Spatially Developing Modes: The Darcy–Bénard Problem Revisited

In this paper, the instability resulting from small perturbations of the Darcy–Bénard system is explored. An analysis based on time–periodic and spatially developing Fourier modes is adopted. The system under examination is a horizontal porous layer saturated by a fluid. The two impermeable and isothermal plane boundaries are considered to have different temperatures, so that the porous layer is heated from below. The spatial instability for the system is defined by taking into account both the spatial growth rate of the perturbation modes and their propagation direction. A comparison with the neutral stability condition determined by using the classical spatially periodic and time–evolving Fourier modes is performed. Finally, the physical meaning of the concept of spatial instability is discussed. In contrast to the classical analysis, based on spatially periodic modes, the spatial instability analysis, involving time–periodic Fourier modes, is found to lead to the conclusion that instability occurs whenever the Rayleigh number is positive.

Analysis of tear film spatial instability for pediatric myopia under treatment

In Taiwan, the prevalence of myopia in children between 6 and 18 years old is over 80%, and high myopia accounts for over 20%, which turned out to be in the leading place worldwide. Orthokeratology and low-dose atropine are proven treatments to reduce myopia progression, though the potential corneal disturbances remain an issue in young populations. The alteration of the tear film is widely discussed but there is no consensus to date, so we aim to investigate the tear film spatial instability in children with myopia control using atropine or orthokeratology. Thirty-eight treatment-naïve participants and 126 myopic children under treatments were enrolled. The ocular surface homeostasis, spatial distribution of tear break-up, and high-order aberrations (HOAs) of the corneal surface were assessed. We found out that myopic children treated with either atropine or orthokeratology showed ocular surface homeostasis similar to that in treatment-naïve children. Nevertheless, children treated with orthokeratology presented higher HOAs (p < 0.00001) and a tendency of the first tear break-up zone at the inner half of the cornea (p = 0.04). This unique spatial instability of the tear film associated with myopia treatment might provide a more focused way of monitoring the pediatric tear film instability.

Statistical Stability and Spatial Instability in Mapping Forest Tree Species by Comparing 9 Years of Satellite Image Time Series

Mapping forest composition using multiseasonal optical time series remains a challenge. Highly contrasted results are reported from one study to another suggesting that drivers of classification errors are still under-explored. We evaluated the performances of single-year Formosat-2 time series to discriminate tree species in temperate forests in France and investigated how predictions vary statistically and spatially across multiple years. Our objective was to better estimate the impact of spatial autocorrelation in the validation data on measurement accuracy and to understand which drivers in the time series are responsible for classification errors. The experiments were based on 10 Formosat-2 image time series irregularly acquired during the seasonal vegetation cycle from 2006 to 2014. Due to lot of clouds in the year 2006, an alternative 2006 time series using only cloud-free images has been added. Thirteen tree species were classified in each single-year dataset based on the Support Vector Machine (SVM) algorithm. The performances were assessed using a spatial leave-one-out cross validation (SLOO-CV) strategy, thereby guaranteeing full independence of the validation samples, and compared with standard non-spatial leave-one-out cross-validation (LOO-CV). The results show relatively close statistical performances from one year to the next despite the differences between the annual time series. Good agreements between years were observed in monospecific tree plantations of broadleaf species versus high disparity in other forests composed of different species. A strong positive bias in the accuracy assessment (up to 0.4 of Overall Accuracy (OA)) was also found when spatial dependence in the validation data was not removed. Using the SLOO-CV approach, the average OA values per year ranged from 0.48 for 2006 to 0.60 for 2013, which satisfactorily represents the spatial instability of species prediction between years.

Statistical Stability and Spatial Instability in Mapping Forest Tree Species by Comparing 9 Years of Satellite Image Time Series

Mapping forest composition using multiseasonal optical time series is still challenging. Highly contrasted results are reported from one study to another suggesting that drivers of classification errors are still under-explored. We evaluated the performances of single-year Formosat-2 time series to discriminate tree species in temperate forests in France and investigated how predictions vary statistically and spatially across multiple years. Our objective was to better estimate the impact of spatial autocorrelation in the validation data on measurement accuracy and to understand which drivers in the time series are responsible for classification errors. The experiments were based on ten Formosat-2 image time series irregularly acquired during the seasonal vegetation cycle from 2006 to 2014. Due to lot of clouds in the year 2006, an alternative 2006 time series using only cloud-free images has been added. Thirteen tree species were classified in each single-year dataset based on the SVM algorithm. The performances were assessed using a spatial leave-one-out cross validation (SLOO-CV) strategy, thereby guaranteeing full independence of the validation samples, and compared with standard non-spatial leave-one-out cross-validation (LOO-CV). The results show relatively close statistical performances from one year to the next despite the differences between the annual time series. Good agreements between years were observed in monospecific tree plantations of broadleaf species versus high disparity in other forests composed of different species. A strong positive bias in the accuracy assessment (up to 0.4 of Overall Accuracy) was also found when spatial dependence in the validation data was not removed. Using the SLOO-CV approach, the average OA values per year ranged from 0.48 for 2006 to 0.60 for 2013, which satisfactorily represents the spatial instability of species prediction between years.