The objective identification of OFBs in meteorological data has been a topic of scientific inquiry for more than 30 years. Uyeda and Zrnić (1986) and Hermes et al. (1993) use detections of wind shifts in terminal Doppler radar velocity measurements to isolate regions of strong radial shear associated with OFBs. Smalley et al. (2007) include the “fine line” reflectivity structure of biological- and precipitation-sized particles to identify OFBs via image template matching. Chipilski et al. (2018) considered the OFB objective identification in numerical models using similar image processing techniques, but with additional dynamical constraints on vertical velocity magnitudes and mean sea level pressure tendency. Objective OFB identification has not been demonstrated to date with the new ABI observations of the GOES-R satellite series. Identification via satellite imagery would be valuable for local deep convection nowcasting algorithms, which use boundary presence as a predictor field (Mueller et al., 2003; Roberts et al., 2012), and for operational centers around the world that may not have access to ground-based Doppler radar data.

Traditionally, forecasters have identified OFBs in satellite imagery by visually identifying the quasi-linear low-level cloud features and backtracking them to an associated deep convection source. Previous objective motion derivation algorithms are not designed to yield the dense wind fields, in which motion is estimated at every image pixel, necessary for identifying and tracking features such as OFBs (Bedka et al., 2009; Velden et al., 2005). In fact, the original image window-matching atmospheric motion vector (AMV) algorithms produce winds only over targets deemed acceptable for tracking by preprocessing checks on the number of cloud layers in a scene, brightness gradient strength, and patch coherency. The targets are further filtered with post-processing checks on acceleration and curvature through three-frame motion and deviation from numerical model flow (Bresky et al., 2012; Nieman et al., 1997; Velden et al., 1997; More in Sect. 2.2). These practices were followed for a very practical reason – AMV algorithms were tailored for model data assimilation. In the formation of the model analysis, observational data must be heavily quality-controlled, with outliers removed, to minimize data rejection. Here, information such as OFBs would be rejected due to the detailed space–time structure of actual convection, which is typically poorly represented by the numerical model.

Deriving two-dimensional flow information at every point in the imagery would require either modification of previous AMV schemes or post-processing of the AMV data via objective analysis (e.g., Apke et al., 2018). The latter typically will not capture motion field discontinuities, resulting in incorrect flows near feature edges (Apke et al., 2016). To capture such discontinuities in a dense flow algorithm, new computer vision techniques, such as the gradient-based methods of optical flow, must be adopted.

Optical flow gradient-based techniques derive motion within fixed windows, thus eliminating the reliance on models for defining a search region. A core assumption of many optical flow techniques is brightness constancy (Horn and Schunck, 1981). Considering two image frames, brightness constancy states that the image intensity I at some point x=x,yT is equal to the image intensity in the subsequent frame at a new point, x+U, where, with a translation model, U=u,vT represents the flow components of the image over the time interval (Δt) between the two images: 1Ix,t=I(x+U,t+Δt). Equation (1) can be linearized to solve for the individual flow components, u and v: 2∇I⋅U+It=0. where ∇I=[Ix,Iy] represents the intensity gradients in the x and y direction, and It represents the temporal gradient of intensity. For one image pixel, Eq. (2) contains two unknowns with a simple translation model for U; therefore, it cannot be solved pointwise. One well-known approach to solving this so-called “aperture problem” is the Lucas–Kanade method, hereafter the LK method, which considers a measurement neighborhood of the intensity space and time gradients (e.g., Baker and Matthews, 2004; Bresky and Daniels, 2006). The use of neighborhoods, or image windows, to derive optical flow is called a local approach. Another seminal approach was introduced by Horn and Schunck (1981; the HS method), which solves the aperture problem by adding an additional smoothness constraint to the brightness constancy assumption and minimizing an energy magnitude between two images: 3EU=∫∫Ω∇I⋅U+It2+α(|∇2u|2+|∇2v|2)dx, where EU represents an energy functional to be minimized over all image pixels Ω, α is a constant weight used to control the smoothness of the flow components u(x) and v(x), and ∇2=[∂/∂x,∂/∂y]T. This derivation is called a global approach, whereby the optical flow u(x)and v(x) at each pixel is found that minimizes the quantity of Eq. (3) by deriving the Euler–Lagrange equations and numerically solving the linear system of equations with Gauss–Seidel iterations.

