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.

The spatial scale in Earth observations and simulations refers to the observation footprint or model unit in which a geophysical variable is observed or modelled (scale is used below as an abbreviation for spatial scale). Scale is traditionally defined in terms of distance, which is not adequate both because distance is a one-dimensional quantity while scale generally refers to a two- or three-dimensional space and because the scale may change in a very complicated manner (for example, from an irregular observation footprint to a square observation footprint). Generally, the scale is not explicitly expressed in the dynamics of a geophysical variable, partially because a rigorous definition of scale is difficult to find, except for an intuitive conception (Goodchild and Proctor, 1997) and certain qualitative classifications of scale (Vereecken et al., 2007). This reflects the complexity of scale and consequently requires a more rigorous mathematical conceptualisation of scale.

The scale transformation of a geophysical variable may result in significant errors (Famiglietti et al., 2008; Crow et al., 2012; Gruber et al., 2013; Hakuba et al., 2013; Huang et al., 2016; Li and Liu, 2017; Ran et al., 2016). These errors are mainly caused by the strong spatial heterogeneities (Miralles et al., 2010; Li, 2014) and irregularities (Atkinson and Tate, 2000) that are associated with geophysical variables across different scales, and are also closely related to dynamic variations, e.g. in hydrological (Giménez et al., 1999; Vereecken et al., 2007; Merz et al., 2009; Narsilio et al., 2009), soil (Ryu and Famiglietti, 2006; Lin et al., 2010) and ecological (Wiens, 1989) processes. How to elucidate the scale transformation by developing mathematical tools has yet to be fully addressed.

Data assimilation could be an ideal tool to explore the scale transformation because it presents a unified and generalised framework in Earth system modelling and observation (Talagrand, 1997). Geophysical data are typically observed by various Earth observations; thus, updating the observation data in a data assimilation system may result in scale transformations between the observation space and system state space. If observation operator is strongly non-linear and complex, the errors caused by the scale transformation are even more serious (Li, 2014). An important concept that is related to the scale transformation in data assimilation is “representativeness error”, which is associated with the inconsistency in the spatial and temporal resolutions between states, observations and operators (Lorenc, 1986; Janjić and Cohn, 2006; van Leeuwen, 2014; Hodyss and Nichols, 2015), and the missing physical information that is related to a numerical operator compared to the ideal operator (van Leeuwen, 2014), such as the discretisation of a continuum model or neglect of necessary physical processes. The representativeness error and instrument error make up the observation error of data assimilation. Under the Gaussian assumption, they are independent of each other (Lorenc, 1995; van Leeuwen, 2014). This study will not consider the instrument error when formulating the scale transformation in data assimilation.

Recently, approaches have been developed to assess the representativeness error. Janjić and Cohn (2006) studied the representativeness error by treating system state as the sum of resolved and unresolved portions. Bocquet et al. (2011) used a pair of operators, namely, restriction and prolongation, to connect the relationship between the finest regular scale and a coarse scale, and determined the representativeness error using a multi-scale data assimilation framework. van Leeuwen (2014) considered two complicated cases, i.e. conducting the observation vector in a finer resolution compared with system state vector and assimilating the retrieved variables. Their solutions were formulated using an agent in observation or state space, and a particle filter was proposed to treat the non-linear relationship between observations, states and retrieved values. Hodyss and Nichols (2015) also estimated the representativeness error by investigating the difference between the truth and the inaccurate value that is generated by forecasting model.

Although these approaches explored the structure of the representativeness error and offered various solutions, improvements are still necessary to investigate the exact expression of the errors caused by scale transformation in data assimilation. The authors believe that these approaches are optimal in linear systems but may not be suitable when observations are heterogeneous and sparse, or when operators are non-linear between states and observations, although the general equations in the non-linear case were given. Without taking heterogeneities and non-linear operators into account, the representativeness error cannot be fully understood. However, heterogeneity varies depending on the situation and is difficult to formulate in a general theoretical study.

Data assimilation studies based on stochastic processes (Apte et al., 2007; Miller, 2007) or a stochastic dynamic model (Miller et al., 1999; Eyink et al., 2004) have been proposed recently. Compared to deterministic models, stochastic data assimilation is more applicable in an integrated and time-continuous theoretical study (Bocquet et al., 2010) and creates an infinite sampling space of the system state (Apte et al., 2007). Although the theorems of calculus that are based on stochastic processes (or stochastic calculus) are different from those of ordinary calculus, these advantages suggest that stochastic data assimilation offers a more general framework to study scale transformation.

We attempt to explore the mathematic definitions of scale and scale transformation, and then formulate the errors caused by the scale transformation on stochastic data assimilation in a general theoretical study. The next section introduces the basic concepts and theorems of measure theory, stochastic calculus and data assimilation. In Sect. 3, we present the definitions of scale and scale transformation. The posterior probability of system state is also reformulated by scale transformation in a stochastic data assimilation framework. In the final section, the contributions and deficiencies of this study are discussed.

