Optimized Dynamic Mode Decomposition for Reconstruction and Forecasting of Atmospheric Chemistry Data

TL;DR

The paper tackles reconstructing and forecasting high-dimensional global atmospheric chemistry data by applying optimized Dynamic Mode Decomposition (optDMD) and Bagging optDMD (BOP-DMD) to GEOS-Chem outputs. By solving a nonlinear regression with variable projection for $\mathbf{X}' \approx \mathbf{A}\mathbf{X}$ and constraining eigenvalues via $\Re(\omega_k) \le 0$, the authors obtain stable, physically interpretable spatio-temporal modes that capture multiscale chemical dynamics. They demonstrate substantially improved reconstruction and forecasting accuracy over classical and exact DMD, and quantify temporal uncertainty using ensembles, highlighting higher variability in longer-lived or higher-order modes. The approach enables real-time, low-cost interpretation and forecasting of atmospheric chemistry and offers a pathway to integrate observations and extend to longer timescales while providing uncertainty estimates.

Abstract

We introduce the optimized dynamic mode decomposition algorithm for constructing an adaptive and computationally efficient reduced order model and forecasting tool for global atmospheric chemistry dynamics. By exploiting a low-dimensional set of global spatio-temporal modes, interpretable characterizations of the underlying spatial and temporal scales can be computed. Forecasting is also achieved with a linear model that uses a linear superposition of the dominant spatio-temporal features. The DMD method is demonstrated on three months of global chemistry dynamics data, showing its significant performance in computational speed and interpretability. We show that the presented decomposition method successfully extracts known major features of atmospheric chemistry, such as summertime surface pollution and biomass burning activities. Moreover, the DMD algorithm allows for rapid reconstruction of the underlying linear model, which can then easily accommodate non-stationary data and changes in the dynamics.

Figures (16)

• Figure 1: The spatial grid for atmospheric chemistry data sets on the left panel. The data ${\bf x}(t_k)$ is collected into snapshot matrices ${\bf X}$ which are used to regress to the best exponential (linear) solution $\mathop{\mathrm{arg\rm{}min}}\limits_{ \boldsymbol{\omega}, \boldsymbol{\Phi}_{\bf b} } \| \mathbf{X} - \boldsymbol{\Phi}_{\bf b} {\bf T}(\boldsymbol{\omega}) \|_F$, where $\boldsymbol{\Phi}_{\bf b}$ are the weighted DMD modes and ${\bf T}$ is a matrix of exponentials for fitting the data (\ref{['eq:optdmd']}).
• Figure 2: Shifting the data for each cell in time to align the local time zones across a latitude to the prime meridian$\mathrm{\left(Lon=0^\circ\right)}$ local time, shown here for ${\mathbf{O_3}}$ tendency data for $\mathrm{Lat}=30^\circ$. The bottom left panel is the raw data for the 3 highlighted cells, the bottom center panel is this data shifted in time, and the bottom right panel shows isolated day time values only.
• Figure 3: Comparing $30$ day reconstruction results for Classical and Optimized DMD at the surface of${\mathbf{NO}}$preprocessed data at $\mathrm{Lat}=30^\circ$. The results are for absolute concentration or $\mathbf{CONC}$ data; the top panel shows the preprocessed data, the middle panel shows the reconstruction from the Classical $\mathrm{DMD}$, and the bottom panel shows the reconstruction from Optimized $\mathrm{DMD}$. The Classical $\mathrm{DMD}$ is unable to capture the dynamics for the absolute concentration data and it decays down to zero. The Optimized $\mathrm{DMD}$ reconstructs the data and resolves the dynamics accurately.
• Figure 4: Summary of the BOP-DMD architecture reproduced with permission from sashidhar2022bagging. The data snapshots $\mathbf{x}(t_k)$ are collected over $m$ snapshots into the matrix $\mathbf{X}$. Columns of $\mathbf{X}$ are randomly sub-selected into the matrix $\mathbf{X}^{(k)}$ to build an optimized DMD model. Each DMD model $\mathbf{x}^{(k)} = \mathbf{\Phi}^{(k)}\exp(\mathbf{\Omega}^{(k)}t)\mathbf{b}^{(k)}$ is used to compute the statistics (mean and variance) of the DMD parametrizations $\mathbf{\Phi,~\Omega,~b}$ which are used in building a the BOP-DMD ensemble solution with Uncertainty Quantification (UQ).
• Figure 5: Comparing the spectrum for $40$ day reconstruction results for Classical and Optimized DMD at the surface of${\mathbf{OH}}$preprocessed data. On the left 4 panels are the eigenvalues of ${\mathbf{OH_{CONC}}}$data; on the right 4 panels are the eigenvalues of ${\mathbf{OH_{TEND}}}$ at $\mathrm{Lat}=30^\circ$. The top panels show the spectrum from Optimized $\mathrm{DMD}$ with no constraints, the second set of panels show the spectrum from Optimized $\mathrm{DMD}$ with linearized constraints that the eigenvalues be on the left-half plane, the third set of panels show the spectrum from Optimized $\mathrm{DMD}$ with linearized constraints that the eigenvalues be imaginary, and the bottom panels show the spectrum from Classical or Exact $\mathrm{DMD}$.