Readers can contrast the HS method with the optical flow algorithm used in GOES AMVs, referred to as “patch matching” (PM; Fortun et al., 2015). In PM, a target (e.g., a 5×5 pixel box) identified as suitable for tracking is iteratively searched for in a sequential image within a reasonable search area (Fig. 1a). The motion is identified by which candidate target (e.g., another 5×5 pixel box displaced by the optical flow motion) in the sequential image best matches the initial target, typically by minimizing the sum of square error between the target and the candidate brightness values (Daniels et al., 2010; Nieman et al., 1997). The reader can draw similarities to the HS method by formulating the PM approach as an energy equation to be minimized: 4EU=∑n∈TIxn,t-Ixn+U,t+Δt2, where the minimum in E is found by computing Eq. (4) at every candidate target in the search region. As E is only minimized within the target area T, PM represents a local method.

Research and extensive validation have shown that, with quality control, PM provides a valuable resource to derive and identify winds in satellite imagery (Velden and Bedka, 2009). However, there are several types of motions for which PM would fail (Fig. 1b), many of which occur frequently in satellite OFB observations. AMVs found with Eq. (4) make two key assumptions: (1) that the brightness remains constant between sequential images at time t and t+Δt and (2) that the motion U is constant within the target. The first assumption, brightness constancy, fails when there are excessive illumination changes in a sector that are not due to motion. These illumination changes may be due to evaporation or condensation or simply due to changes in solar zenith angle throughout the day in visible imagery. The HS method also uses assumption (1), though it is relaxed when combined with the smoothness constraint. Assumption (2), which is not made in the HS method or other global methods, implies that the PM method has no way to handle rotation, divergence, or deformation in an efficient manner unless it is known a priori. Assumption (2) also fails to account for motion discontinuities, such as those near cloud edges or within transparent motions. Furthermore, as there is no other constraint aside from constant brightness, PM methods struggle when there is little to no texture in the target and candidates. Quality control schemes are thus necessary to remove sectors that are poorly tracked with Eq. (4) in most AMV approaches.

PM was a popular method for AMVs over other optical flow approaches prior to the GOES-R era due to its simplicity, computational efficiency, and capability to handle displacements common in low-temporal-resolution satellite imagery (Bresky and Daniels, 2006). Linearizing the brightness constancy assumption in Eq. (2) means that large and nonlinear displacements (typically > 1 pixel between images) will not be captured (Brox et al., 2004). Thus, most optical flow computations initially subsample images to the point at which all the displacements are initially less than 1 pixel (Anandan, 1989; discussed more in Sect. 3.1), which can cause fast-moving small features to be lost. Note that reducing the temporal resolution of GOES imagery (e.g., 10 min vs. 5 min scans) increases the displacement of typical meteorological features between frames. Furthermore, constancy assumptions are more likely violated with reduced temporal resolution since image intensity changes more through the evaporation and condensation of cloud matter over time. Thus, for the spatial resolution of ABI, it is impractical to consider optical flow gradient-based methods at temporal resolutions coarser than 5 min for several mesoscale meteorological phenomena, including OFBs. Very spatially coarse images do not need to be initially used with faster scanning rates, such as super rapid scan 1 min information (Schmit et al., 2013) or the 30 s temporal resolution mesoscale mode of ABI (Schmit et al., 2017).

While the HS method is designed for deriving dense flow in imagery sequences, it also does not account for motion discontinuities in the flow fields. Hence, it suffers from incorrect flow derivations near cloud edges and would perform poorly for OFB detection and tracking. Black and Anandan (1996) offer an intuitive solution to this problem, whereby the energy functional is designed to minimize robust functions that are not sensitive to outliers. 5EU=∫∫Ωρd∇I⋅U+It+ρs|∇2u|2+|∇2v|2dx The robust function data term for the HS method is simply ρdr=r2 and smoothness ρsr=r, which implies that energy functionals increase quadratically for r outliers. Other robust functions can also be minimized with similar gradient descent algorithms to Gauss–Seidel iterations, while being less sensitive to outliers (Press et al., 1992; Black and Anandan, 1996). Robust functions are popular in recent optical flow literature (Brox et al., 2004; Sun et al., 2014), and a similar approach adopted here is discussed further in the Methodology section. The reader is referred to works by Barron et al. (1994), Fleet and Weiss (2005), Sun et al. (2014), and Fortun et al. (2015) for more comprehensive reviews on optical flow background and techniques.