The scale greatly depends on the geometric features of a certain observation footprint or model unit. The model unit is a specified subspace where a geophysical variable evolves in the model space; it could be a point, a rectangular grid, or an irregular unit such as a response unit (watershed, landscape patch, etc.). We offer a solution in which the definition of scale uses measure theory and the expression of a geophysical variable as a stochastic process uses stochastic calculus. Therefore, we first introduce several basic concepts of measure theory and stochastic calculus.

Let

If

Let

An

Next, the

Actually the outer measure does not match the two conditions of a measure,
but one can define the outer measure

The

In the two-dimensional case (

Additional details regarding measure theory can be found in the literature (for example, Billingsley, 1986; Bartle, 1995).

We then introduce some necessary concepts and theorems of stochastic calculus without proofs; their detailed derivations can be found in the literature (Itô, 1944; Karatzas and Shreve, 1991; Shreve, 2005).

Stochastic calculus based on Brownian motion produces an

As distinguishing features of stochastic calculus, the quadratic variation and drift can be regarded as stochastic versions of the variance and expectation, respectively. That is, the variance and expectation are instances of their stochastic counterparts within a certain integral path. Therefore, rather than being constants, the quadratic variation and drift are given in terms of probability.

We use the well-accepted Bayesian theory of data assimilation (Lorenc, 1995; van Leeuwen, 2015) to investigate its time- and scale-dependent errors. State and observation are first assumed to be one-dimensional.

A non-linear forecasting system can be described by

If a new observation is available at time

Previous studies (e.g. Janjić and Cohn, 2006; Bocquet et al., 2011)
described the origins of the components of

According to Bayesian theory, the posterior PDF of the state based on the
addition of a new observation into the system is

We define the scale based on the measure theory that was introduced in
Sect. 2. The relationship between Lebesgue measure in

Measure of a single-point observation: when the observation footprint is very small and homogeneous, we assume that its footprint approaches zero, and its measure is accordingly zero under the condition of the Lebesgue measure.

Measure along a line: the measure is a one-dimensional Lebesgue measure.

Measure of a rectangular pixel (for example, remote sensing observation):

Measure of a footprint-scale observation: the footprint is any bounded closed
domain

All of the above measures depend mainly on the shape and size of

Now, we can generalise the above examples by defining the

We represent the scale as

We can further define

If two scales follow the one-dimensional rule, they are geometrically
similar. This rule simplifies scale as a one-dimensional variable that
corresponds to the scale transformations between most remote sensing images
with various spatial resolutions. For example,

Layer 1 in Fig. 1 shows the relationship between the Lebesgue measure and
scale. The measure space

Diagram of the relationships among a Lebesgue measure, scale and geophysical variable.

Instead of using Eqs. (5) and (6), which are discrete in time, we use Ito
process-formed expressions with the one-dimensional infinitesimals d

A geophysical variable can be regarded as a real function

In Fig. 1, layer 2 presents a heterogeneous geophysical variable in the
entire region. If we aggregate layer 2 into layer 1 and let each pixel
intensity be the value for a geophysical variable in that pixel, then the
measure space

If the statistical properties of the geophysical variable are available, we
can construct an explicit stochastic equation for it. We introduce the
time-dependent Ito process Eq. (1) to define the geophysical variable
process:

Equation (9) can be regarded as a continuous-time version of Eq. (5), i.e.
the estimation of the state is equal to the integral of Eq. (9) over a time
interval. Here,

The state in the forecasting step can be expressed by Eq. (9) because only time is involved. In the analysis step of data assimilation, the state does not pertain to time, and we assume that the scale has a quantifiable effect on the errors in this step; thus, both the states and observations can be defined by Eq. (10).

First, we provide the following lemma.

Note that in the definition of Brownian motion, the parameter starts at zero.
However, the scale is realistically greater than zero, which means that it
cannot be directly applied in Brownian motion. Therefore, Lemma 1 is logical
because it implies that

In the following content, we use Brownian motion with a parameter that starts
at

In assumption 1, the one-dimensional rule ensures that scale changes in a sense of geometrical similarity (for example, from a larger square observation footprint to a smaller square observation footprint, or from C2 to C3 as presented in Fig. 1). Therefore, based on assumption 1, scale only varies in one-dimensional space, meaning that the corresponding scale transformation is an integral over one-dimensional space.

Assumption 2 indicates that the model unit and state scale are supposed to be the same and both invariant in space and time. Thus, there is no scale transformation in the forecasting step; thus, Eq. (9) can adequately describe this step.

