ESMGFZ EAM Products for EOP Prediction: Toward the Quantification of Time Variable EAM Forecast Errors

Time variations in the orientation of the solid Earth are largely governed by the exchange of angular momentum with the surface geophysical fluids of atmosphere, oceans, and the land surface. Modeled fields of atmospheric winds, atmospheric surface pressure, ocean currents, ocean bottom pressure, and terrestrial water storage allow to calculate effective angular momentum (EAM) functions that provide highly reliable information about changes in the orientation of the Earth’s rotation axis and rotational speed. The influence of atmospheric surface pressure and winds on variations in Earth rotation has been already demonstrated by Barnes et al. (1983), who introduced EAM for the first time. EAM summarizes the angular momentum changes from mass redistributions in any of the Earth subsystem’s atmosphere, ocean, and land hydrology to the overall effect on the orientation of the solid Earth with respect to the celestial reference frame (Brzezinski, 1992). In addition to the matter term related to the time variable mass distribution, EAM also contains the motion term derived from relative angular momentum changes such as vertically integrated zonal and meridional atmospheric winds. EAM functions intrinsically consider a partly decoupled rotation of the Earth's core, the effect of elastic Earth surface deformations under atmospheric pressure, and rotational deformations.

EAM can be also calculated from atmospheric, oceanic, and hydrological model forecasts. Comparing such EAM forecasts with subsequently calculated EAM analysis shows significant high prediction skills for about 5 days into the future. For longer prediction horizons, EAM forecast prediction skills decrease rapidly for atmospheric angular momentum (AAM), but for oceanic angular momentum (OAM) and hydrological angular momentum (HAM), there is still some useful information on longer prediction horizons, indicating that low-frequency oceanic and hydrospheric processes can substantially extend the window of predictability (Dill et al., 2013). In recent years, more and more studies included such EAM forecasts into Earth’s orientation prediction approaches. Especially, the short-term predictions of variations in the Earth’s spin rate universal time–coordinated universal time (UT1–UTC) are dominated by the third component of AAM forecasts. UT1 prediction errors were reduced by 20% at a forecast horizon of 5 days (Bell et al., 1991). However, it also showed that including the AAM forecasts sometimes degraded the UT1 prediction skill due to systematic differences between AAM forecasts and the UT1 series (Freedman et al., 1994). Smoothing of the AAM data to reduce the sub-daily variability helped to mitigate those effects, albeit on the expense of losing temporal detail. Using neural network prediction approaches, the system can be trained to ignore AAM forecast errors while extracting only the reliable information (Modiri et al., 2020, Kiani et al., 2022a).

Not only UT1 and length of day (LOD) predictions, but also the polar motion predictions could be improved by AAM forecasts. Moreover, OAM and HAM forecast data have become very valuable for better Earth orientation parameter (EOP) prediction (Dill & Dobslaw, 2010). Although EAM forecasts have typically a forecast horizon of only several days, 90-day EOP predictions benefit as well from including EAM forecast data in the EOP prediction approach (Dill et al., 2018).

Nowadays, EAM forecasts are frequently used in many advanced EOP prediction methods (see e.g., the second EOP prediction comparison campaign [EOPPCC], http://eoppcc.cbk.waw.pl/) including machine learning and Kalman filter techniques. All prediction techniques that show as low mean absolute errors as the official prediction of the International Earth Rotation and Reference Systems Service (IERS), called bulletin A, or even better use at least AAM forecasts or all available EAM forecasts (Kur et al., 2022). Best prediction methods also implicitly consider the EAM forecast errors by a combination of AAM forecasts with AAM analysis data (Dill et al., 2019), elimination of AAM wind term forecast errors by a cascade forward neural network (Dill et al., 2021), or generating improved EAM forecasts using machine learning (Kiani et al., 2022b; Gou et al, 2023). Although such new methods can reduce the influence of existing EAM forecast errors, a more detailed picture of the EAM forecast uncertainties is hidden in such approaches. Hence, in this contribution, we will focus more closely on the differences between EAM forecasts and subsequently available EAM analysis data as processed by the Earth system modeling group at the GFZ German Research Centre for Geosciences (ESMGFZ, http://esmdata.gfz-potsdam.de:8080/repository). Knowing EAM forecast uncertainties, they can be directly introduced into the prediction by, for example, a Kalman filter (Freedman et al., 1994) or a probabilistic machine learning method (Kiani & Soja, 2022).

