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This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

With the development of quantitative remote sensing, scale issues have attracted more and more the attention of scientists. Research is now suffering from a severe scale discrepancy between data sources and the models used. Consequently, both data interpretation and model application become difficult due to these scale issues. Therefore, effectively scaling remotely sensed information at different scales has already become one of the most important research focuses of remote sensing. The aim of this paper is to demonstrate scale issues from the points of view of analysis, processing and modeling and to provide technical assistance when facing scale issues in remote sensing. The definition of scale and relevant terminologies are given in the first part of this paper. Then, the main causes of scale effects and the scaling effects on measurements, retrieval models and products are reviewed and discussed. Ways to describe the scale threshold and scale domain are briefly discussed. Finally, the general scaling methods, in particular up-scaling methods, are compared and summarized in detail.

The advance of remote sensing technology in the 20th century has provided a powerful means to conduct regional and global measurements. Remote sensing technology can quickly access a wide range of real-time land surface spatial information and provides an effective way for resource surveys, environmental monitoring and disaster prediction. With the help of remote sensing technology, one can get geo-information quickly, accurately, efficiently and comprehensively. Undoubtedly, remote sensing will play an increasingly important role in the field of geosciences.

At present, remote sensing technology has entered an era of quantitative analysis. Thus, the important issues – scale effects and scaling – have already become one of the most important research focuses of remote sensing [

As we know, research about scale has already permeated various disciplines, such as hydrology [

Obviously, research is suffering from a severe scale discrepancy between data sources and the models used. Both data interpretation and model application become difficult due to scale issues. Openshaw [

To scientists, scale is undoubtedly one of the most important bases of research. What we study cannot be disengaged from the scale. Goodchild and Quattrochi [

The aim of this paper is to demonstrate scale issues from the points of view of analysis, processing and modeling and to provide technical assistance when facing scale issues in remote sensing. The next sections will be organized as follows. In Section 2, some basic definitions will be given, as researchers may not seem to have agreed on the meaning of such concepts. They are the basis of analysis the scale issues. In Section 3, the mechanism of scale effects, which may benefit the scaling model, is briefly discussed. The main causes of scale effects and the scaling effects of the measurements, retrieval models and products will be reviewed and discussed. These are the key points in resolving scale issues in remote sensing. If the scale effects could be estimated correctly, the scaling model would become easier. One of the great advantages of remote sensing is its capacity for providing data at multi-scales, which is increasingly used to evaluate the influence of scale effects for identifying structure or patterns and modeling results. This raises a problem: where is the interval in which the phenomenon or the structures are nearly invariable or slowly variable? And, what is the validation scope of the retrieval models? In Section 4, the quantitative descriptions of scale threshold (invariance of scale) and scale domain are given. The analysis of scale threshold and the scale domain is imperative for understanding the dynamics of landscapes. In addition, they can be thought of as the linkage point between heterogeneity and homogeneity, which is closely related to the scale effects. Demonstrated in Section 5 are a few general scaling approaches, especially the up-scaling method, to characterize the influence of scale. Although the hypotheses and starting points are not the same, all roads lead to the same end result. These scaling methods can enhance upscaling from one scale to another. Under different circumstances, however, we can choose different methods.

In the field of scientific research, scale is often one of most involved terminologies. The scale can refer both to the magnitude of a study (e.g., its geographic extent) and also to the degree of detail (e.g., its level of geographic resolution) [

Observation scale can be called a “measurement scale”. It depends on the method or the characteristics of the instrument and can be thought of as measurement units (i.e., intervals or areas or volumes) at which data is measured or sampled. To remote sensing, the measurement scale refers to the description of resolution, time interval, spectral range, solid angle or polarization direction. As the limitation of data collection and storage capacity, the smaller measurement scale usually corresponds to the smaller geographic scale and vice versa.

Modeling scale is the scale at which the model is built or derived in order to give reliable output. Both the measurement scale and the operational scale may influence the modeling scale. Observations sampled at a measurement scale are used as input for models, so the measurement scale must coincide with the modeling scale. If the measurement scale is smaller or larger than the modeling scale, it should be scaled. Again, a model needs to reveal the process; the modeling scale should also coincide with the operational scale. Similarly, it also needs to be scaled.

