Based on the assumption that similar conditions trigger a comparable deformation response , conditions and responses may be categorized. The current slope conditions are then matched against historic conditions, and the deformation response is assumed to be the same. Extrapolation of the response to previously unencountered conditions is typically impossible with these models. However, the system will therefore also not yield unrealistic results and could be considered bound to the previously encountered deformation signal.

The simplest, linear, model is the weighted sum of the quantified conditions at the slope. However, the slope response may not be linear and is typically not immediate. Neural networks may be used to estimate any signal by the formation of a network of interlinked nodes that ingest and combine the conditions on the slope in subsequent layers of nodes . A time series passed to a single input neuron is equal to a weighted sum of the time series plus a bias.

As more hidden layers of neurons are introduced to the system, the direct link to the (time series) input is lost as combinations are made. Furthermore, an activation function may be applied to scale the output of each node, especially to normalize the response and filter outliers, at the cost of introducing non-linearity to the system. The number of parameters, degrees of freedom of the model, is associated with the number of input variables. When historic observations are supplied as additional observations, they will each require their own model parameters and increase the degrees of freedom in the model.

State aware models, such as recurrent neural networks , maintain a track record of the state of the landslide instead and iterate over the input time series in successive model runs. Individual observations are fed into the system, with the system maintaining track of their contribution to the current state of the landslide. These models resemble a bucket model, a simplified representation of the water storage in the subsoil. However, unlike in a traditional (soil moisture/ground water) bucket model, all variables are taken into account, even if they do not directly represent water. Furthermore, unlike regular neural networks, the number of trainable parameters is not dependent on the length of the history supplied to the model but on the number of memory cells and time series.

Models based on recurrent neural networks suffer from computational difficulties during optimization, where gradients may vanish . Therefore, they are typically replaced by models based on long short-term memory (LSTM) nodes that do not suffer from this due to built-in normalization. Each LSTM “bucket” is capable of weighting, retaining, and clearing a memory of previous inputs and as such tracks the system state.

The challenge specific to forecasting and nowcasting is the absence of information on the future slope conditions. The latest information available to the system is the current conditions and the last estimation of the system state. Auto-regressive models predict these conditions as well so that subsequent forecasts may use these environmental conditions in their models. However, precipitation especially is governed by external influences and may not be predictable from the other forcing parameters in the system. As an alternative, forecasts may be included in the model. However, this would require forecasts for all input variables. Therefore, such a system was deemed not suitable for this application.

Special attention should be paid to the robustness of the model. Even 10 years of daily observations will result in a time series of less than 4000 reference observations, much less than desirable for use in more complex machine learning models such as neural networks . If too few training data are provided, the abundance of input data creates unique combinations of conditions and outputs. This will lead to excellent performance during training but reduced performance during testing and application and is known as overfitting.

There are infinite data-driven modelling possibilities, and the generic character of many data-driven models suits the diversity in available remote sensing variables. However, due to the limited length of the time series, in comparison to typical machine learning studies, one should stay close to the physics and processes, to limit the freedom of the model towards a solution. Therefore, one has to ensure a balance between the number of parameters to be estimated and the training and validation data available.

The model variable selection is based on the analysis of factors of influence , and is mainly of a data-driven nature. Pre-disposing or causal factors, such as topography, that are necessary for a landslide to form are considered static in this study. Therefore, the focus is on the dynamic conditions leading up to landslide instability and deformation as well as triggering factors. The selection of variables is listed in Table .

Our method is designed with the intent to be generally applicable. Therefore, except for the deformation, remote sensing products were used, as they are likely to be available elsewhere as well. Where available, redundant products that represent the same or similar quantities were included to assess their relative performance in the nowcasting model. The correlation between the products is limited (μ|ρ|=0.16, max⁡|ρ|=0.7, Fig. ), indicating differences between the products of the same quantity. Effects that may not be observed directly, such as soil moisture under snow, require some form of modelling or reanalysis. These quantities, not directly available from remote sensing, are taken from reanalysis models “ECMWF reanalysis, version 5” (ERA5) and the “Global Land Evaporation Amsterdam Model” (GLEAM).

The desired output of our model is a daily, 4 d ahead prediction of the landslide deformation rate at benchmarks “D_WS_1” and “D5_1”. Reference, training, and validation samples are provided by the automated total station located on the Wattenberg, opposite Vögelsberg (Fig. ). Deformation measurements were performed hourly from 4 May 2016 to 28 June 2020 and aggregated to 1482 daily averages to reduce noise. The noise in the signal was further reduced by a 32 d moving average filter, the results of which are shown in Fig. . The time series at the 51 other benchmarks (Fig. ) were not used in the modelling.

