Table 4 Chosen numerical values of the regularization parameters. The values are valid for all the models built in this paper insofar as the regularization terms are applicable to the specific model

Internal field

Time-dependent

\(\lambda _t = {1.0}\,{\left( \frac{\hbox {nT}}{\hbox {yr}^{3}}\right) ^{-2}}\), \(\lambda _{t_\text {s}} = {0.03}\,{\left( \frac{\hbox {nT}}{\hbox{yr}^{2}}\right) ^{-2}}\), \(\lambda _{t_\text {e}} = {0.03}\,{\left( \frac{\hbox {nT}}{\hbox {yr}^{2}}\right) ^{-2}}\),

\(\lambda _{0} = 60\), \(\lambda _{m} = 0.65\)

External field

RC-baseline corrections

\(\lambda _{\text {ext}} = {4 \times 10^5}\,{\left (\frac{\hbox {nT}}{\hbox {yr}}\right )^{-2}}\)

\(\hbox {Calibration}^{1}\)

CryoSat-2 FGM1

\(\lambda _b = {9.1\times 10^2}\,{\left (\frac{\hbox {eu}}{\hbox {yr}}\right )^{-2}}\), \(\lambda _s = {9.1\times 10^{10}}\,{\left (\frac{\hbox {eu}}{\hbox {nTyr}}\right )^{-2}}\), \(\lambda _u = {2.8\times 10^2}\,{\left (\frac{1{}^{\circ }}{\hbox {yr}}\right )^{-2}}\)

CryoSat-2 FGM2

\(\lambda _b = {9.1\times 10^2}\,{\left (\frac{\hbox {eu}}{\hbox {yr}}\right )^{-2}}\), \(\lambda _s = {9.1\times 10^{10}}\,{\left (\frac{\hbox {eu}}{\hbox {nTyr}}\right )^{-2}}\), \(\lambda _u = {2.8\times 10^2}\,{\left (\frac{1{}^{\circ }}{\hbox {yr}}\right )^{-2}}\)

CryoSat-2 FGM3

\(\lambda _b = {9.1\times 10^2}\,{\left (\frac{\hbox {eu}}{\hbox {yr}}\right )^{-2}}\), \(\lambda _s ={9.1\times 10^{10}}\,{\left (\frac{\hbox {eu}}{\hbox {nTyr}}\right )^{-2}}\), \(\lambda _u ={2.8\times 10^2}\,{\left (\frac{1{}^{\circ }}{\hbox {yr}}\right )^{-2}}\)

GRACE-A

\(\lambda _b = {1.2\times 10^3}\,{\left (\frac{\hbox {eu}}{\hbox {yr}}\right )^{-2}}\), \(\lambda _s = {1.2\times 10^{13}}{\left (\frac{\hbox {eu}}{\hbox {nTyr}}\right )^{-2}}\), \(\lambda _u = {3.7\times 10^8}\,{\left (\frac{1{}^{\circ }}{\hbox {yr}}\right )^{-2}}\)

GRACE-B

\(\lambda _b = {1.2\times 10^3}\,{\left (\frac{\hbox {eu}}{\hbox {yr}}\right )^{-2}}\), \(\lambda _s = {1.2\times 10^{13}}\,{\left (\frac{\hbox {eu}}{\hbox {nTyr}}\right )^{-2}}\), \(\lambda _u = {3.6\times 10^8}\,{\left (\frac{1{}^{\circ }}{\hbox {yr}}\right )^{-2}}\)