long_full_4dvar_M1QN3.yml

Warning

Before running a real inversion with a new model configuration, always test the adjoint first. The Toy Gaussian model is already validated, so you can skip this step here.

Step 4 — Explore the outputs#

After convergence, workdir ($outdir/inversion_long_full_4dvar_M1QN3/) contains:

workdir/
├── controlvect/
│   ├── flux/CH4/
│   │   └── controlvect_flux_CH4.nc   optimized fluxes (x, xb, x_phys, xb_phys)
│   └── simulator/
│       ├── cost.txt                  cost function J at each iteration
│       └── gradcost.txt              gradient norm ‖∇J‖ at each iteration
└── obsoperator/
    ├── fwd_-001/obsvect/             prior simulated concentrations
    └── fwd_-002/obsvect/             posterior simulated concentrations

Open controlvect/flux/CH4/controlvect_flux_CH4.nc in any NetCDF viewer (ncview, xarray) to compare the prior fluxes xb_phys with the optimised fluxes x_phys.

Note

A perfect retrieval is unlikely with only 5 stations and a short 5-day window. Try increasing nstations in the data vector to see how observational density affects the inversion quality.

Posterior uncertainties (Monte Carlo)#

The YAML includes a mode/montecarlo block that estimates posterior uncertainties by running the inversion on perturbed copies of the observation vector. The result is stored in controlvect/flux/CH4/controlvect_flux_CH4.nc as the variable pa_std (posterior standard deviation).

To disable this (faster run), comment out the montecarlo block in the YAML.

Going further#

  • Other inversion algorithms — See Comparing inversion algorithms with the toy Gaussian model for 4D-VAR with the conjugate-gradient minimizer, the analytical solution, and the EnSRF ensemble method.

  • Spatial resolution of the control vector — The control vector can work at full-pixel resolution (hpixels), aggregated spatial bands (ibands), or a single global scalar (global). All three variants are available in Examples for dummy (look for _full_, _bands_, _global_ in the YAML names).

  • Article-quality benchmark — The CI article and article-uncertainties jobs run all algorithms across all resolutions and generate publication-quality figures. See Article-quality inversion benchmark (CI jobs).

  • All dummy YAML examples — Examples for dummy