1. Introduction
Variability of the Earth climate system can be decomposed into a broad spectrum with a continuous distribution (Hasselmann 1976). Separation of signals within a spectral band is the first step in the process of understanding physical mechanisms driving the variability associated with that signal. Numerical methods routinely used in signal analysis (e.g., spectral and weight-based filters) are sensitive to the sample size. These methods work well when applied to time series of observations and climate model simulations but have limitations when applied to model forecast data or real-time monitoring based on observations. One particular example of such limitations is the intraseasonal variability (ISV) of the tropics (20–100 days), which is the target of subseasonal-to-seasonal (S2S) prediction and operational forecast systems. Due to the theoretical limit of predictability of ISV, the length of the S2S forecasts is between 35 and 45 days, which creates challenges for removal of low-frequency variability (>100 days). When analyzing forecast data, using a conventional method for extracting the ISV from total anomaly fields requires blending of forecast with observation to extend the length of the forecast time series and allow for removal low-frequency variability associated with El Niño–Southern Oscillation (ENSO). The padding with observations varies from 90 days (e.g., Gottschalck et al. 2010) to 2 years (Janiga et al. 2018) prior to initialization of the forecast. This step introduces artificial or spurious features that affect the forecast skill. The limited ability to properly diagnose the ISV in the forecast models restricts our ability to understand modeling capabilities for predicting variability of these time scales and narrows the opportunities for improving forecasting systems. Evaluation of ISV in the forecast is important not only for tropical predictions but also for the prediction of atmospheric teleconnections between the tropics and extratropics, which vary from high-impact weather events to large-scale barotropic structures (Stan et al. 2017). While forecast data need to be extended prior to the initialization of the forecast, real-time monitoring based on observations requires extrapolation of data into the future in order to extract the ISV signal present at the current state.
Machine learning and artificial intelligence (ML/AI) approaches have the potential to perform signal processing that overcomes the limitations of conventional statistical approaches. Artificial neural networks (ANN; Rumelhart et al. 1986) have been already used in geosciences for identification and classification of patterns and signals of climate variability (Li et al. 2016; Liu et al. 2016; Barnes et al. 2019; Toms et al. 2020; Yoo et al. 2020; Toms et al. 2021; Labe and Barnes 2021; Mayer and Barnes 2021), improving the signal-to-noise ratio of seismological datasets (Chen et al. 2019), and detection of errors in model generated datasets (Moghim and Bras 2017; Dutta and Bhattacharjya 2022). As ML/AI is based on identifying hidden regularities embedded in the data (Flach 2012), they have potential to succeed in extracting the ISV of tropical atmosphere because a substantial portion of this variability is explained by regularities in the form of large-scale waves (∼10 000 km) and modes that manifest in basic-state variables (e.g., pressure, wind, temperature) as well as in physical phenomena (e.g., rainfall, cloudiness) aggregated across multiple scales by organizing mechanisms. The equatorial waves are known as convectively coupled equatorial waves (e.g., Rossby waves, inertia–gravity waves, mixed-Rossby gravity waves, and Kelvin waves). The dominant modes include the Madden–Julian oscillation (MJO; Madden and Julian 1971, 1972), the boreal summer intraseasonal oscillation (BSISO; Lau and Chan 1986), and the 30–90-day mode (Jiang and Waliser 2009). The review of Serra et al. (2014) offers a detailed description of tropical variability.
Convolutional neural networks (CNNs or ConvNets) represent a class of deep learning ANN. It has been demonstrated that CNNs can be used to approximate any continuous function to certain accuracy, which depends on the depth of the network (Zhou 2020). Methods in this class attempt to extract information using stacked layers of nonlinear information processing algorithms distributed in a hierarchical architecture (LeCun et al. 2015). The literature describes many variants of the CNN architectures (Gu et al. 2018); however, they share similar basic components such as an input layer, a hidden layer, and an output layer. The convolutional layer is one of the hidden layers and is the layer learning feature properties during the training. The feature maps are computed by the convolutional kernels. In this study a one-dimensional (1D) CNN model is used, which is suitable for time series analysis (Wibawa et al. 2022). In 1D CNN models the kernel is a vector. The 1D CNN model is applied to construct a bandpass filter for tropical ISV. To demonstrate the accuracy of the method, the CNN-based filter is first compared to a conventional Lanczos digital filter (Duchon 1979) and then it is applied to problems for which conventional filters have challenges due to the limited sample size and instead surrogate methods are adopted.