Daily precipitation information is provided by the Integrated Multi-satellitE Retrievals for GPM (IMERG) algorithm of the Global Precipitation Measurement mission (GPM) . “Early” results are provided with sub-day delay and are therefore especially suitable for an operational nowcasting model. For comparison, daily precipitation from the ECMWF ERA5 Land reanalysis is included as well . Snow properties are covered by two products of the ERA5 Land reanalysis: snow water equivalent and snowmelt.

Soil moisture, especially at depth, cannot be observed directly from space at a high enough resolution for this application. The low-latency, operational products from the Copernicus Land Service, Soil Water Index and Surface Soil Moisture, are frequently unavailable either due to unfavourable slope topography or due to snow cover. Alternatives are provided by SMAP L4 , “Global Land Evaporation Amsterdam Model” (GLEAM) , and ERA5 Land . Evaporation estimates are taken from GLEAM as well. Air temperature, a proxy indicator of evaporation and snowmelt, is included from ERA5 Land ().

The model is fed with the 11 variables defined in Sect.  (Table ). Except for the deformation time series, all sources consist of gridded products with wide area coverage. In this study, only the data point closest to the active part of the Vögelsberg landslide was used. To match the time resolution of the deformation measurements, the model is run at daily intervals. Observations available at shorter intervals are aggregated to daily means first. Where data are missing, for example due to sensor failure, the values are filled with the data from the previous day (forward filling), as would be possible in an operational scenario. Furthermore, two modelled time series were added to the system: an antecedent precipitation index (API) as a basic hydrological model and a random, seasonal noise signal.

The antecedent precipitation index (API, API/API, V10) was designed to estimate the water present in the watershed . The API is included to determine if such variable could support the deformation nowcasting model. Precipitation less than 0.1 mm was ignored, in addition a 10 % direct evaporation loss, and a 4 % daily storage loss is assumed. These parameters were chosen based on an estimate of the hydrological setting by an experienced landslide hydrologist. The API at time step t is calculated as 1APIt=max⁡(0,p-0.1)⋅0.9+0.96⋅APIt-1, with p being the daily precipitation sum. The API, calculated from the operational GPM precipitation data (GPM/precipitationCal), is shown in Fig. .

A random variable with seasonal characteristics is added to the variable selection to analyse the effect of spurious correlation on the model. The random variable, fake/fake (V11), based on Brownian motion is tuned to match a typical seasonal characteristic in the 32 d history relevant to the model. The auto-correlation behaviour is illustrated in Fig.  and resembles the dynamics of both the surface temperature as provided by ERA5 and the soil moisture from SMAP for the first 2–3 months. Longer correlation periods are not relevant for our model.

All variables are offset to become zero-mean and scaled by the standard deviation. Therefore, all input variables are on approximately equal scale and represented as deviations from their average condition. The normalization parameters, mean and standard deviation, should be kept fixed while new data are added to remain consistent with the scaling of the time series used during training. The data set is fed to the model as a time-stamped collection of daily observations, illustrated in Fig. .

Our model is a shallow neural network with only a single hidden layer . This hidden layer consists of a single long short-term memory (LSTM) node that resembles a bucket model for the water storage in the subsoil. The model is supplied with a 32 d history of observations, equal to the length of the moving average filter, longer than the lag time for snow (0–8 d), and sufficient to cover most of the 20–60 d lag time for rainfall at the Vögelsberg landslide found by . From a predefined, optimized initialization, the model is cycled for each day of preceding observations, feeding the observations into memory before a prediction is made based on the final bucket values (m). The model is illustrated in Fig. , as function of environmental conditions (x, Table ), at each of the n=32 d preceding the nowcast, the LSTM node and four neurons of a single benchmark, one for each prediction day. This last, output, layer is repeated for both benchmarks (“D_WS_1” and “D5_1”) to be predicted, while the LSTM memory (m) is shared between the benchmarks to reduce the number of parameters.

In total, for a network configuration with a single memory cell (m), 68 parameters have to be estimated. The LSTM node, with one hidden state, requires 52 parameters to be estimated for the 11 variables (Table ). A total of 16 parameters are required for the output, eight for each of the deformation time series: one bias and one scaling parameter per day for the final state of the LSTM node. The number of parameters to be estimated is independent of the history length.

Four parameters are added per extra prediction day (two benchmarks, one bias and weight each). An extra memory cell requires 8h+4x+1 extra parameters, with h the current number of hidden nodes and x the number of input variables, while only four parameters are added for each additional input variable. Hence, extra memory always requires more parameters than extra input variables. Therefore, to limit the number of parameters in the model and minimize the risk of overfitting, the addition of a variable to the model should be preferred over the addition of a memory cell.

