Earth Rotation Parameters Prediction and Climate Change Indicators in it

As complex variations of the Earth’s rotation, there are commonly relative regular and irregular signals coupling in EOP data series, such as the trend, annual, Chandler terms and high-frequency trembles in polar motion and the trend, interannual, seasonal and sub-seasonal oscillations in ΔLOD. For the predictions of these stable signals, we adopt the LS model expressed by polynomial trend and harmonic oscillations, and a stochastic process AR model can be employed for the predictions of irregular variations (Xu et al. 2015, 2012).

The LS model could be presented as 1 mregular(t)=a+bt+cccos(2πtT1)+cssin(2πtT1)+dccos(2πtT2)+dssin(2πtT2)+… \[\begin{matrix} {{m}_{regular}}(t)=a+bt+{{c}_{c}}\cos (\frac{2\pi t}{{{T}_{1}}})+{{c}_{s}}\sin (\frac{2\pi t}{{{T}_{1}}}) \\ +{{d}_{c}}\cos (\frac{2\pi t}{{{T}_{2}}})+{{d}_{s}}\sin (\frac{2\pi t}{{{T}_{2}}})+\ldots \\ \end{matrix}\] where

m(t) is the EOP data time series,

The rest of the LS fitting terms are then processed by the AR model; for a stationary random sequence m_iregular, the AR model is expressed as 2 mregular(t)=∑i=1Pφim(t−i)+ai \[{{m}_{regular}}(t)=\underset{i=1}{\overset{P}{\mathop \sum }}\,{{\varphi }_{i}}m(t-i)+{{a}_{i}}\] where

a is the zero-mean white noise,

In the LS + Convolution method, the EAM from ESMGFZ is selected as the input excitation series (using only the main contributions AAM and OAM). Firstly, the EAM series are predicted by the commonly used LS + AR model. Secondly, the interannual, seasonal and sub-seasonal terms of EOP are calculated from the EAM predictions by Liouville convolution equation. Meanwhile, the rest of EOP trend terms are extrapolated by the polynomial LS model. Finally, the total EOP predictions are combined by the excitation calculations and trend extensions (throughout the paper, this is abbreviated as the LS + Convolution method).

Liouville convolution equation describes the relationship between theoretical EOP m(t) and excitation function ψ(t) (Eubanks 1993; Gross 1992) as follows: 3 { m(t)=m(0)eiσct−iσc∫0tψ(t′)eiσc(t−t′)dt′m3(t)=ψ3+constant \[\{\begin{array}{*{35}{l}} \mathbf{m}(t)=m(0){{e}^{i{{\sigma }_{c}}t}}-i{{\sigma }_{c}}\underset{0}{\overset{t}{\mathop \int }}\,\psi ({{t}^{\prime }}){{e}^{i{{\sigma }_{c}}(t-{{t}^{\prime }})}}d{{t}^{\prime }} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\mathbf{m}}_{3}}(t)={{\psi }_{3}}+cons\tan t \\ \end{array}\] where

A, C are the principal moments of inertia,

Considering the viscoelasticity of the Earth, the real number σ_c should be replaced by the complex one σc=2πTc(1+i2Qc). Here, T_c = 433d and Q_c = 60 represent the period and attenuation factor of Chandler wobble, respectively (Seitz et al. 2005).

The excitation function ψ(t) in Equation 3 can be expressed as 4 {ψ1=1Ω2(C−A)(Ω2ΔI13+Ωh1+ΩΔI˙23+h˙2−Γ2)ψ2=1Ω2(C−A)(Ω2ΔI23+Ωh2−ΩΔI˙13−h˙1−Γ1)ψ3=1Ω2(C)(−Ω2ΔI33−Ωh3+Ω∫0tΓ3dt′) \[\begin{matrix} {{\psi }_{1}} & =\frac{1}{{{\Omega }^{2}}(C-A)}({{\Omega }^{2}}\Delta {{I}_{13}}+\Omega {{h}_{1}}+\Omega \Delta {{{\dot{I}}}_{23}}+{{{\dot{h}}}_{2}}-{{\Gamma }_{2}}) \\ {{\psi }_{2}} & =\frac{1}{{{\Omega }^{2}}(C-A)}({{\Omega }^{2}}\Delta {{I}_{23}}+\Omega {{h}_{2}}-\Omega \Delta {{{\dot{I}}}_{13}}-{{{\dot{h}}}_{1}}-{{\Gamma }_{1}}) \\ {} & {{\psi }_{3}}=\frac{1}{{{\Omega }^{2}}(C)}({{\Omega }^{2}}\Delta {{I}_{33}}-\Omega {{h}_{3}}+\Omega \underset{0}{\overset{t}{\mathop \int }}\,{{\Gamma }_{3}}d{{t}^{\prime }}) \\ \end{matrix}\] where

ψ(t) is the excitation source induced by surface fluid and luni-solar torques,

Regarding that there are no high-frequency terms (with period shorter than 2 days) in the conventional EOP observations, the theoretical EOP m(t) is equal to the observed EOP without more translations (Gross 1992).

Among many indicators of the prediction precision, the RMSE is a quality measure and is represented in Equation 5. 5 RMSEj=1n∑i=1n(Pij−Oij)2 \[RMS{{E}_{j}}=\sqrt{\frac{1}{n}\underset{i=1}{\overset{n}{\mathop \sum }}\,{{(P_{i}^{j}-O_{i}^{j})}^{2}}}\] where

O are the EOP observations,

Based on the correlations between EOP observations and geophysical excitation sources expressed by Liouville equation, we can calculate the EOP from excitation function directly. The excitation function series utilized in this study is the EAM from ESMGFZ, mainly arising from atmospheric and oceanic angular momentum changes. The time span of EAM series is from 1976 till now with a sampling interval of 3 h and is then averaged to 1 day to be consistent with that of EOP series. To illustrate the prominent contributions from surface fluid forcing, the interannual, seasonal and sub-seasonal variations extracted from IERS C04 EOP observations are compared in Figure 1 with the convoluted EOP from EAM series.