This paper is organized as follows: In section 2, we introduce data and methods used in this study along with the description of the 1D CCN-based filter. In section 3, we present the results of applying the CNN filter to extract the intraseasonal variability (30–90 days) from fields relevant to tropical variability such as zonal wind stress (a basic-state variable) and outgoing longwave radiation at the top of the atmosphere (a good indicator of tropical deep convection and associated rainfall). In section 4 we present the downstream impact of the CNN-based filter on calculating the MJO components of the zonal wind stress and outgoing longwave radiation (OLR). We summarize the main findings and discuss perspectives in section 5.
2. Data and methods
a. Data
For wind stress we use a combination of the high-resolution (0.25° latitude–longitude) Blended Sea Winds stress product (Zhang et al. 2006; Peng et al. 2013) and Advanced Scatterometer (Bentamy and Fillon 2012) for the period 1988–2016. Daily means of NOAA interpolated OLR with a horizontal resolution of 2.5° latitude–longitude (Liebmann and Smith 1996) are used for the period 1980–2022. These satellite derived datasets are selected instead of reanalysis because the latter is more likely to be affected by models’ ability to simulate the ISV of the tropics. Daily anomalies for these variables are calculated by removing the climatological mean, defined as the daily average over all years in each dataset. For the CNN model described in the next section, the data are partitioned into a training period (1 January 1988–31 December 2012), a validation period (1 January 2013–31 December 2014), and a testing period (1 January 2015–31 December 2016). To maintain independence of training data from testing/validation data, the climatology is computed only using the training period.
b. 1D CNN
The weights of the convolutional kernel (wk) at each grid point are estimated by training the 1D CNN model. The kernel size p needs to be set before the training process starts and remains fixed. In the CNN model used here, the assumption is that the convolution operation of a daily signal with a kernel of length p will retain signals with a period greater than p days.
The architecture of the CNN model consists of an input layer, a subtract layer separating two convolutional layers, and an output layer. The schematic of the model is shown in Fig. 1. In the input layer, the time series of daily anomaly maps with dimension (latitude, longitude) are parsed into time series at each grid point. In this layer, the dimension of the input dataset is reshaped into (time, grid), where grid = latitude × longitude. Data scaling (e.g., standardization and normalization) sometimes recommended as a preprocessing step in using neural network models (Wang et al. 2006) is not performed because the model uses only one input variable and the output variable has the same units as the input variable. In the first convolutional layer, each time series of daily anomalies is passed through a convolutional layer with the kernel size p = 90. The size of the kernel is determined by the intention to design a filter applicable to operational seasonal forecasts, which have a typical length of 90 days (e.g., NCEP CFSv2; Saha et al. 2014). This convolution operation retains signals with variability greater than 90 days. Next, the output of the linear operation such as convolution is passed through the subtraction layer. In this layer, the output of the first convolutional layer is subtracted from the input. There are no learnable parameters in this layer. After subtraction, the remaining signal retains variability of less than 90 days. Then this 90-day high-pass-filtered signal is passed to the second convolutional layer with the kernel size p = 30, and the convolution operation will retain variability between 30 and 90 days. Trials using different lengths for the second kernel indicate a small influence of this kernel size on the accuracy of the model; the length of the first kernel has the largest impact (not shown). The convolutional layers do not change the dimension of the output, and the padding is set to zero. In the output layer, the filtered time series at each grid point are mapped into times series of daily filtered maps with dimension (latitude, longitude). Finally, the output time series at each grid point are compared to the 30–90-day Lanczos-filtered time series at the same grid point. In this algorithm, the filtering is done independently for each grid point, that is, no spatial pattern information is used in the training.
The architecture of the CNN-based filter. All layers have the same size, which is the sample size. The dashed–dotted line denotes the kernel size. In the first convolutional layer, the kernel size p = 90 and in the second convolutional layer p = 30. Grid = lon × lat. Time represents the number of samples (days). In each layer, a rectangle represents one grid point. The horizontal arrows show the workflow of the algorithm excluding the hidden layers in the convolutional layers. The gray line connecting the input layer and the subtraction layer denotes that input data can be passed to the subtract layer from the input layer.
Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0079.1
Training and validation loss for the OLR data at a grid point located at 160°E on the equator.
Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0079.1
The CNN model is trained independently for each grid point and dataset. For both datasets, the training period is 1 January 1988–31 December 2012, the validation period is 1 January 2013–31 December 2014, and the testing period is 1 January 2015–31 December 2016. The training period provides 9132 samples or days (time dimension in Fig. 1) at each grid point (grid dimension in Fig. 1). Because ISV is present year-round, all days are suitable for being used in the training. The impact of sample size on the training error of deep ANNs is an open question and is problem dependent (Chattopadhyay et al. 2020).
c. Evaluation metrics
3. Results
Intraseasonal variability of the tropics
The CNN-based filter, was applied to extract the ISV in the 30–90-day range from the daily anomalies of the zonal component of the wind stress vector and OLR. The results for the wind stress anomalies and OLR are shown in Fig. 3 during the first year of the testing period, that is, 1 January 2015–31 December 2015. For a complete description of each field, the total daily anomalies are also included. The signal filtered using the conventional Lanczos filtering serves as the truth. The Lanczos filtering is applied to the whole year of data, which requires additional data before the first and last date of the analyzed period. For example, to obtain a filtered time series that begins on 1 January 2015 the time series on which Lanczos filter is applied begins 90 days prior, that is, 2 October 2014. If the end date of the filtered time series is 31 December 2015, the end date of the unfiltered time series is 31 March 2016. For the zonal wind stress, the filtered signal based on the CNN method is shown in Fig. 3c along with the 30–90-day bandpass-filtered signal (Fig. 3b) obtained using the conventional Lanczos filtering method.
Hovmöller diagrams averaged over 7.5°S–7.5°N for the testing period 1 Jan 2015–31 Dec 2015. (top) Zonal wind stress (N m−2) and (bottom) OLR (W m−2). (a),(e) The total daily anomaly of the fields. (b),(f) The 30–90-day filtered anomalies using the Lanczos filter. (c),(g) The 30–90-day filtered anomalies using the CNN-based filter. (d),(h) The Lanczos filtered anomalies minus the CNN-based filtered anomalies.
Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0079.1
The direct comparison (Figs. 3b,c) and the difference between the filtered signals obtained using the conventional and ML/AI methods (Fig. 3d) show a good agreement between the patterns in the ISV signal with small differences in the amplitudes. The amplitudes of zonal wind stress anomalies obtained using the CNN filter are 37% larger (smaller) than the amplitudes of filtered zonal wind anomalies obtained using the Lanczos method. This number is calculated by dividing the absolute value of maximum difference between filtered amplitudes using the two methods by the absolute value of maximum (minimum) amplitude of Lanczos-filtered anomalies. Results for the OLR anomalies are shown in Figs. 3e–h for the same period (1 January 2015–31 December 2015) as for the zonal wind stress. The OLR based results also suggest a good agreement between the two methods. The difference in the amplitude of the filtered OLR produced by the two methods is in the same ballpark (31.5%) as for the zonal wind stress.
Subsets of years used for k-fold cross validation.
Hovmöller diagrams of the effectiveness of the CNN-based filtering model measured by (a) ME and (b) standard deviation (σ) obtained by resampling the training, validation, and testing periods into six folds. The domain is an average over 7.5°S–7.5°N.
Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0079.1
As in Fig. 3h, the mean error (Fig. 4a) and standard deviation (Fig. 4b) also emphasize the model’s limitations at the ends of the time series. The errors manifest in the first 30 days, after which the CNN models become equivalent to each other.
To further evaluate the ability of the CNN-based filter to isolate the ISV in the 30–90-day range, Figs. 5 and 6 show additional metrics constructed using results based on the zonal wind stress (Fig. 5) and OLR (Fig. 6) for the testing period. Based on the maximum variance of the 30–90-day filtered anomalies, two regions are selected: 125°–160°E, 7.5°S–7.5°N for the zonal wind stress and 160°–190°E, 7.5°S–7.5°N for the OLR. The comparison between the zonal wind stress (Fig. 5a) and OLR (Fig. 6a) filtered time series using the conventional and CNN-based filtering methods reveals a notable difference at the beginning of the time series.
(a) Time series of 30–90-day filtered anomalies using the Lanczos filter (red solid curve) and the CNN-based filter (blue dashed line) averaged over 125°–160°E; 7.5°S–7.5°N for the testing period 1 Jan 2015–31 Dec 2016 along with their correlation coefficient (r). (b) The power spectrum density times frequency of filtered anomalies shown in (a). (c) IOA and (d) RMSE between the 7.5°S and 7.5°N averaged filtered anomalies using the Lanczos and CNN-based filters during the testing period 1 Jan 2015–31 Dec 2016.
Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0079.1
As in Fig. 5, but for OLR anomalies averaged over 160°–190°E, 7.5°S–7.5°N.
Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0079.1
There are other cases where the end of the time series also shows differences between the two filtering methods. This difference is related to the constraint in the convolutional layer to maintain the length of the time series. The power spectra (Figs. 5b and 6b) of the two time series show that the CNN filtering method captures very well the spectral peak centered around 35 days and slightly overestimates the amplitude of the spectral peak located close to 60 days. The CNN method also shows an adequate removal of the spectral power existing in data outside of the intended spectral window. The RMSE (Figs. 5c and 6c) and IOA (Figs. 5d and 6d) between the time series constructed using the two methods also reveal a good spatial agreement at all longitudes in a tropical channel between 7.5°S and 7.5°N. RMSE is used as an indicator of the outliers and IOA is a measure of the degree to which the model’s predictions are free of errors (Willmott 1981).
Consistent with the IOA, R2 for the zonal wind stress (Fig. 7a) is lower than for the OLR (Fig. 7b). One can speculate the better fit for the OLR is due to a stronger MJO signal in this field compared to the wind (e.g., Waliser et al. 2009) and/or a better satellite product for the OLR than surface wind stress.
R2 of the filtered anomalies using the Lanczos and CNN-based filters during 1 Jan 2015–31 Dec 2016 for (a) zonal wind stress and (b) OLR. Time series at each zonal grid point represent the average between 7.5°S and 7.5°N.
Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0079.1
4. Applications of the CNN-based filter
The intraseasonal anomalies are used for model evaluations, to construct metrics and diagnostics for studying the properties of climate system on S2S time scale, and to characterize the tropical oscillations such as the MJO and BSISO, the two dominant modes of tropical ISV. Lybarger et al. (2020) introduced a metric designed to characterize the interaction between the MJO component of the wind stress and a low-frequency (∼48 months) oscillation of the tropical Pacific, ENSO. The metric is used to evaluate the ENSO forecast skill of a seasonal forecast system. The key element of the metric is the MJO component of the wind stress. The first step in calculating the MJO component in the forecast anomalies of the wind stress is to extract the ISV variability from the forecast anomalies and then project the observed patterns of MJO [the first four empirical orthogonal functions (EOFs)] onto the ISV anomalies. Because the length of the seasonal forecasts is 90 days, conventional filtering methods for extracting ISV cannot be applied. For example, a Lanczos filter requires 181 days, and a frequency of (90 days)−1 cannot be extracted by a Fourier analysis. Thus, proxy methods have been developed based on the total (unfiltered) daily anomalies of the wind stress. We used the CNN-based filtered anomalies to compute the MJO component of the zonal wind stress (
Longitude–time plots of (a) IOA between
Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0079.1
In all years, the IOA based on the CNN filter is larger than the IOA when using unfiltered anomalies. In the western and central Pacific, the IOA of the CNN-based method is much larger than the IOA of unfiltered anomalies. In the eastern Pacific, the difference between the IOAs is slightly smaller than in the other regions. This smaller difference in the eastern Pacific could be explained by the fact that MJO is more active in the western Pacific than in the east, where low-frequency variability (e.g., ENSO) is the dominant signal.
Another application that can benefit from using the CNN-based filter is the real-time monitoring of the MJO (e.g., Gottschalck et al. 2010; Kikuchi et al. 2012; Kikuchi 2020). We applied the method described by Kikuchi (2020) to extract the MJO signal in the OLR anomalies using a proxy method for computing the daily filtered anomalies and the CNN-based filtering method. In the proxy method, first, anomalies are constructed by subtracting from each daily value the climatological mean and three harmonics of the climatological annual cycle. Second, these anomalies are filtered by subtracting the mean of the previous 40 days from each daily anomaly. In both methods, the MJO signal is then extracted from the ISV by projecting the daily filtered anomalies onto the two extended EOFs (EEOFs) precomputed by Kikuchi (2020) and then multiplying the resulting principal components (PCs) time series with the EEOFs (OLRMJO = EEOF1 × PC1 + EEOF2 × PC2). Results from the two methods are compared in Fig. 9, which shows the OLR-filtered anomalies and the reconstructed OLRMJO.
Hovmöller diagrams of daily OLR filtered anomalies (shading) and MJO signal (contours) averaged over 7.5°S–7.5°N for the period 1 Sep 2018–31 Mar 2019. Above the gray dashed line, anomalies are filtered using a 25–90-day Lanczos filter. Below the gray dashed line, anomalies are filtered using (a) the proxy method (see text for details) and (b) the CNN-based method (right). The gray line denotes the last date for which data would be available for the proxy method. In both cases the MJO is computed following Kikuchi (2020) method.
Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0079.1
In the top part of the figures (above the gray dashed line), anomalies are filtered using a 25–90-day bandpass Lanczos filter. In the bottom part of the figures (below the gray dashed line), anomalies are filtered using the proxy method (Fig. 9a) and the CNN-based method (Fig. 9b). It is easy to see that CNN-based filtered anomalies and the MJO signal remain coherent with the existing structures and the amplitude of the ISV (filtered anomalies) is not distorted as in the case of proxy method. Relative to the period when the Lanczos filter was applied, the amplitudes of the anomalies become stronger for the period when the proxy filtering method is used. The MJO is characterized by the amplitude and phase. The amplitude is typically measured by the principal components
(top) Comparison of MJO amplitude (PC1 and PC2) and (bottom) phase space for the period 1 Sep 2018–31 Mar 2019. The black line denotes the Lanczos filter, and the calculation is done assuming availability of data in the future. The blue and red lines correspond to the proxy and CNN calculations. The vertical line on 1 Jan 2019 denotes the date after which no data would be available for applying the Lanczos filtering.
Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0079.1
A comparison of the power spectra of the filtered anomalies and MJO signal (Fig. 11) illustrates the efficiency of the CNN-based method at filtering the intended frequencies with the right power spectral density and not introducing as many spurious frequencies.
Power density spectra using a 25–90-day Lanczos filter (red solid line), Kikuchi et al. (2012) proxy method (blue dashed line), and CNN-based method (red dashed line). (a) Daily OLR anomalies. (b) MJO component of the daily OLR anomalies. Power spectra are calculated for the time series of area average over 160°–190°E; 7.5°S–7.5°N between 1 Sep 2018 and 31 Mar 2019.
Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0079.1
5. Conclusions
In this study we have demonstrated that CNN methods can be applied to construct a bandpass filter for intraseasonal variability (30–90 days) of the tropical atmosphere. The CNN-based filter contains two convolutional layers with 90 weights in the first convolutional layer and 30 weights in the second convolutional layer. Our results suggest that the CNN-based filter is a robust new technique for signal processing of geophysical data. The CNN-based filter yields similar results to the Lanczos filter and with no loss of data at the beginning and end of the analyzed period. The Lanczos filter uses 181 weights (or points), therefore 90 points (days in this case) are lost at the beginning and end of the time series. In fact, a Lanczos filtering cannot be applied to time series with the length of 90 days if the cutoff frequency is (90 days)−1. The CNN-based filter only shows a small reduction in accuracy at the ends of time series The number of weights used by the Lanczos filter are determined to ensure a reduction of the Gibbs phenomena that occurs in the vicinity of a discontinuity when the Fourier analysis is carried out (Duchon 1979). Since the CNN-based filter does not use a Fourier analysis, generation of Gibbs waves is not possible.
The CNN-based filter works well when applied to basic-state variables (wind stress) and phenomena-based variables (OLR). For both variables, the relative difference from the conventional Lanczos filtering method is in the ballpark of 30%–40%. The IOA for the state variable is also almost on par with that for OLR. By further using the filtered data to successfully construct the MJO signal we demonstrate that the classification determined by the CNN-based filter is based on physical principles as shown in the power spectrum information and MJO phase diagram.
The CNN-based filter can be applied to extracting ISV from the forecasts. Current extracting methodologies (e.g., Gottschalck et al. 2010; Janiga et al. 2018) have some disadvantages such as the contamination of the MJO signal from the higher-frequency variability and low-frequency variability associated with ENSO. The filter described in this study works for seasonal forecasts. Further developments are needed to make it applicable to S2S forecasts, which are shorter, for example, 45 days (Vitart et al. 2017). The 90-day requirement embedded in the CNN-filter presented in this study prevents its application to shorter time series.
Acknowledgments.
The authors acknowledge support from the NOAA/WPO through Grant NA20OAR4590316. CS was also partially supported by Laboratory Directed Research and Development (LDRD) funding from Argonne National Laboratory provided by the director of the Office of Science of the U.S. Department of Energy (Contract DEAC02-06CH11357) though a joint appointment. The study was partially supported by resources provided by the Office of Research Computing at George Mason University and funded in part by grants from the National Science Foundation (Awards 1625039 and 2018631). The authors appreciate the feedback provided by three anonymous reviewers.
Data availability statement.
The CNN is built using Keras, a deep learning framework that wraps Tensorflow. The source code for the CNN-filter has been made publicly available (https://github.com/cristianastan2/AIES-Deep-Learning-Filter). QSCAT data are available at https://www.remss.com/missions/qscat/. DASCAT data are deposited online (http://apdrc.soest.hawaii.edu/datadoc/ascat.php). OLR data are deposited online (https://climatedataguide.ucar.edu/climate-data/outgoing-longwave-radiation-olr-avhrr). The EEOFs used for computing the MJO signal in the OLR are deposited online (http://iprc.soest.hawaii.edu/users/kazuyosh/Bimodal_ISO.html).
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