Operational scale refers to the scale at which a certain process is supposed to operate. It can also be called the “scale of action”. For example, thunderstorms may happen in an area of dozens of square kilometers. The operational scale of thunderstorms may be dozens of kilometers. It can be defined either as spatial extent (the lifetime), period (cycle) or the correlation length (integral scale), depending on the nature of the process [

Geographic scale, which is also called “coverage”, refers to the spatial extent of research. It determines the biological organization level on which the surface property is observed, such as the leaves (a few centimeters), the canopy (10 to 100 m), the landscape (100 m to a few kilometers) or the region (about 100 km) [

Policy scale is the scale at which the decisions are made or the policy is implemented [

Cartographic scale is defined simply as the ratio between distance on the map and on the ground. It is often used to represent the spatial distribution of research results. Generally speaking, a smaller cartographic scale corresponds to a larger geographic scale and may show fewer instances of features or less detail when compared to a larger cartographic scale.

The six meanings of scale described above are mutually related. It is indispensable to determine the desirable scale before investigation. But, how do you select the suitable scale in remote sensing? Generally speaking, the choice of scale may only depend on the goals of the study if you do not take other factors (i.e., manpower, finance and time) into account. Commonly, the policy scale is determined first. And then, the operational scale is decided based on the previous knowledge of the research. Due to the fact that the policy scale is selected by the decision-making department and the operational scale is the natural characteristic of the process, they may have nothing to do with remote sensing. Remote sensing may only be used to provide knowledge to reveal the actual operational scale. By comparion, the observation scale, modeling scale and geographic scale are more closely related to remote sensing. They are more or less determined by the application of remote sensing. The smaller observation scale is not always correct. For example, the optimum observation scale for classification in land use and land cover is the scale where the variability within classes is at its minimum and the variability between classes is at its maximum. On the whole, the geographic scale should be large enough to characterize the image spatial variability or structures. At the same time, the observation scale and modeling scale should be smaller than the operational scale and be mutually consistent with each other. The result is not reliable when the observation scale and the modeling scale are totally different. Finally, the cartographic scale is determined to show the results and images which serve for decision-making after the research.

Although “scale” is a widely used term and has different meanings in various disciplines, in general, it can be thought of as having multi-dimensionality, complexity and variability. Firstly, scale has a multi-dimensional nature [

Secondly, scale has complex hierarchies. It is the reflection of the level of the organization of nature, which results in the research targets varying with scales. For example, small-scale hydrological studies may mainly focus on the scale of vegetation and soil; meso-scale studies may focus on the response of the hydrology unit to the changes of land surface; while large-scale studies may be mainly about the interaction of the atmosphere and the land surface. These phenomena and processes occurring at different scales may interact with each other, consequently, many regional or global changes, such as pollution, the greenhouse effect and biodiversity may be rooted in local scale or small-scale environmental problems. Similarly, large-scale changes (such as global climate change and ocean circulation anomalies), in turn, will influence small-scale phenomena and processes. This shows that both large-scale and small-scale studies are equally important. Large-scale describes the abstract features or macro-structure, and small-scale characterizes the details. On the one hand, we can understand the macro-changes and the general trend through large-scale studies; on the other hand, we may find the mechanism of the development of the process and give reasonable explanations through small-scale studies.

Finally, scale may also have variability, that is, the targets at different scales will show different characteristics. The isothermal surface would become non-isothermal. The spectral curve of emissivity may become smoother when the spectral scale is coarser. It has increased the difficulty of scale analysis.

