# Parameter inversion

FastIsostasy relies on simplifications of the full GIA problem and might therefore need a calibration step to match the data, typically obtained from observations or from a "golden-standard" 3D GIA model. By means of an unscented Kalman inversion (Huang *et al.*, 2021; Huang and Huang, Feb 2021), one can, for instance, infer the appropriate field of effective mantle viscosity that matches the data best. Whereas this is known to be a tedious step, FastIsostasy is developped to ease the procedure by providing a convenience struct `InversionProblem`

. We demonstrate this on a low-resolution grid since (1) the underlying unscented Kalman filter requires many simulations and (2) estimating high-resolution viscosity field might lead to overfit the problem. The effective viscosity field we estimate(Wiens *et al.*, 2022) can be loaded by using `load_dataset`

with appropriate depths of the layer boundaries:

```
using CairoMakie
using FastIsostasy
Omega = ComputationDomain(3000e3, 5)
c = PhysicalConstants()
lb = [100e3, 200e3, 300e3]
_, eta, eta_itp = load_dataset("Wiens2022")
loglv = cat([eta_itp.(Omega.X, Omega.Y, z) for z in lb]..., dims = 3)
lv = 10 .^ loglv
p = LayeredEarth(Omega, layer_boundaries = lb, layer_viscosities = lv)
heatmap(p.effective_viscosity)
```

To make this problem more exciting, we shift the center of the ice load to $ (-1000, -1000) \: \mathrm{km} $ where the viscosity field displays a less uniform structure. For the sake of simplicity, the data to fit is obtained from a FastIsostasy simulation with the ground-truth viscosity field.

```
R, H = 1000e3, 1e3
Hcylinder = uniform_ice_cylinder(Omega, R, H, center = [-1000e3, -1000e3])
Hice = [zeros(Omega.Nx, Omega.Ny), Hcylinder, Hcylinder]
t_out = years2seconds.(collect(1e3:1e3:2e3))
pushfirst!(t_out, t_out[1]-1e-8)
t_Hice = copy(t_out)
true_viscosity = copy(p.effective_viscosity)
fip = FastIsoProblem(Omega, c, p, t_out, t_Hice, Hice)
solve!(fip)
```

```
┌ Warning: Duplicated knots were deduplicated. Use Interpolations.deduplicate_knots!(knots) explicitly to avoid this warning.
│ k1 =
│ 3-element Vector{Float64}:
│ 3.15576e10
│ 3.15576e10
│ 6.31152e10
└ @ Interpolations ~/.julia/packages/Interpolations/91PhN/src/gridded/gridded.jl:77
```

Now that we have the displacement field, we can recover the viscosity field from which it results. We therefore pass an `InversionConfig`

and an `InversionData`

to an `InversionProblem`

. Let's look at the initialized viscosity field:

```
config = InversionConfig(N_iter = 15)
data = InversionData(copy(fip.out.t[2:end]), copy(fip.out.u[2:end]), copy(Hice[2:end]),
config)
paraminv = InversionProblem(deepcopy(fip), config, data)
function plot_viscfields(paraminv)
estim_viscosity = copy(true_viscosity)
estim_viscosity[paraminv.data.idx] .= 10 .^ get_ϕ_mean_final(
paraminv.priors, paraminv.ukiobj)
cmap = cgrad(:jet, rev = true)
crange = (19.5, 21.5)
fig = Figure(size = (1800, 1000), fontsize = 40)
axs = [Axis(fig[1,i], aspect = DataAspect()) for i in 1:2]
[hidedecorations!(ax) for ax in axs]
heatmap!(axs[1], log10.(true_viscosity), colormap = cmap, colorrange = crange)
heatmap!(axs[2], log10.(estim_viscosity), colormap = cmap, colorrange = crange)
Colorbar(fig[2, :], vertical = false, colormap = cmap, colorrange = crange,
width = Relative(0.5))
return fig
end
fig1 = plot_viscfields(paraminv)
```

By performing a Kalman inversion, we can achieve a close match between ground truth and estimated viscosity field:

```
solve!(paraminv)
fig2 = plot_viscfields(paraminv)
```

This remains an academic example, where we try to recover a known parameter field from data generated by the model itself. Nonetheless, the user should get a proof-of-concept and a scheme of how to implement such a procedure themself.