Formulation of scale transformation in a stochastic data assimilation framework

Abstract. Understanding the errors caused by spatial-scale transformation in Earth observations and simulations requires a rigorous definition of scale. These errors are also an important component of representativeness errors in data assimilation. Several relevant studies have been conducted, but the theory of the scale associated with representativeness errors is still not well developed. We addressed these problems by reformulating the data assimilation framework using measure theory and stochastic calculus. First, measure theory is used to propose that the spatial scale is a Lebesgue measure with respect to the observation footprint or model unit, and the Lebesgue integration by substitution is used to describe the scale transformation. Second, a scale-dependent geophysical variable is defined to consider the heterogeneities and dynamic processes. Finally, the structures of the scale-dependent errors are studied in the Bayesian framework of data assimilation based on stochastic calculus. All the results were presented on the condition that the scale is one-dimensional, and the variations in these errors depend on the differences between scales. This new formulation provides a more general framework to understand the representativeness error in a non-linear and stochastic sense and is a promising way to address the spatial-scale issue.

[1]  Jeffrey P. Walker,et al.  Upscaling sparse ground‐based soil moisture observations for the validation of coarse‐resolution satellite soil moisture products , 2012 .

[2]  Robert N. Miller.  Topics in data assimilation: Stochastic processes , 2007 .

[3]  Kiyosi Itô.  109. Stochastic Integral , 1944 .

[4]  G. C. Pomraning.  Radiative transfer and transport phenomena in stochastic media , 1998 .

[5]  A. Stuart,et al.  Sampling the posterior: An approach to non-Gaussian data assimilation , 2007 .

[6]  Dongryeol Ryu,et al.  Multi‐scale spatial correlation and scaling behavior of surface soil moisture , 2006 .

[7]  Olivier Talagrand,et al.  Assimilation of Observations, an Introduction (gtSpecial IssueltData Assimilation in Meteology and Oceanography: Theory and Practice) , 1997 .

[8]  G. Eyink,et al.  A mean field approximation in data assimilation for nonlinear dynamics , 2004 .

[9]  C.,et al.  Analysis methods for numerical weather prediction , 2022 .

[10]  Preprint Mps.  The Error of Representation: Basic Understanding , 2014 .

[11]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[12]  Marc Bocquet,et al.  Bayesian design of control space for optimal assimilation of observations. Part II: Asymptotic solutions , 2011 .

[13]  M. Wild,et al.  Spatial representativeness of ground‐based solar radiation measurements , 2013 .

[14]  Robert G. Bartle,et al.  The Elements of Integration and Lebesgue Measure: Bartle/The Elements , 1995 .

[15]  Robert N. Miller,et al.  Data assimilation into nonlinear stochastic models , 1999 .

[16]  W. Otten,et al.  Soil architecture and preferential flow across scales , 2010 .

[17]  Tae Sup Yun,et al.  Upscaling of Navier–Stokes equations in porous media: Theoretical, numerical and experimental approach , 2009 .

[18]  P. Atkinson,et al.  Spatial Scale Problems and Geostatistical Solutions: A Review , 2000 .

[19]  Chunlin Huang,et al.  Representativeness errors of point-scale ground-based solar radiation measurements in the validation of remote sensing products , 2016 .

[20]  Xin Li,et al.  Characterization, controlling, and reduction of uncertainties in the modeling and observation of land-surface systems , 2013, Science China Earth Sciences.

[21]  J. Parajka,et al.  Scale effects in conceptual hydrological modeling , 2009 .

[22]  W. Rawls,et al.  Scaling properties of saturated hydraulic conductivity in soil , 1999 .

[23]  Y. Ran,et al.  Spatial representativeness and uncertainty of eddy covariance carbon flux measurements for upscaling net ecosystem productivity to the grid scale , 2016 .

[24]  Stephen E. Cohn,et al.  Treatment of Observation Error due to Unresolved Scales in Atmospheric Data Assimilation , 2006 .

[25]  W. Crow,et al.  Estimating Spatial Sampling Errors in Coarse-Scale Soil Moisture Estimates Derived from Point-Scale Observations , 2010 .

[26]  Y. Knyazikhin,et al.  Stochastic Modeling of Radiation Regime in Discontinuous Vegetation Canopies , 2000 .

[27]  J. Wiens.  Spatial Scaling in Ecology , 1989 .

[28]  R. Bartle.  The elements of integration and Lebesgue measure , 1995 .

[29]  Peter Jan van Leeuwen.  Representation errors and retrievals in linear and nonlinear data assimilation , 2015 .

[30]  Andrew C. Lorenc.  Atmospheric Data Assimilation and Quality Control , 2002 .

[31]  M. Bocquet,et al.  Beyond Gaussian Statistical Modeling in Geophysical Data Assimilation , 2010 .

[32]  S. Shreve.  Stochastic calculus for finance , 2004 .

[33]  T. Harter,et al.  Upscaling Hydraulic Properties and Soil Water Flow Processes in Heterogeneous Soils: A Review , 2007 .

[34]  E. Kassianov,et al.  Stochastic radiative transfer in Markovian mixtures: Past, present, and future , 2011 .

[35]  Michael F. Goodchild,et al.  Scale in a digital geographic world , 1997 .

[36]  Peter Jan van Leeuwen,et al.  Nonlinear Data Assimilation for high-dimensional systems , 2015 .

[37]  Wouter Dorigo,et al.  Characterizing Coarse‐Scale Representativeness of in situ Soil Moisture Measurements from the International Soil Moisture Network , 2013 .

[38]  T. Jackson,et al.  Field observations of soil moisture variability across scales , 2008 .