With the development of scale research, scientists have found that the dominant factors which affect the processes change with the scale. Marceau and Hay [

Apart from the concept of scale, we have to pay attention to the terms, “scaling” and “scale effects”. Scaling, is just defined as transferring information across scales [

In order to analyze the mechanism of scale effects, firstly, we need to abstract the retrieval process from reality, which is concluded as follows:

Here, _{n}_{n}_{2} and distributed products < _{1}. The other is to use < _{2} and < _{1} through the retrieval model and the inverse model to generate the corresponding ones, thereby producing lumped products < _{2} and equivalent measurements < _{1}. It is difficult to determine which are best. We can only select the appropriate one by real situations. For example, the goal of scaling for leaf area index (LAI) is to make the values derived from coarse resolution sensor data equal to the arithmetic average of values derived independently from fine resolution sensor data [_{1} may be more suitable because it follows the law of conservation of matter. Otherwise, < _{2} may be more advisable. The product of temperature is an example. Here is the other thing we need to pay more attention to. The aggregation may not be area-weighted, the aggregation of radiance in a heterogeneous terrain region should consider both the area and the local slope angle effects [_{1} and < _{2}, and < _{1} and < _{2} may be the focus of scale research. From the discussion above, the research on scale effects and scaling in remote sensing should begin around the points of view of measurements, retrieval models and products.

The main causes of scale effects can be summed up in three main reasons from the perspective of analysis, processing and modeling.

The first reason is the limitation of measurement. Any measurement equipment has its own scale representation. It can only reflect the specific information within the scope of observation. An infrared radiometer at ground level can merely represent the temperature at the scale of points; however, the Large Aperture Scintillometer (LAS) can reflect the exchange of energy at the scale of a region. Zhang

The second reason is the scale applicability of the retrieval models. The retrieval models do not explicitly express the characteristics of scale; however, they may be suitable for homogeneous surface or point measurements [

The third reason is the heterogeneity of land surface and the characteristics of linearity or nonlinearity of the retrieval models. These two factors affect the scale effects together. If the measurements are homogeneous, it would not cause scale effects no matter whether the retrieval models are linear or not. The heterogeneity could be thought of as the inherent nature of land surfaces, which are a mosaic of different cover types. In other words, heterogeneity would be considered as the surface properties vary over the observed scene [

The mechanism analyses of scale issues undoubtedly involve several questions. The first one is what the main causes of scale issues are in remote sensing, which has already been analyzed above. The following one is what the effects of scale on the measurements, models and products are.

With measurements, what we are more concerned about may be the mean, variance and correlation lengths. Bloschl

To the retrieval models, the effects of scale may be difficult to analyze. We may select suitable models at the corresponding scale; for example, the radiative transfer model of vegetation at the leaf scale [

To the products, the effects of scale have already been widely discussed. There is conflicting conclusions in the literature as to whether products are scale dependent or scale free. The main reason is that the scaling effects are usually dependent on the real application. If the retrieval model that is used is linear, there may be no scale effects, yet when the retrieval model has a uniform form for all the land covers, for example, if simply mapping the reflected solar radiation to the surface albedo, we can argue that the reflected solar radiation parameterization is scale invariant. Otherwise, the nonlinearity of albedo with topography and spectral dependence of albedo and the reflected solar radiation would be scale dependent. This demonstrates that a different parameterization and different assumptions related to the retrieval model can lead to different conclusions for the same physical process [

After discussing on the main causes of scale issues and the effects of scale on the measurements, models and products, the analysis of scale threshold and scale domain becomes an another critical problem waiting to be resolved. It is the basis of understanding scale issues. At present, one great advantage of remote sensing is the capacity to provide data at various resolutions. It may become easier to identify the scale thresholds below by identifying which biophysical or geophysical variables are spatially dependent and whether they become less dependent or independent. Here, we may take the scale domain as the appropriate scale for a given geographical environment. The relevant knowledge of scale threshold and scale domain may benefit the understanding of the validation scope of the retrieval model, dynamics of landscapes. In the following, several representative methods will be presented. Although these approaches may not necessarily apply to all cases, they in fact provide effective ways to cope with the problems.