Based on assumption 3, the analysis step is related to the scale. The scale
transformation is only involved in the process of mapping the state vector
from state space to observation space. According to Eq. (10), the state and
observation in the analysis step are

Based on the above discussion, the integral forms of the state are

The Bayesian equation of data assimilation (Eq. 7) produces the posterior PDF

Then, we calculate the posterior PDF. The scale-dependent observation
operator is

Assumption 1 suggests that the observation and state spaces have the same
probability measure; thus, the Brownian motions in these two spaces are
equivalent. Equation (19) can also be rewritten by replacing

The quadratic variation of Eq. (20) is

In particular, if

In Eq. (24), the integral

The significance of Eqs. (20)–(25) is that the effect of scale on the
posterior PDF can be determined quantitatively. In addition to the model
error and instrument error (both were not introduced explicitly in this study
because they have little influence on the error caused by scale
transformation), a new type of error in data assimilation was discovered in
the analysis step. The expectation of the posterior PDF may vary with the
scale of the state space if

To explicitly show how the stochastic scale transformations impact
assimilation, we introduce an illustrative example based on the scales
presented in Fig. 1. Assume that in the analysis step, the state has the
standard scale

Now the scales of state space and observation space obey the one-dimensional
rule, and we further presume that the measure space

If

To formulate the likelihood function in the case that the observation is different from the state, the SRTE will be employed in the following text. The SRTE is a stochastic integral-differential equation that describes the radiative transfer phenomena through a stochastically mixed immiscible media. Scientists have developed analytical or numerical methods for finding the stochastic moments of the solution, such as the ensemble averaged and the variance of the radiation intensity (Pomraning, 1998; Shabanov et al., 2000; Kassianov and Veron, 2011).

Consider the general expression of the SRTE (leaving out the scattering and
emission),

To tie into more substantial random optical properties of the transfer
media, such as absorption and scattering, the optical depth

The analytical solution of Eq. (26) is

SRTE can be considered as a concrete instance of a stochastic observation
operator by defining

Integrating both sides of Eq. (28) yields the general solution of the
radiation intensity,

Considering that the optical depth

Our study offered a stochastic data assimilation framework to formulate the errors that are caused by scale transformations. The necessity of the methodology, the difference from previous works by other investigators, and the advantages and limitations of this study are discussed as follows.

The reasons that the methodology focuses on a stochastic framework are as
follows. First, the stochastic data assimilation framework is essentially
consistent with the concepts of scale and scale transformation; both are
associated with corresponding measure spaces

The significant innovation of this work is as follows. We developed a more
rigorous formulation of the scale and scale transformation based on Lebesgue
measure, which places the related concepts in a rigorous mathematical
framework and then provides a new understanding of the errors caused by scale
transformation. In addition, due to the Ito process-formed state and
observation, a stochastic data assimilation framework was proposed by
considering the non-linear operators, heterogeneity of a geophysical variable
and a general Gaussian representativeness error. The scale transformation is
also non-linear if the one-dimensional rule is not applied. Additionally, Ito
process-formed state and observation offer the drift rate (i.e.

As a theoretical exploration towards scale transformation and stochastic data assimilation, there is still much room for improvement. First, we reduced the scale transformation by the one-dimensional rule, and let the variables in data assimilation evolve regularly according to assumptions 1–3; thus, only the ideal result was investigated. Therefore, an in-depth and comprehensive exploration should be conducted in the future to describe other situations in the real world. However, the use of either an arbitrary scale transformation or the geophysical variable without ignoring the drift rates will obtain lengthy results. Therefore, the second improvement focuses on how to make the formulation more concise. Lastly, noting that all the results in our framework were given in terms of probability, it is necessary to implement real-world applications of these theoretical results, such as introducing some concrete dynamic models to formulate the Ito process-formed geophysical variable of scale.

In this study, we mainly addressed two basic problems associated with scale transformation in Earth observation and simulation. First, we produced a mathematical formalism of scale and scale transformation by employing measure theory. Second, we demonstrated how scale transformation and its associated errors could be presented in a stochastic data assimilation framework.

We revealed that the scale is the Lebesgue measure with respect to the observation footprint or model unit. The scale is related to the shape and size of a footprint, and scale transformation depends on the spatial change between different footprints. We then defined the geophysical variable, which further considers the heterogeneities and physical processes. A geophysical variable consequently expresses the spatial average at a specific scale.

We formulated the expression of scale transformation and investigated the error structure that is caused by scale transformation in data assimilation using basic theorems of stochastic calculus. The formulations explicate that the first-order differential of the non-linear observation operator should be considered in representativeness error, and the uncertainty of representativeness error is directly associated with the difference between scales. A concrete physical models (SRTE) was introduced to demonstrate the results when observation operator is non-linear.

This work conducted a theoretical exploration of formulating the errors caused by scale transformation in a stochastic data assimilation framework. We hope that the stochastic methodology can benefit the study of these errors.

No data sets were used in this article.

Basic notations.

New notations.

The authors declare that they have no conflict of interest.

We thank the executive editor of NPG, Olivier Talagrand, and his kind help and valuable comments on our manuscript. We also thank Peter Jan van Leeuwen and another anonymous reviewer for their valuable comments and suggestions. This work was supported by the NSFC projects (grant numbers 91425303 and 91625103) and the CAS Interdisciplinary Innovation Team of the Chinese Academy of Sciences. Edited by: Olivier Talagrand Reviewed by: Peter Jan van Leeuwen and one anonymous referee