Parameterization, sensitivity, and uncertainty of 1-D thermodynamic thin-ice thickness retrieval
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Abstract: Retrieval of Thin-Ice Thickness (TIT) using thermodynamic modeling is sensitive to the parameterization of the independent variables (coded in the model) and the uncertainty of the measured input variables. This article examines the deviation of the classical model’s TIT output when using different parameterization schemes and the sensitivity of the output to the ice thickness. Moreover, it estimates the uncertainty of the output in response to the uncertainties of the input variables. The parameterized independent variables include atmospheric longwave emissivity, air density, specific heat of air, latent heat of ice, conductivity of ice, snow depth, and snow conductivity. Measured input parameters include air temperature, ice surface temperature, and wind speed. Among the independent variables, the results show that the highest deviation is caused by adjusting the parameterization of snow conductivity and depth, followed ice conductivity. The sensitivity of the output TIT to ice thickness is highest when using parameterization of ice conductivity, atmospheric emissivity, and snow conductivity and depth. The retrieved TIT obtained using each parameterization scheme is validated using in situ measurements and satellite-retrieved data. From in situ measurements, the uncertainties of the measured air temperature and surface temperature are found to be high. The resulting uncertainties of TIT are evaluated using perturbations of the input data selected based on the probability distribution of the measurement error. The results show that the overall uncertainty of TIT to air temperature, surface temperature, and wind speed uncertainty is around 0.09 m, 0.049 m, and −0.005 m, respectively.
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Figure 1. The geographic locations of the sonar sites (a), the IceBridge footprints (b), the IABP buoys (c), and the wind speed stations (d) used in this study.
Figure 3. Relative deviation of the model output TIT using each test scheme (T1, T2, …, T14) from the TIT obtained using the default scheme. The solid points are mean values of the deviation, and the vertical lines represent one standard deviation. The colors denote the upper limit of the different TIT bins, from 0.1 m to 0.5 m.
Figure 4. TIT from the 1-D model versus the ULS data, obtained for different TIT intervals. The solid dots represent the mean value from the model, and the error bars represent 0.5 standard deviation. Dotted line indicates diagonal line.
Figure 5. Comparison between the model-retrieved TIT using the combination of schemes T5 and T9 and the TIT from the ULS, IceBridge (IB), and SMOS/SMAP products, respectively. Cmb. denotes the combination scheme. The interval between the data on the horizontal scale is 0.05 m. The error bars represent 0.5 standard deviation. Dotted line indicates diagonal line.
Figure 6. Comparisons of ERA5
$ {T}_{{\mathrm{a}}} $ vs. IABP$ {T}_{{\mathrm{a}}} $ (a), MODIS$ {T}_{{\mathrm{s}}} $ vs. IABP$ {T}_{{\mathrm{s}}} $ (c), and ERA5$ u $ vs. NWS-observed (Obs.)$ u $ (e), and their corresponding probability distributions of the differences, with fitted curves (black lines) (b, d, and f). The error bars represent one standard deviation. The numbers between brackets in a and e denote the data counts. a and c share the same data counts. The numbers in b, d, and f at the peak of each curve are the mean value of the differences. Dotted line in a indicates diagonal line.
Figure 7. Plot between
$ {T}_{{\mathrm{s}}} $ from MODIS and ERA5 (a), and histogram of the difference between the measurements (b). The shadow colors in a show the counts of data. The error bars represent one standard deviation. The black line in b is the fitted Gaussian curves, and the number at the peak is the mean difference. Dotted line in a indicates diagonal line.
Figure 8. Plots showing the retrieved TIT obtained using the original Ts, Ta, and u as inputs (values in the horizontal axis) and the retrieved TIT with perturbations added representing the error in measurements from Ta (with respect to IABP Ta) (a), Ts (with respect to IABP Ts) (b), u (with respect to the NWS/NDBC observations) (c), and Ts (with respect to ERA5 skin temperature) (d). The error bars represent one standard deviation. The shadow colors show the counts of the data points. Dotted lines indicate diagonal lines.