The relevance of optical flow in satellite meteorological research continues to increase now that scanning rates of sensors such as the ABI are routinely at sub-5 min timescales, making motion easier to derive objectively (Bresky and Daniels, 2006; Héas et al., 2007; Wu et al., 2016). The dense motion estimation within fine-temporal-resolution data has yet to be used for feature identification. Optimizing optical flow for this purpose, and its specific application to OFBs, is the aim of this study. The next section outlines our approach to this end.

As recently overviewed in Fortun et al. (2015), there are several optical flow approaches that provide dense motion estimates that account for the weaknesses highlighted in Fig. 1b. Many have their own advantages and drawbacks in terms of computational efficiency, flexibility, and capability to handle large displacements, motion discontinuities, texture-less regions, and turbulent scenes. We selected an approach here by Brox et al. (2004) (Hereafter B04), given its simplicity, current availability of open-source information, and excellent documentation. The reader is cautioned, however, that dense optical flow is a rapidly evolving field, and research is currently underway to improve present techniques. While dense optical flow validation for satellite meteorological application research like OFB identification is taking place, the reader is referred to the Middlebury (Baker et al., 2011), the MPI Sintel (Butler et al., 2012), and the KITTI (Geiger et al., 2012) benchmarks for extensive validation statistics of the most recent techniques using image sequences for more general applications.

The B04 approach handles the drawbacks described in Fig. 1b and more, whereby the brightness constancy assumption is no longer linearized, i.e., 6EU=∫∫ΩρdIx+U,t+Δt-Ix,t2+γ∇2Ix+U,t+Δt-∇2Ix,t2+αρs|∇2u|2+|∇2v|2dx. Following B04, within the data robust function, we now have also included a gradient constancy assumption, which is weighted by a constant γ to make the derived flow more resilient to changes in illumination. Avoiding linearization of constancy assumptions improves the identification of large displacements between images. The Charbonnier penalty is used for the data and smoothness robust functions following Sun et al. (2014), 7ρdr2=ρsr2=r2+ϵ2, with ϵ representing a small constant present to prevent division by zero in minimization, set to 0.001. The values for U are found by solving the Euler–Lagrange equations of Eq. (6) with numerical methods: 8Eu-dEuxdx-dEuydy=0,9Ev-dEvxdx-dEvydy=0, with reflecting boundary conditions and subscripts that imply the derivatives. Equations (8) and (9) are solved with a nested fixed-point successive over-relaxation iteration scheme described in B04 and summarized in Fig. 2. The reader is referred to Chap. 4 of Brox (2005) for details on the full discretization of the derivatives in the successive over-relaxation scheme. Here, only the spatial dimensions are used for the smoothing term, though it is possible to include the time dimension with this system as well.

A difficulty in solving Eqs. (8) and (9) is that the successive over-relaxation scheme may converge on a local minimum of E(U) rather than finding the global minimum. The typical approach to find the global minimum is to compute optical flow with coarse- to fine-scale warping iterations (e.g., Anandan, 1989). Coarse- to fine-scale warping iterations work by subsampling the initial image at the native resolution to a coarser spatial resolution and computing the flow initially at the coarsest resolution in the image pyramid. The U results from the coarse image flow are then used as the first-guess field for the next finest scale on the image pyramid (Fig. 3), and the second image is warped accordingly. The warping step ensures that estimated displacements at every step in the image pyramid remain small.

The B04 scheme includes coarse- to fine-scale warping iterations at every outer iteration k. This means that the first iteration is run on a subsampled image, and the subsampling is reduced by a scale factor at every k until the image reaches the native resolution at the final k=nK. Images at every k in this subsampling are found using a Gaussian image pyramid technique with bicubic interpolation. The flow values of the image at k-1 are also upscaled accordingly at k with bicubic interpolation (the initial flow guess is u=v=0 at k=0). For improved computation of spatial derivatives, the initial image is also smoothed with a 9×9 pixel kernel Gaussian filter with a standard deviation set to 1.5 pixels. The specific settings used for the coarse- to fine-warped flow scheme here are shown in Table 1.

