Theoretical analysis of errors when estimating snow distribution through point measurements

In recent years, marked improvements in our knowledge of the statistical properties of the spatial distribution of snow properties have been achieved thanks to improvements in measuring technologies (e.g., LIDAR, terrestrial laser scanning (TLS), and ground-penetrating radar (GPR)). Despite this, objective and quantitative frameworks for the evaluation of errors in snow measurements have been lacking. Here, we present a theoretical framework for quantitative evaluations of the uncertainty in average snow depth derived from point measurements over a profile section or an area. The error is defined as the expected value of the squared difference between the real mean of the profile/field and the sample mean from a limited number of measurements. The model is tested for one- and two-dimensional survey designs that range from a single measurement to an increasing number of regularly spaced measurements. Using high-resolution (~ 1 m) LIDAR snow depths at two locations in Colorado, we show that the sample errors follow the theoretical behavior. Furthermore, we show how the determination of the spatial location of the measurements can be reduced to an optimization problem for the case of the predefined number of measurements, or to the designation of an acceptable uncertainty level to determine the total number of regularly spaced measurements required to achieve such an error. On this basis, a series of figures are presented as an aid for snow survey design under the conditions described, and under the assumption of prior knowledge of the spatial covariance/correlation properties. With this methodology, better objective survey designs can be accomplished that are tailored to the specific applications for which the measurements are going to be used. The theoretical framework can be extended to other spatially distributed snow variables (e.g., SWE – snow water equivalent) whose statistical properties are comparable to those of snow depth.

Theoretical framework for estimating snow distribution through point measurements

Michael Lehning, Ernesto Trujillo Gomez

In recent years, marked improvements in our knowledge of the statistical properties of the spatial distribution of snow properties have been achieved thanks to improvements in measuring technologies (e.g. LIDAR, TLS, and GPR). Despite of this, objective and quantitative frameworks for the evaluation of errors and extrapolations in snow measurements have been lacking. Here, we present a theoretical framework for quantitative evaluations of the uncertainty of point measurements of snow depth when used to represent the average depth over a profile section or an area. The error is defined as the expected value of the squared difference between the real mean of the profile/field and the sample mean from a limited number of measurements. The model is tested for one and two dimensional survey designs that range from a single measurement to an increasing number of regularly-spaced measurements. Using high-resolution (~1 m) LIDAR snow depths at two locations in Colorado, we show that the sample errors follow the theoretical behavior. Furthermore, we show how the determination of the spatial location of the measurements can be reduced to an optimization problem for the case of the predefined number of measurements, or to the designation of an acceptable uncertainty level to determine the total number of regularly-spaced measurements required to achieve such error. On this basis, a series of figures are presented that can be used to aid in the determination of the survey design under the conditions described, and under the assumption of prior knowledge of the spatial covariance/correlation properties. With this methodology, better objective survey designs can be accomplished, tailored to the specific applications for which the measurements are going to be used. The theoretical framework can be extended to other spatially distributed snow variables (e.g. SWE) whose statistical properties are comparable to those of snow depth.

2015

Theoretical Framework for the Assessment of Uncertainties in Snow Depth Measurements

Michael Lehning, Ernesto Trujillo Gomez

In recent years, marked improvements in our knowledge of the statistical properties of the spatial distribution of snow properties have been achieved thanks to improvements in measuring technologies (e.g., LIDAR, TLS, and GPR). Despite of this, objective and quantitative frameworks for the evaluation of uncertainties in snow measurements have been lacking. Here, we present a theoretical framework for quantitative evaluations of the uncertainty of point measurements of snow depth when used to represent the average depth over a profile section or an area. We define the error as the expected value of the squared difference between the real mean of the profile/field and the sample mean from a limited number of measurements. The model is tested for one and two dimensional survey designs that range from a single measurement to an increasing number of regularly-spaced measurements. Using high-resolution (~ 1m) LIDAR snow depths at three locations in Colorado, we show that the sample errors follow the theoretical behavior. Furthermore, we show how the determination of the spatial location of the measurements can be reduced to an optimization problem for the case of the predefined number of measurements, or to the designation of an acceptable uncertainty level to determine the total number of regularly-spaced measurements required to achieve such error. On this basis, a series of figures are presented that can be used to aid in the determination of the survey design under the conditions described, and under the assumption of prior knowledge of the spatial covariance/correlation properties. With this methodology, better objective survey designs can be accomplished, tailored to the specific applications for which the measurements are going to be used. Although we discuss the proposed methodology in the context of snow depth, the theoretical framework can be extended to other spatially distributed snow variables (e.g., SWE and/or snow depth) or hydrologic variables (e.g., soil moisture) whose statistical properties can be compared to those of snow depth. Applications of the methodology will be discussed, such as the selection of the location of snow/meteorological stations in mountain environments or survey design to be able to capture watershed scale means.