Moellering and Tobler [

The wavelet transform method, which is a relatively new mathematical technique, has already been widely used in various disciplines. It uses a localized function in time or space. The wavelet’s size can be adjusted and shifted to analyze a data set. Thus, we can investigate features of interest in the data set at an appropriate scale, for example, broad features at a large scale and fine features at a small scale. With the help of the wavelet transform method, we can find where changes in a data set take place and simultaneously measure how large these changes are. Percival [

In order to choose an appropriate scale for a particular application, Woodcock and Strahler [

The semivariogram is often used as a tool to measure the difference in property values at two sample locations as a function of the distance between these locations. It provides the mean characteristics of spatial heterogeneity at the image scale. There are three features to characterize the semivariogram: nugget, sill and range. These features can be used to characterize and quantify the spatial heterogeneity of a land surface [_{c}_{c}_{c}_{c}

The term “fractals”, first proposed by Mandelbrot [

In order to solve scaling problems and compensate for scaling effects, several authors have already developed some frameworks. These frameworks provide a few systematic approaches to characterize the influence of scale on the measurements, retrieval models and products. In order to better understand these approaches, we classify them into three main categories which will be briefly summarized below.

Scaling methods for measurements are easier to deal with because the measurements recorded by remote sensors usually capture the radiance emitted or reflected by the surface. In such cases, the Area-Weighted Scaling Methods (AWM) may be applicable in a flat region. Otherwise, the influence of slope angle should be taken into account [

Generally speaking, we may only know the retrieval model at a specific scale. That is to say, we need to scale the retrieval models to the other scales through the appropriate assumptions and simplifications in order to not only consider the scaling effects but also provide scale invariant algorithms. Raffy [

Compared to the scaling methods of measurements and retrieval models, the scaling methods for products are more widely studied in research. These scaling methods provide a few systematic approaches to characterize the land surface heterogeneity and compensate for scaling effects. Here, we just emphasize the spatial domain. Other scaling methods applied in different domains can be realized by certain approaches. For example, scaling in the temporal domain may consider the general diurnal patterns of meteorological variables [

ERM is simply used to empirically calibrate the relationship of products between fine and coarse scales [

TSEM is based on Taylor’s theorem of linearizing the retrieval model around the arithmetic average of measurements [

Generally, the variance and covariance within one pixel are important for the traditional textural parameters that can capture the spatial variability of the surface. However, they are difficult to be estimated due to the fact that the concurrencies of high and low resolution images are often not available. In addition, they may not be used to discriminate the situations where the surface heterogeneity can be caused either by the cover type changes or by density change within the same cover type. If the surface heterogeneity is caused by cover type changes, the linear models which have different forms for different cover types may also cause scale effects. In contrast, if the surface heterogeneity is caused only by density change, the non-linearity of the model would generally cause very small scale effects [

SFSM is based on the simple scaling and multiscaling characteristics of products. Dubayah

Scale has already been recognized as a crucial concept in the description of the hierarchical structure of our world. Undoubtedly, remote sensing will advance the development of scale research. In reviewing the scale issues in remote sensing from analysis, modeling and demonstrating perspective, we found that there is no universal scaling method. Each has specific problems and limitations, although they could provide the possibility for solving scaling problems and compensating for scaling effects. The decision as to which method to use depends on real situations. The reason why no one method is proved to be effective may mainly stem from the heterogeneity of land surface and the nonlinearity of the retrieval models. We may expect the advancement in the understanding of the scale threshold and scale domain would bring rapid progress of scale research in remote sensing.

Although important steps have already been made, the research concerning scale is still in the initial stage and the scaling methods are not mature. In the future, we may pay more attention to: 1) How to effectively scale data with the concurrence of both local and large scale data. The optimal scale to estimate land surface parameters, the relationship between scale domain and coefficients of scaling can be deeply investigated when the data at different scales is available. 2) How to characterize the inner pixel heterogeneity without the concurrence of local data. We may use other spectral bands (i.e. optical, thermal or microwave sensors) or combine the Land Model with time-series data to estimate the heterogeneity within the pixel. 3) How to well combine the TSEM and CPM and absorb the advantages of both methods. The effect of heterogeneity caused either by the change of density or the discontinuity on scaling can be resolved. 4) How to well define the input variables, models and output parameters. Different parameterizations may result in different conclusions. Furthermore, the clear separation of system errors from the retrieval model and scale effects is also the key perspective of scale research. 5) How to effectively validate products at different scales. This needs the development of scale research. We will face the challenges to bridge the gap between theory and application.