Many ESMGFZ EAM forecasts are characterized by continuously growing forecast errors with increasing lead time. This particularly applies not only to all OAM, HAM, and sea-level angular momentum (SLAM) components, X₁, X₂, X₃, mass, and motion terms, but also to the X₁ and X₂ mass term forecasts of the atmosphere, which all show excellent prediction skills with Brier skill scores (von Storch and Zwiers, 1999) well above 0.8, even beyond day 6. In contrast to this good performance, AAM X₁ and X₂ motion term forecasts are characterized by substantially lower prediction skills. Brier skill scores frequently drop below zero during the first forecast days already (see Fig. 3 in Dobslaw & Dill, 2017). Quasi-periodic variations present in the forecasts that are not replicated in the later available analysis data substantially degrade EAM forecast skills, and consequently EOP prediction.

When plotting the differences between AAM forecasts and associated analyses over 7 years (Figure 1), we find that AAM forecast equatorial wind terms (AAM X₁ and X₂ motion) deviate largely from the analysis in an apparently systematic way, especially in the very first forecast epochs. In contrast, for the other AAM terms and also for all OAM and HAM terms, the differences between forecast and analysis (see Figures 2 and 3) do not contain such features, but show a rather random distribution that gradually increases with prediction length. For this study, we consider the differences between forecast and analysis as a measure of the forecast error.

Figure 1.

Figure 2.

Figure 3.

In the AAM X₁ and X₂ motion term forecasts, we found artificial quasi-periodic signals with initial amplitudes larger than the increasing stochastic forecast error, with an average period of 1.071 days in X₁ and 1.098 days in X₂. From day to day, this artificial signal is excited irregularly in the numerical weather prediction system of the European Center for Medium Weather Forecast (ECMWF) with seemingly arbitrary amplitude and phase. This signal, if excited, is somewhat dampened out with increasing forecast length.

When calculating the average of all daily forecast differences, we obtain almost zero for all individual forecast epochs in the case of a stochastic error distribution. This is not the case for AAM X₁ and X₂ motion term forecasts. We denote this kind of error distribution as systematic error.

2.3.

Stochastic error – level of uncertainty prediction

The stochastic error in EAM forecasts generally increases with forecast length; however, we found very different error levels in forecasts of subsequent days. A simple linearly increasing error level would not describe sufficiently the error level of individual days. Therefore, we started to characterize those different error patterns using a feed forward neural network.

Subsequently, we do not train the network with the differences between forecasts and analysis as for the systematic error, but instead use only the absolute value of those differences. We set up a feed forward network using daily 6-day forecasts and the analysis data of the last 36 days. We calculated 30 sets of absolute values from the difference between 30 individual 6-day forecasts and the related analysis data. Including the initial value before the forecast starts, they have 49 epochs with three-hourly resolution each. In addition, the network is fed with one new EAM forecast, 6 days with 48 3-hourly epochs, for which we want to estimate the unknown forecast uncertainty level (Figure 4). The first and second hidden layers consist of 5 and 10 neurons, respectively. The target is defined as the level of uncertainty, which is the absolute value of the true forecast error. Figure 5 shows exemplarily the predicted level of uncertainty (blue) against the true forecast error (red) for eight different epochs of AAM X1 mass term forecasts. Transferable results were obtained for all the other components of EAM mass and motion terms. We are not able to predict the forecast uncertainty for all sub-daily variations in detail, but the uncertainty prediction gives a very good first guess of the variable uncertainty level for each different AAM forecast. We were able to capture also the totally different error evolution over the whole range of each 6-day forecast between consecutive days.

Figure 4.

Figure 5.

While analyzing EAM forecast errors contained in the ESMGFZ prediction products, we found predominantly normal error distributions in the OAM and HAM mass and motion terms. As expected, the error increases generally with prediction length. This error is introduced in the ocean and hydrological model forecast runs by the rapidly decreasing prediction skill after day 5 of the atmospheric forecast data, which is the most important input to both the ocean and the land surface model. Long-term processes excited in the ocean and hydrological model are able to enhance the model-derived EAM forecast prediction skill after 5 days into the future. However, we also found large systematic error contributions in the AAM X₁ and X₂ motion terms that dominate the very first forecast epochs. This error originates in the artificial wind signals that are excited in the ECMWF atmospheric weather forecast system after the end of the data assimilation window that constrains the ECMWF model. We were able to identify the patterns of such systematic error contributions with a machine learning approach. Such a systematic error can be predicted for every new forecast depending on the characteristics of a few past forecasts. Correcting for the systematic error, the prediction skill of AAM equatorial wind terms is significantly improved, leading finally to improved polar motion predictions during the first 10 forecast days. The remaining error follows a more or less stochastic error distribution. The stochastic error part cannot be predicted and removed, but we were able to estimate at least the level of uncertainty for the EAM forecasts, apart from sub-daily variations. One might overcome this drawback by using only daily values of EAM forecasts for daily sampled EOP predictions. As soon as we implement the EAM forecast error estimation in the routinely processing environment, we will provide these uncertainty estimates along with our deterministic EAM forecast products for further use in EOP prediction methods like, for example, Kalman filters.