Table 1. Summary of the previous uncertainty in TIR-TIT retrieval
Reference Data source (data volume) Uncertainty analysis scheme Finding Yu and Rothrock (1996) AVHRR (13 images) $ {T}_{{\mathrm{s}}} $ ± 1 K, $ {T}_{{\mathrm{a}}} $ ± 1.6 K,
$ u $ ± 2 m/s(1) The uncertainty of the TIT caused by the error in the variables increases with the ice thickness. (2) The uncertainty of the cumulative distribution is no more than 3% for ice thinner than 0.2 m. Willmes et al. (2010) AVHRR + NCEP
(1 image)$ {T}_{{\mathrm{a}}} $ ± 5 K, $ u $ ± 3 m/s Results in max. TIT errors of ±20% (for TIT ≤ 0.5 m) Wang et al. (2010) MODIS (case study) $ {T}_{{\mathrm{s}}} $ ± 2 K, $ u $ ± 1 m/s For TIT < 0.3 m, TIT is −0.172 m or +0.179 m when $ {T}_{{\mathrm{s}}} $ + 2 K or Ts − 2 K; TIT is + 0.166 m or −0.133 m when $ u $ + 1 m/s or −1 m/s. Mäkynen et al. (2013) MODIS + HIRLAM
(199 images)– The largest TIT uncertainty comes from air temperature. Adams et al. (2013) MODIS + NCEP/COSMO (two-winter data) $ {T}_{{\mathrm{s}}} $ ± 1.6 K, $ {T}_{{\mathrm{a}}} $ ± 4.5 K;
$ u $ ± 1.3 m/s, NCEP vs. COSMO(1) For all the variables’ uncertainties, the TIT varies by ±0.37 m for TIT of ≤ 0.5 m. (2) The NCEP $ {T}_{{\mathrm{a}}} $ leads to overestimated ice thicknesses, in comparison to COSMO $ {T}_{{\mathrm{a}}} $. Zeng et al. (2016) MODIS + ERA-interim
(four images)Altering the value of $ {T}_{{\mathrm{a}}}-{T}_{{\mathrm{s}}} $ (1) For Ta − Ts > +3 K, the Ta − Ts error of only +1 K causes the TIT to
vary by +0.1 m. (2) For Ta − Ts ≤ 0 K or Ta − Ts > +2 K, the TIT decreases by 1 cm to 2 cm or 9 cm when u increases from 0 m/s to 12 m/s.Note: The symbols are specified in Table A1 in Appendix. The data sources of AVHRR or MODIS are used for $ {T}_{{\mathrm{s}}} $, and the National Centers for Environmental Prediction (NCEP), High Resolution Limited Area Model (HIRLAM), or European Centre for Medium-Range Weather Forecasts (ECMWF) Re-analysis (ERA)-Interim reanalysis data are the other meteorological input variables of the 1-D model. In the third column, the regular font denotes the analysis scheme for the errors in the input variables, the values on the right side of the positive and negative signs “±” are the analyzed variable’s errors, and the bold font is for comparing the sources of the different input variables. − means no uncertainty analysis scheme. 
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Table 2. Specifications of the sonar measurements used in this study
Data source Data form Number of sites Data period Number of samples Nominal error/cm NPEO 10-min ice draft data 2 2002, 2005 10 10 BGEP daily average draft statistics 34 2003–2021 737 5–10 IOS-EBS 4-min ice draft data 1 2003 1724 5 IOP 5-min ice draft data 5 2005–2008 3305 10 Note: NPEO is the abbreviation for the North Pole Environmental Observatory (Morison, 2009). BGEP denotes the Beaufort Gyre Exploration Project. IOS and EBS are the short forms for the Institute of Ocean Sciences and the Eastern Beaufort Sea, respectively (Melling and Riedel, 2008). IOP denotes the Integrative Observational Platforms from University of Washington.