We are grateful to the anonymous referees for comments on a draft of this paper. This work was supported by the National Natural Science Foundation of China under Grant 40425012 and by Chinese 973 Program under Grant 2007CB714402.

The relationship of measurements, retrieval model and products at different scales.

Comparison of the six meanings of scale used in the field of scientific research.

Observation scale | The measurement units at which data is measured or sampled | Referring to the description of resolution, time interval, spectral range, solid angle or polarization direction. |

Modeling scale | The scale at which the model is built or derived | In order to better reveal the process, the modeling scale should be coincided with both the observation scale and the operational scale. |

Operational scale | The scale of action at which a certain process is supposed to operate. | Depending on the nature of the process. Variability lower than modeling scale may be lost if the operational scale is smaller than the modeling scale. |

Geographic scale | The spatial extent of research | A larger geographic scale study involves a larger spatial area and a smaller geographic scale study only contains a smaller spatial area. |

Policy scale | The scale at which the decisions are made or the policy is implemented | In order to infer a reliable conclusion, the policy scale should be larger than the operational scale. |

Cartographic scale | The ratio between distance on the map and on the ground | A smaller cartographic scale corresponds to a larger geographic scale and may show fewer instances of features or less detail. |

Comparison of different methods used to quantitatively describe scale threshold and scale domain.

GVM |
- A hierarchical analysis to determine the relative variability and independent contribution at each level. - The data set can be divided into any arbitrary nested scale. |
Its validity remains unclear and more analyses are needed. | [ |

WTM | It can investigate features of interest in the data set at an appropriate scale and find the length scale of the variability. |
- The dimension of data set must be the exponent of 2. - The manner of WTM is dependent on the mother wavelet. |
[ |

LVM | The principle is easy to be understood. |
- It is unrealistic to assume the pixel value of a coarse resolution image is simply the average of finer resolution pixels within the corresponding coarse pixel. - It is dependent on the global variance in the image and the values of local variance cannot be directly compared between different images |
[ |

SVM |
- It can be used to judge whether the geographical scale is large enough to detect the length scales of the landscape. - The loss of image spatial variability at a given spatial resolution can be estimated. |
The second order stationarity hypothesis should be satisfied. | [ |

FM |
- It has a theoretical basis that many curves or surfaces in the world may show the statistical self-similar property. - The more irregular an object, the bigger the fractal dimension. The turning points of fractal dimension may contain some important information. |
No agreement has been reached on the definition of fractal dimension which can be used to determine the characteristic scale. | [ |

Summaries of general scaling methods used in remote sensing.

Scaling methods for measurements | AWM |
- Simple principle. - Easy usage. |
- May be only suitable for flat regions. |
[ |

FRPM |
- Simple principle. - The analytical solution to scale measurements. |
- The representative parameters may have no specific physical meanings. - It is difficult to get representative parameters when facing a large number of input arguments. |
[ | |

Scaling methods for retrieval models | CGM |
- Regardless of whether or not retrieval models are continuous or derivable. |
- Does not take into account the actual distribution of parameters. The weights for lower and upper bounds of a retrieval model may be inappropriate. - Needs a large amount of computing time and a special algorithm to retrieve convex hull, when facing a large number of input arguments. |
[ |

PSM |
- More accurate. |
- It is difficult to derive when facing a large number of input arguments. |
[ | |

Scaling methods for products | ERM |
- Simple principle. - Easy usage. |
- Less accurate. |
[ |

TSEM |
- Better basis of mathematics. - Easy usage. |
- The retrieval model and its derivatives must be continuous in the domain. - It may cause greater error when the model is strongly non-linear. - The model needs to use the local variance as input, which may usually not be available. |
[ | |

CPM |
- Taking the discontinuity as the main cause of scale effects. - Easy usage. |
- Neglects the heterogeneity within certain land types. - Has no theoretical or physical basis. |
[ | |

SFSM |
- Simple principle. - Grasps the simple scaling and multi-scaling characteristics of surface nature. - Scaling products without other prior knowledge. |
- The scale domain is not fully understood. |
[ |