Figure 1.

As apparent from Figure 1a and b, after removing the trend terms, the polar motion sequences mainly contain the annual and Chandler oscillations and can be explained well by the EAM convolutions. Compared to polar motion, the EAM contributions to ΔLOD in Figure 1c are not that prominent. After removing the tidal and trend terms, ΔLOD series can be accounted for by EAM contributions mostly. The remaining discrepancy may be ascribed to EAM model errors and other sources associated with interior processes (Hsu et al. 2021; Dobslaw and Dill, 2018). In addition, the same conclusions have been confirmed with other angular momentum data sets given by the National Center for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis project and Estimating the circulation and Climate of the Ocean (ECCO) model (Chen et al. 2019; Bizouard et al. 2011).

As climate change indicators in Earth’s rotation, LOD variations on interannual scales have been discussed extensively. Besides, the latest study reveals that there are three signals (at ~6, ~7 and ~8 years) in ΔLOD interpreted as internal causes (Hsu et al. 2021), and these fluctuations need to be removed to obtain the interannual ΔLOD solely related to climatic variations. Considering all these effects from the trend terms, tidal variations and fluctuations due to internal causes, the time series of climate-related ΔLOD are displayed in Figure 2 after a rigorous data processing flowchart (see Xu et al. 2022 for more details). To show consistency between the climate-related ΔLOD and ENSO indices, the Nino 3.4 of Oceanic Niño Index (ONI) from NOAA during the same time span is also presented in Figure 2.

Figure 2.

Figure 2 shows that the climate-related ΔLOD and the ENSO indices exhibit similar positive and negative fluctuations, illustrating the speed decelerations and accelerations in Earth’s rotation and corresponding to the El Nino (warm) and La Nina (cold) events in climate change, respectively. According to the NOAA determination conditions (sea surface temperature anomaly [SSTA] ≥0.5°C or SSTA ≤−0.5°C for more than 5 months), there were 11 El Nino and 14 La Nina events over decades and three extreme El Nino events occurred during 1982–1983, 1997–1998 and 2015–2016 (Lambert et al. 2017). These events are identified in the observational record every time the equatorial Pacific SSTA is greater than 2.0°C for more than 5 months. To be specific, the clearly descending areas in interannual ΔLOD and ENSO indices are enclosed with a red square frame, pointing to the latest two La Nina events during 2020– 2022. The minimum SSTA of approximately −1.3°C and −1.0°C occurred in November 2020 and December 2021, which could be both identified as intermediate strength. Furthermore, these two La Nina events forced the interannual ΔLOD to minimum values of approximately -0.22 and -0.19 ms, separately.

To evaluate EOP predictions carried out by the LS + AR and LS + Convolution methods, we estimate accuracy of 1–90 days’ results over the second EOP PCC period of 07.2021–03.2022. These predictions were produced weekly, and the RMSE values of different forecasting spans are listed in Table 1. In order to illustrate the results more intuitively, the 1–90 days’ RMSE values calculated from the two models are graphically shown in Figure 3. The red and dot line represents the RMSE for the LS + Convolution predictions, and the blue and dot line for the LS + AR predictions.

Figure 3.

From the closely related prediction RMSE results in Table 1 and Figure 3, a preliminary finding could be summarized: in short-term EOP predictions within 10 days, the LS + Convolution (red dot line) method performs slightly better, while the traditional LS + AR model (blue dot line) shows significant improvements in medium-term predictions over 10–90 days.

Regarding the consistent climate change information shown in interannual ΔLOD and ENSO indices in Section 3.2, here we try to investigate the climate change forecasts with ΔLOD long-term predictions, focusing on the period 2020–2023. Based on the accuracy estimates listed in Table 1, the LS + AR model is selected to obtain the long-term ΔLOD predictions. To make the prediction of the ENSO events more convincing, we have tried two forecasting tests of the latest two La Nina during 2020–2021 and 2021–2022 from ΔLOD predictions of 1 year over the event. The climate-related variations extracted from different ΔLOD observations and predictions are compared in Figure 4.

Figure 4.

Figure 4 illustrates the climate-related variations extracted from ΔLOD observations (blue, green and black curves) and predictions of 1 year into the future (yellow, orange and red curves). As can be seen in black curve (c), the latest two La Nina events during the periods 2020–2021 and 2021–2022 are marked with blue, with minimum values of approximately −0.22 and −0.19 ms, respectively. Compared to the variations extracted only from ΔLOD observations, the first forecasting test indicates the latest 2020–2021 La Nina event with a weaker strength (yellow curve (a), −0.16 ms), while the second test predicts the latest 2021–2022 La Nina with a bigger amplitude (orange curve (b), −0.25 ms). In summary, these two tests prove that the climate-related variations extracted from ΔLOD predictions could be indicators of climate change. After forecasting tests of the latest two ENSO events, the climate-related terms extracted from ΔLOD predictions of 1 year over 2022–2023 are denoted in the red cure (c). It can be seen that another stronger follow-up La Nina is predicted in the next year (marked with pink, −0.27 ms). Considering that the latest 2021–2022 La Nina is still in the process of development, this predicted minimum point may be a second trough continuing from the 2021–2022 phenomenon. The latest two La Nina events were suggested to be potential indicators of the Earth’s rotation acceleration and series of extreme weather events (e.g. cold waves and strong rainfalls) in the recent two years; therefore, the next predicted trough needs more investigations and attention.