There are two steps to the objective OFB identification process. First, a linear feature or sharp boundary is identified in visible or infrared imagery. In some cases, the first step alone is enough to identify OFBs subjectively. The second step is tracking that feature back in time to see where it originated from (typically, near an area with deep convection). In the case of near-stationary convection and low-level flow, a forecaster might also use radial-like propagation in this decision-making process; however, since convection geometry and low-level flow vary from storm to storm, only the first two steps are considered here. This approach aims to mirror the subjective process, leveraging the information content of optical flow to do so.

To handle the first step of line feature identification, a simple image line detection scheme was performed by convolving the original brightness field with a set of line detection kernels, so 10L=∑i=14ai⋆G(R), where ⋆ is the convolution operator, G is a Gaussian smoothing function (using a 21×21 kernel and standard deviation of 5 pixels), R is the reflectance factor (radiance times the incident Lambertian-equivalent radiance, or the “kappa factor”; Schmit et al., 2010), L is the resulting line detection field, and ai represents the two-dimensional line detection kernels defined as follows. a1=-1-1-1222-1-1-1a2=-12-1-12-1-12-1a3=2-1-1-12-1-1-12a4=-1-12-12-12-1-1 The resulting L field exhibits higher intensities wherein line features exist (Gonzalez and Woods, 2007). A threshold of L≥0.02 was used here to indicate that a pixel contained a line feature. This method was compared to a subjective interpretation of boundary location for validation.

To address the second step of the process, the constrained optical flow approach described in Sect. 3.1 was used to track the boundary pixels (both objectively and subjectively identified) back in time for 3 h. The values of motion at each step in the backwards trajectory were determined with bilinear interpolation of the optical-flow-derived dense vector grid. If a back-traced pixel of the linear feature arrived within a 50 km great-circle distance of a 10.35 µm brightness temperature (BT10.35) pixel lower than 223 K (-50 ∘C; using previous satellite imagery matched to the back-trajectory time), the original point was considered an OFB. The area subtended by the 50 km great circles derived from BT10.35 is hereafter referred to as the “deep convection area.” While this brightness temperature threshold is subjective and can vary from case to case, it was found to produce a reasonable approximation of deep convection areas when compared to ground-based radar information for the case study described in the subsequent sections.

The objective OFB identification methodology is tested using a case study from 5 July 2018 over the southwestern United States. This event featured a distinct OFB and associated dust storm that was well-sampled by various ground- and space-based sensors. GOES-16 was in mode 3, generating one image over the study area every 5 min (continental US, or CONUS, ABI scan domain, NOAA, 2019). Optical flow computations employ the GOES-16 (GOES-East) ABI red band (0.64 µm; ABI channel 2), provided at a nominal sub-satellite spatial resolution of 500 m, but closer to 1 km at the case study location. This channel is used at native resolution, though it can be subsampled with a low-pass filter such that future versions can implement color information from the blue and near-infrared bands (e.g., Miller et al., 2012). This means that the optical flow approach here is daytime only. A similar B04 approach can be used on infrared data as well for day–night independent information, though for detecting OFBs in the low levels, proxy visible products would perform best. As described above, the clean longwave infrared band (10.35 µm; ABI channel 13) is used as first-order information on optically thick cloud-top heights and to assess the convective nature of the observed scene (BT10.35 < 223 K).

High-frequency Automated Surface Observing Stations (ASOSs; NOAA, 1998), recording temperature, pressure, wind speed, and direction once every minute, complement the satellite imagery. The Weather Surveillance Radar-1988 Doppler (Crum and Alberty, 1993) dual-polarimetric data also sampled the OFB event from the KIWA radar near Phoenix, AZ. To highlight the OFBs and the presence of dust, horizontal reflectivity and the correlation coefficient are used (Van Den Broeke and Alsarraf, 2016). Finally, for information on the full 3D dynamics of the case study, a numerical model representation of the environment was collected from the High Resolution Rapid Refresh system (HRRR; Benjamin et al., 2016). The combination of these model and observation datasets is employed to confirm the presence of a distinct convective OFB rather than some other quasi-linear feature, such as a bore or elevated cloud layer.