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Table 3. Station ID and geographic locations of the wind speed observation stations
Data source Station ID Latitude and longitude Temporal resolution NWS 99950 71.00°N, 8.46°E 1 h 99710 74.50°N, 19.00°E 1 h 99720 76.51°N, 25.01°E 1 h 99740 78.91°N, 28.89°E 1 h 99752 76.47°N, 16.54°E 1 h 99790 78.07°N, 13.63°E 1 h 99927 80.06°N, 16.24°E 1 h 99935 80.65°N, 25.00°E 1 h 99938 80.10°N, 31.46°E 1 h NDBC ULRA2 63.87°N, 160.78°W 6 min PRDA2 70.40°N, 148.53°W 6 min
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Table 4. The parameterized variable test schemes
Test scheme Parameter Default scheme Parameterization scheme Parameter name Equation number T1 atmospheric emissivity ($ {\varepsilon }_{{\mathrm{a}}} $) $ {\varepsilon }_{{\mathrm{a}}\_\mathrm{J}\mathrm{X}06} $ (+${E}_{\mathrm{A}\mathrm{E}96}^{'}$) $ {\varepsilon }_{{\mathrm{a}}\_\mathrm{EF}61}(+{E}_{\mathrm{AE}96}^{'}) $ Eq. (A1) [+ Eq. (A6)] T2 atmospheric emissivity ($ {\varepsilon }_{{\mathrm{a}}} $) $ {\varepsilon }_{{\mathrm{a}}\_\mathrm{J}\mathrm{X}06} $ (+${E}_{\mathrm{A}\mathrm{E}96}^{'}$) $ {\varepsilon }_{{\mathrm{a}}\_\mathrm{K}\mathrm{L}94} $ Eq. (A2) T3 atmospheric emissivity ($ {\varepsilon }_{{\mathrm{a}}} $) $ {\varepsilon }_{{\mathrm{a}}\_\mathrm{J}\mathrm{X}06} $ (+${E}_{\mathrm{A}\mathrm{E}96}^{'}$) $ {\varepsilon }_{{\mathrm{a}}\_\mathrm{J}\mathrm{X}06} $ (+${E}_{\mathrm{M}78}$) (denoted as $ {\varepsilon }_{{\mathrm{a}}\_{\mathrm{J/M}}} $) Eq. (A3) [+ Eq. (A4)] T4 atmospheric emissivity ($ {\varepsilon }_{{\mathrm{a}}} $) $ {\varepsilon }_{{\mathrm{a}}\_\mathrm{J}\mathrm{X}06} $ (+${E}_{\mathrm{A}\mathrm{E}96}^{'}$) $ {\varepsilon }_{{\mathrm{a}}\_\mathrm{JX}06} $ (+${E}_{\mathrm{AE}96}$) ($ {\varepsilon }_{{\mathrm{a}}\_{\mathrm{J/A}}} $) Eq. (A3) [+ Eq. (A5)] T5 air density ($ {\rho }_{{\mathrm{a}}} $, $ \mathrm{k}\mathrm{g}/{\mathrm{m}}^{3} $) $ {\rho }_{{\mathrm{a}}}=1.3 $ kg/m3 $ {\rho }_{{\mathrm{a}}\_{\mathrm{GL}}} $ Eq. (A7) T6 specific heat of air [$ {c}_{{\mathrm{p}}} $, $ \mathrm{J}/(\mathrm{k}\mathrm{g} \cdot \mathrm{K}) $] $ {c}_{{\mathrm{p}}}= $ 1004 J/(kg·K)$ {{c}}_{\mathrm{pw}} $ Eq. (A8) T7 latent heat of sublimation ($ {L}_{{\mathrm{s}}} $,$ \mathrm{J}/\mathrm{k}\mathrm{g} $) $ {L}_{{\mathrm{s}}}=2.5\times {10}^{6} $ J/kg $ {L}_{{\mathrm{sw}}} $ Eq. (A9) T8 ice conductivity [$ {k}_{{\mathrm{i}}} $, $ \mathrm{W}/(\mathrm{m} \cdot \mathrm{K}) $] $ {k}_{{\mathrm{i}}}=2.03 $ W/(m·K) $ {k}_{{\mathrm{i}}\_\mathrm{U}64} $ (+$ {k}_{0\_\mathrm{N}64}+{S}_{{\mathrm{i}}\_\mathrm{J}94} $) ($ {k}_{{\mathrm{i}}\_{\mathrm{UNJ}}} $) Eq. (A10) [+Eq. (A11) + Eq. (A15)] T9 ice conductivity [$ {k}_{{\mathrm{i}}} $, $ \mathrm{W}/(\mathrm{m} \cdot \mathrm{K}) $] $ {k}_{{\mathrm{i}}}=2.03 $ W/(m·K) $ {k}_{{\mathrm{i}}\_\mathrm{U}64} $ (+$ {k}_{0\_\mathrm{S}78}+{S}_{{\mathrm{i}}\_\mathrm{J}94} $) ($ {k}_{{\mathrm{i}}\_{\mathrm{USJ}}} $) Eq. (A10) [+Eq. (A12) + Eq. (A15)] T10 ice conductivity [$ {k}_{{\mathrm{i}}} $, $ \mathrm{W}/(\mathrm{m} \cdot \mathrm{K}) $] $ {k}_{{\mathrm{i}}}=2.03 $ W/(m·K) $ {k}_{{\mathrm{i}}\_\mathrm{U}64} $ (+$ {k}_{0\_\mathrm{C}10}+{S}_{{\mathrm{i}}\_\mathrm{J}94} $) ($ {k}_{{\mathrm{i}}\_{\mathrm{UCJ}}} $) Eq. (A10) [+Eq. (A13) + Eq. (A15)] T11 ice conductivity [$ {k}_{{\mathrm{i}}} $, $ \mathrm{W}/(\mathrm{m} \cdot \mathrm{K}) $] $ {k}_{{\mathrm{i}}}=2.03 $ W/(m·K) $ {k}_{{\mathrm{i}}\_\mathrm{U}64} $ (+$ {k}_{0\_\mathrm{C}10}+{S}_{{\mathrm{i}}\_\mathrm{C}74} $) ($ {k}_{{\mathrm{i}}\_{\mathrm{UCC}}} $) Eq. (A10) [+Eq. (A13) + Eq. (A14)] T12 snow depth ($ {h}_{{\mathrm{s}}} $, m) $ {h}_{{\mathrm{s}}}=0 $ m $ {h}_{{\mathrm{s}}\_\mathrm{D}71}\;(+{k}_{{\mathrm{s1}}}) $ Eq. (A16) T13 snow depth ($ {h}_{{\mathrm{s}}} $, m) $ {h}_{{\mathrm{s}}}=0 $ m $ {h}_{{\mathrm{s}}\_\mathrm{M}19}\;(+{k}_{{\mathrm{s1}}}) $ Eq. (A17) T14 snow conductivity [$ {k}_{{\mathrm{s}}} $, $ \mathrm{W}/(\mathrm{m} \cdot \mathrm{K}) $] $ {k}_{{\mathrm{s}}1}=0.33 (+{h}_{{\mathrm{s}}\_{\mathrm{M}}19}) $ $ {k}_{{\mathrm{s2}}}=0.21\;(+{h}_{{\mathrm{s}}\_\mathrm{M}19}) $ [+ Eq. (A17)] Note: All the symbols and equations in the third and fourth columns are introduced in Appendix.
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Table 5. The mean deviation of TIT for the 14 parameterization schemes with respect to the default scheme
Test scheme Mean deviation of TIT/% TIT=0–0.1 m TIT=0.2–0.3 m TIT=0.4–0.5 m TIT=0–0.5 m T1 11.18 12.81 14.34 12.49 T2 16.11 22.20 17.20 19.25 T3 0.10 −0.15 −0.46 −0.16 T4 −0.10 −0.14 −0.20 −0.14 T5 −2.99 −3.40 −1.92 −3.00 T6 −0.03 −0.01 0.00 −0.01 T7 −0.09 −0.08 −0.02 −0.07 T8 −9.77 9.31 10.50 4.79 T9 −23.40 −1.40 0.95 −5.95 T10 −13.84 10.09 11.65 4.29 T11 −10.72 9.93 11.70 5.05 T12 −8.09 −23.52 −38.09 −26.02 T13 −8.09 −33.34 −35.63 −28.23 T14 −9.25 −41.79 −46.52 −36.44 Note: Calculations are for the three shown ice thickness bins as well as the average TIT over the entire thickness range of 0–0.5 m.
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Table 6. The sensitivity of the model’s output to increments in TIT of 0.1 m for the 14 parameterization equation schemes
Test scheme Sensitivity/% Test scheme Sensitivity/% T1 0.33 T8 3.64 T2 0.79 T9 4.36 T3 −0.07 T10 4.53 T4 −0.01 T11 3.94 T5 0 T12 −4.48 T6 0 T13 −5.03 T7 0 T14 −6.80
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Table 7. Deviation of the model TIT with respect to the ULS TIT based on the different parameterization schemes
Scheme Deviation of the model TIT/% Ice bin =
0–5 cmIce bin =
5–10 cmIce bin =
10–15 cmIce bin =
15–20 cmIce bin =
20–25 cmIce bin =
25–30 cmIce bin =
30–35 cmIce bin =
35–40 cmIce bin =
40–45 cmDef. 9.35 1.92 0.94 0.28 0.06 −0.04 −0.25 −0.18 −0.32 T1 10.65 2.43 1.05 0.39 0.16 0.00 −0.18 −0.13 −0.25 T2 10.21 2.13 1.02 0.48 0.14 0.05 −0.11 −0.05 −0.25 T5 10.73 2.27 1.02 0.54 0.15 0.04 −0.14 −0.03 −0.26 T8 9.62 1.99 0.98 0.31 0.09 −0.03 −0.23 −0.17 −0.30 T9 10.59 2.28 1.04 0.45 0.16 0.04 −0.17 −0.09 −0.24 T12 10.38 2.48 1.04 0.33 0.09 −0.03 −0.23 −0.16 −0.28 T13 10.17 2.64 1.18 0.28 0.02 −0.08 −0.25 −0.31 −0.29 T14 10.40 2.36 1.11 0.24 −0.04 −0.09 −0.27 −0.31 −0.34 Note: The deviation was calculated according to Eq. (7) in Section 3.
