Spatial scale invariance of aggregated dynamics – Application to crops cycle observed from space

Abstract:

Observational data is always associated to specific time and space scales. When the observed area of study is homogeneous, the same dynamics can be expected at different observed scales. It is generally not the case. This is a common obstacle when comparing data or products of different resolution. This question is investigated here considering the cycles of rainfed crops observed from space in semi-arid regions. In such context, the rainfed crops are coupled to the climatic dynamics in a synchronized way, the observational signal can thus be seen as an aggregation of phase synchronized dynamics.

In the first part of this work, a case study is implemented. Rössler chaotic systems are used for this purpose as elementary oscillators relating to homogeneous behavior. The ‘observational’ signal is obtained by aggregating additively the signals of several elementary chaotic systems. Analytically, it is found that the aggregated signal can be approximated by the Rössler system itself but with some parameterization changes. This result can be generalized to any system for which a canonical approximation is possible. Using the global modeling technique [1], this theoretical result is then illustrated practically, by showing that an approximation of the Rössler dynamics can be retrieved, without any a priori knowledge, from the aggregated signal.

In the second part, the cycle of cereal crops observed from space in semi-arid conditions is investigated from real observational data (the GIMMS product of Normalized Difference Vegetation Index [2] is used for this purpose). A low-dimensional chaotic model could recently be obtained from a spatially aggregated signal which presents properties never observed from real data before: a toroidal and weakly dissipative dynamics [3]. These unusual properties are then retrieved at various places and scales.

[1] Mangiarotti S., Coudret R., Drapeau L. & Jarlan L., 2012. Polynomial search and Global modelling: two algorithms for modeling chaos. Physical Review E, 86(4), 046205.

[2] Tucker C. J., Pinzon J. E., Brown M. E., Slayback D. A., Pak E. W., Mahoney R., Vermote E. F. & Saleous N. E., 2005. Int. J. Remote Sensing, 26, 4485.

[3] Mangiarotti S., Drapeau L. & Letellier C., 2014. Two chaotic global models for cereal crops cycles observed from satellite in Northern Morocco. Chaos, 24, 023130.