2014

The Uncertainty of Seasonal Snow Energy- and Mass Balance Simulations as a Function of Typical Input Uncertainty

Mathias Thierry Pierre Bavay, Charles Fierz, Michael Lehning

The assessment of water resources and their change in time as well as many other applications from hydrology to meteorology require the successful modelling of the dynamics of the seasonal snow cover. Especially if extrapolation in time, e.g. for climate change scenario assessment, is aimed for, the use of energy balance models has been proven to be more trustworthy than simpler temperature index models. The disadvantage of the energy balance method is the requirement of a variety of input quantities, which are not always but become increasingly accessible. What is missing, however, is an in-depth assessment of the influence of typical errors in the individual input quantities on model performance. This contribution presents a sensitivity study, which compares the influence of typical errors and uncertainties in radiation, wind, temperature and humidity input, both measured and modeled, with errors caused by insufficient knowledge of solid precipitation rates. The analysis is carried out with the widely used SNOWPACK model and focuses on the build-up and melt of a mid-latitude seasonal snow cover. While mass input uncertainties still dominate the overall performance especially at high elevations (alpine zone), errors in wind and radiation can cause very significant effects as well. Wind is considered to be more critical in this context, because its spatial variation is both more pronounced and more difficult to estimate than the spatial variation in short- and longwave radiation. In absence of reliable radiation input, its value can successfully be calculated or parameterized if e.g. cloud information is available. Errors in air temperature input or a failure of a correct assessment of its spatial variability can be locally very important because of the feed-back mechanism via atmospheric stability. For the overall mass balance of a seasonal snow cover, humidity errors are less important. Notable exceptions are applications such as the correct assessment of surface hoar formation or prediction of melt rates during rain on snow events, for which good humidity input is required. A main conclusion is that input data through a combination of measurement stations and weather models are now widely available with sufficient quality to even allow a small-scale distributed application of energy balance models.

2013

The Uncertainty of Seasonal Snow Energy- and Mass Balance Simulations as a Function of Typical Input Uncertainty

Mathias Thierry Pierre Bavay, Charles Fierz, Michael Lehning

2013

Theoretical framework for estimating snow distribution through point measurements

Michael Lehning, Ernesto Trujillo Gomez

In recent years, marked improvements in our knowledge of the statistical properties of the spatial distribution of snow properties have been achieved thanks to improvements in measuring technologies (e.g. LIDAR, TLS, and GPR). Despite of this, objective and quantitative frameworks for the evaluation of errors and extrapolations in snow measurements have been lacking. Here, we present a theoretical framework for quantitative evaluations of the uncertainty of point measurements of snow depth when used to represent the average depth over a profile section or an area. The error is defined as the expected value of the squared difference between the real mean of the profile/field and the sample mean from a limited number of measurements. The model is tested for one and two dimensional survey designs that range from a single measurement to an increasing number of regularly-spaced measurements. Using high-resolution (~1 m) LIDAR snow depths at two locations in Colorado, we show that the sample errors follow the theoretical behavior. Furthermore, we show how the determination of the spatial location of the measurements can be reduced to an optimization problem for the case of the predefined number of measurements, or to the designation of an acceptable uncertainty level to determine the total number of regularly-spaced measurements required to achieve such error. On this basis, a series of figures are presented that can be used to aid in the determination of the survey design under the conditions described, and under the assumption of prior knowledge of the spatial covariance/correlation properties. With this methodology, better objective survey designs can be accomplished, tailored to the specific applications for which the measurements are going to be used. The theoretical framework can be extended to other spatially distributed snow variables (e.g. SWE) whose statistical properties are comparable to those of snow depth.

2015

Theoretical Framework for the Assessment of Uncertainties in Snow Depth Measurements

Michael Lehning, Ernesto Trujillo Gomez

In recent years, marked improvements in our knowledge of the statistical properties of the spatial distribution of snow properties have been achieved thanks to improvements in measuring technologies (e.g., LIDAR, TLS, and GPR). Despite of this, objective and quantitative frameworks for the evaluation of uncertainties in snow measurements have been lacking. Here, we present a theoretical framework for quantitative evaluations of the uncertainty of point measurements of snow depth when used to represent the average depth over a profile section or an area. We define the error as the expected value of the squared difference between the real mean of the profile/field and the sample mean from a limited number of measurements. The model is tested for one and two dimensional survey designs that range from a single measurement to an increasing number of regularly-spaced measurements. Using high-resolution (~ 1m) LIDAR snow depths at three locations in Colorado, we show that the sample errors follow the theoretical behavior. Furthermore, we show how the determination of the spatial location of the measurements can be reduced to an optimization problem for the case of the predefined number of measurements, or to the designation of an acceptable uncertainty level to determine the total number of regularly-spaced measurements required to achieve such error. On this basis, a series of figures are presented that can be used to aid in the determination of the survey design under the conditions described, and under the assumption of prior knowledge of the spatial covariance/correlation properties. With this methodology, better objective survey designs can be accomplished, tailored to the specific applications for which the measurements are going to be used. Although we discuss the proposed methodology in the context of snow depth, the theoretical framework can be extended to other spatially distributed snow variables (e.g., SWE and/or snow depth) or hydrologic variables (e.g., soil moisture) whose statistical properties can be compared to those of snow depth. Applications of the methodology will be discussed, such as the selection of the location of snow/meteorological stations in mountain environments or survey design to be able to capture watershed scale means.

2014