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Table 8. Statistics of the difference between the model and ULS TIT based on the different parameterization schemes
Scheme Bias/m RMSE/m MAE/m R2 ρ Def. 0.027 0.103 0.088 0.741 0.861 T1 0.031 0.101 0.086 0.766 0.875 T2 0.032 0.106 0.089 0.693 0.832 T5 0.010 0.100 0.087 0.766 0.875 T8 0.028 0.101 0.088 0.755 0.869 T9 0.018 0.106 0.091 0.663 0.814 T12 0.004 0.120 0.102 0.367 0.606 T13 −0.004 0.120 0.102 0.407 0.638 T14 −0.018 0.118 0.102 0.511 0.715 Note: RMSE is root-mean-square error, MAE is mean absolute error, R2 is the coefficient of determination, ρ is the Pearson’s linear correlation coefficient. The p-value for each scheme is no more than 0.05.
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Table 9. Bias of
$ {T}_{{\mathrm{a}}} $ (ERA5 minus IABP measurements) and$ {T}_{{\mathrm{s}}} $ (MODIS minus IABP) for the different ice bins and temperature binsInput variable Ice bin/m Bias for different temperature bins/K Bias for all temperature bins/K $ \rho $ 240–245 K 245–250 K 250–255 K 255–260 K 260–265 K 265–270 K Ta 0–0.1 – −2.08 −1.17 −1.96 −2.84 −3.59 −2.24 0.86 Ta 0.1–0.2 – −1.28 −2.68 −2.64 −4.91 −4.23 −3.19 0.81 Ta 0.2–0.3 6.62 −0.95 −4.01 −3.39 −5.53 −4.59 −3.81 0.82 Ta 0.3–0.4 3.42 −2.25 −3.39 −4.92 −5.93 −4.98 −3.80 0.76 Ta 0.4–0.5 3.77 −3.13 −3.36 −5.41 −5.57 −4.45 −4.05 0.71 Ta 0–0.5 4.06 −2.11 −3.36 −3.87 −5.31 −4.44 −3.62 0.75 Ts 0.2–0.3 5.71 3.94 −0.50 −3.69 – – −0.57 0.47 Ts 0.3–0.4 5.43 −0.02 −0.62 −5.13 – – −1.67 0.61 Ts 0.4–0.5 3.16 −0.90 −0.59 −6.54 – – −3.78 0.62 Ts 0–0.5 4.72 0.89 0.08 −3.88 – – −1.38 0.53 Note: Correlation coefficients $ \rho $ of the data from the two sources are also shown in the last column. – indicate no data.
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Table 10. Statistics of the difference between ERA5
$ u $ and the observed$ u $ for different ice bins and wind speed binsInput variable Ice bin/m Bias for different wind speed bins/(m·s−1) Bias for all wind speed bins/(m·s−1) $ \rho $ 0–5 m/s 5–10 m/s 10–15 m/s 15–20 m/s u 0–0.1 0.93 −1.10 −3.61 −5.12 −1.17 0.65 u 0.1–0.2 1.16 −0.30 −3.76 −4.74 0.07 0.68 u 0.2–0.3 0.99 −0.54 −2.67 −4.95 −0.15 0.74 u 0.3–0.4 0.81 −0.94 −3.13 −5.71 −0.41 0.72 u 0.4–0.5 0.65 −0.99 −4.48 −3.80 −0.35 0.69 u 0–0.5 0.94 −0.93 −3.54 −5.07 −0.75 0.68
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Table 11. Uncertainty (bias) of TIT due to the uncertainty of the input variables of
$ {T}_{{\mathrm{a}}} $ ,$ {T}_{{\mathrm{s}}} $ , and$ u $ Ice bin/m Bias of TIT/m Ta Ts (vs. IABP) Ts (vs. ERA5) u 0–0.1 0.047 −0.027 0.068 0 0.1–0.2 0.117 −0.100 0.088 −0.002 0.2–0.3 0.127 −0.178 0.065 −0.006 0.3–0.4 0.104 −0.261 0.017 −0.009 0.4–0.5 0.064 −0.346 −0.045 −0.010 0–0.5 0.090 −0.200 0.049 −0.005 Note: Columns 2–5 correspond to the uncertainty of the input variables in Figs 6b, d, f, and 7b.
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