An interpretation of the network is that the development of the slope state in the last 32 d is described by the LSTM node. The state is scaled, and otherwise matched to the individual benchmarks, by the output neurons. The 4 d are an extrapolation of the current state of the system; no prediction of the conditions on the slope is made.

The “mean squared error” was chosen as the loss function. This function that quantifies the difference between the predicted and observed deformation is to be minimized during training. The quality of the prediction is measured on the period not used for training. This function assures that the cumulative deformation over time is realistic, as errors are balanced between overestimation and underestimation. Therefore, the predictions will not show a bias towards acceleration or deceleration. The TensorFlow machine learning framework was chosen to implement the model . The LSTM model is implemented in a stateless fashion: the warm-up phase is repeated for every nowcast. The model was run on a workstation based on an Intel Xeon W-2123 (4 cores, 8 threads, 3.6 GHz) with 32 GB RAM, while model variations were tested on the high-performance computing cluster of the Delft University of Technology. Given the limited size of the region of interest, as well as the limited number of parameters, the full model fits into 1 GB of memory.

During training the model parameters are tuned such that the final model state best describes the deformation prediction. The model is optimized with the Adam optimizer . The model is trained on the loss; after 50 training passes that do not lower the mean squared error over the training period, the model's parameters are fixed. If this steady state is not achieved after 25 000 passes, the training is stopped anyway and the model parameters used as-is.

Due to temporal correlation, training and validation cannot be divided over random chunks or batches according to the “traditional” 30 %–70 % chunks . Therefore, the training data are split into equal years instead, as shown in Fig. . Data outside the training period are used for validation. This includes the period before the training period, when available.

The robustness of the model to the selection of the training data is assessed from the stability of the results when training over the subsequent periods (Fig. ), a variation on cross-validation . Each model iteration starts with the same (random) initial weights but is trained independently from the start. The quality of fit is assessed by evaluation of the loss function, the mean squared error, on the periods not used for training. Finally, the model performance is compared between the training periods. Large deviations of the model quality suggest there are dynamics the system is not capable of describing.

To assess the impact of irrelevant data on the system, as well as the effect of overfitting, the additional, correlated random variable (fake/fake) is used. Overfitting will make the model prone to spurious correlation with this variable, which results in poor performance in the validation stage. Furthermore, to ensure there is no accidental correlation between the seasonal noise and the deformation signal during training and/or validation, the signal was re-rendered for every model run.

All possible combinations of the 11 input variables were tested on the model. With 11 variables this results in 211-1=2047 combinations, as each of the time series may be used or not (2 options), except for the case where no input is used. Furthermore, the model was trained and validated on each of the 10 combinations of training and validation year(s). Each sequence of model training and validation was repeated at least three times, to account for the `luck' introduced by the random initialization of each model. In total, 147 984 model runs were performed.

Due to the complexity of the operations applied to the input signal in the LSTM layer, it is not straightforward to analyse the contribution of the individual components to the final model outcome. As all model variations were tested (Sect. ), it is possible to analyse the influence of the presence of a variable by comparing the quality of the model variations. For this analysis, only model iterations with a training period (Fig. ) that left at least 1 year left for validation were used. Furthermore, all model variations were run multiple times to assess the robustness of the outcome to the random initialization.

Figure  shows the results of this analysis and illustrates the mean squared error over the validation period for all models including each variable. For each variable the minimum and average mean squared errors for the validation period are shown, while the maximum mean squared error is often out of range. The thickness of the line indicates the density of results for that mean squared error, where thicker lines at lower mean squared errors indicate a concentration of models with a high quality of fit.

Models based only on SMAP/sm_profile (V5) score the poorest (highest mean squared error) on average but with the widest distribution, including many solutions with a low mean squared error. The difference in performance between the variables vanishes as more variables are introduced into the model; however, the models including the SMAP soil moisture (V5) time series show a consistently larger range in performance, including models with a low mean squared error. Remarkable is the approximately equal performance between API/API (V10) and fake/fake (V11), where the latter contains no information on the hydro-meteorological processes and is only marginally outperformed by the antecedent precipitation index (API, V10). For models with more than four variables, there is no significant difference in model quality for any of the variables.

The possibilities for data-driven modelling are infinite: our model is only a single realization of the possible combinations of variables and operations. This raises three questions regarding the model selection: (i) how to match model and process, (ii) how to validate and quantify the quality the nowcast, and (iii) how to tune the model implementation.

The major challenge for the model of a deep-seated landslide is the discrepancy between the sub-daily variations of the input (especially precipitation and snowmelt) and a delayed, daily output (accelerated deformation). Therefore, non-time-aware models show erratic behaviour, as the consequence of sudden changes to conditions such as snow cover and (extreme) precipitation that, in reality, do not translate into immediate acceleration. Traditionally, the addition of groundwater physics, smoothing the hydro-meteorological signal, circumvents these peaks. However, the addition of groundwater physics requires knowledge of the geohydrology of the specific slope.

An LSTM node resembles a bucket model and was chosen to capture the delay between precipitation and deformation by modelling the buildup of water in the model. Our results showed that our model was unable to fully capture these hydro-meteorological dynamics. For reference, five alternative models were implemented (Table ) that were designed to better address the diversity of the slope, and/or lower the number of parameters required by the model to prevent overfitting.

The lstm3-32 model contains two additional memory cells (buckets) in the LSTM node compared to the lstm1-32 model previously used. The concept is that the memory cells may represent different systems or layers in the subsurface, potentially interacting with each other. For each subsequent time step, all states are included in the calculation of the new states and could therefore also model interactions between layers in hydrology, such as the transfer between layers.

The rnn1-32 and rnn3-32 models based on a traditional recurrent neural network are similar to their LSTM counterparts, with one and three memory cells respectively. However, unlike an LSTM node, they are unable to “forget” their state on command, and are more susceptible to unstable behaviour. The rnn1lin-32 did not incorporate an activation function and is comparable to a moving average filter with interaction between the variables. For all three models the number of parameters is less than for the equivalent LSTM-based models.

The da-32 model resembles a linear least squares model. Variables are first summarized as their average over their 32 d history and included in eight nodes without bias in the hidden layer of the network. The final predictions are a linear combination of the node values. In a “traditional” linear least squares solution, a direct combination of all input variables, the number of parameters will often outnumber the number of observations available and were therefore not tested.

The performance of each model is shown for comparison in Fig.  as a function of the parameters required. Model performance is typically optimal for models with only a single parameter and is comparable between the models. Like the original model (lstm1-32), each model was re-run multiple times with a random initialization of the seasonal noise (V11) and model parameters to verify the consistency of the output. Most alternative models do not outperform the average deformation rate as predictor for the future deformation rate, as shown in Fig. .

Over the full time span of the measurements, four distinct acceleration periods can be identified (Fig. ). These acceleration periods are especially of interest to an early warning system, as they mark the start of a period of accelerated deformation and associated hazard. Although the periods of accelerated deformation are comparable in length to the periods of continued, but reduced, deformation, the acceleration events are much shorter (Fig. ). Therefore, these periods are underrepresented in error metrics during training and validation. However, training on these four periods alone leaves insufficient variability to describe the system and reliably fit the required model parameters. Furthermore, the episodic deformation behaviour poses a challenge to the prediction system, since the forcing variables on the slope do not reflect such sudden changes observed in the deformation behaviour, as shown in Fig. .

The variable selection in Table  was compiled based on our knowledge of the physics behind the landslide process, as well as the availability and continuity of the data. With the ambition for a future, regional implementation in mind, the variables preferably come from satellite remote sensing observations rather than local, field sensors. However, we did not succeed in a fully remote-sensing-driven operation due to the limited availability of such operational products. Especially deformation observations from space (satellite radar interferometry, InSAR) were found to be promising, but we were unable to replace our local deformation time series with the noisier satellite deformation data.

Our results show that deformation nowcasting is an open challenge. Although well monitored, the Vögelsberg landslide is a complex system and therefore not a straightforward test case. Our results are inconclusive regarding whether our method could work on other deep-seated landslides. More direct dynamics and/or stronger and more frequent acceleration periods would help constrain the system. The inclusion of field data, such as groundwater level , might be another approach to bypass modelling of the most volatile hydrological processes. The ideal slope to further develop a machine-learning-based nowcasting method has the following characteristics: (i) a dynamic deformation behaviour; (ii) is controlled by hydro-meteorological conditions, with limited delay; and (iii) has field monitoring data for reference and training.

For short time series machine learning methods are known to be outperformed by basic statistical methods . Therefore, our current challenge to nowcast deformation time series may be partially solved in the near future by the natural extension of time series. Furthermore, continued development of the (satellite) data products by their providers may enable new possibilities. Desirable improvements include timeliness of delivery of data products as well as their precision and spatio-temporal resolution.

Notable is the recent publication of the first version of the European Ground Motion Service data set , a pan-European InSAR product. This data set will allow for experimental, regional, and weekly nowcasting systems based on a replay of historic observations. Regional applications will enhance training possibilities and may help overcome the hurdle of limited deformation time series, as multiple slopes are monitored simultaneously. However, to “learn” from the differences between slopes and enlarge variation in training data, events have to be largely uncorrelated.