USING SATELLITE DATA TO ESTIMATE SNOW LOADS IN ALASKA. Russell Frith. RECOMMENDED: Rob Lang P.E., Ph.D. Gennady Gienko Ph.D.

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1 USING SATELLITE DATA TO ESTIMATE SNOW LOADS IN ALASKA By Russell Frith RECOMMENDED: Rob Lang P.E., Ph.D. Gennady Gienko Ph.D. Scott Hamel P.E., Ph.D. Chair, Advisory Committee Rob Lang P.E., Ph.D. Chair, Civil Engineering Department APPROVED: Fred Barlow Ph.D. Dean, College of Engineering Helena S. Wisniewski Ph.D. Vice Provost for Research and Graduate Studies Dean, Graduate School Date

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3 USING SATELLITE DATA TO ESTIMATE SNOW LOADS IN ALASKA A THESIS Presented to the Faculty of the University of Alaska Anchorage in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE By Russell Frith, M.S. Anchorage, Alaska December 2015

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5 Abstract Structural engineers rely on published ground snow load values to calculate the forces for which their structures must be designed. Outside the major urban areas of Alaska, those load values were most recently published in 1994, more than 20 years ago. In most locations, no values exist at all. In addition, those loads were based on records available at the time, which were generally insufficient. In many cases, as few as six years of data was available. Determining probability-based snow loads requires both a long record of measured snow-depths, and knowledge about the average snow density for each depth. The values used for the later in calculating the 1994 snow loads have also been questioned, causing disagreement about what loads to use in design. This project assembled depth and weight data from NOAA s Global Historic Climate Network (GHCN) and from the National Snow and Ice Data Center to update the probability-based ground snow load values for as many Alaskan locations as are available. Climate monitoring stations that recorded weight (snow-water equivalent) and satellite observations which estimated snow water equivalency were used to evaluate the existing characterizations of snow density in Alaska. v

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7 Table of Contents Page Signature Page... iii Title Page... iii Abstract...v Table of Contents... vii List of Figures... ix List of Tables... xiii List of Appendices... xv Acknowledgments... xvii Section 1 Introduction... 1 Section 2 Literature Review... 5 Section 3 Scope... 9 Section 4 Methodology Data Management Data Exploration Data Distribution of SWE Satellite SWE Data Extraction Predicting SWE Measurements Using Kriging Exploratory Variogram Analysis Task 1: Construct Sample Variogram Task 2: Generate Prediction Grid Task 3: Generate Map Products Task 4: Validate Kriging Results vii

8 Page Section 5 Results Physically Measured SWE Distributions Local SWE Variograms Simple Kriging Predictions Sensitivity Observations Model Diagnostics Section 6 Summary and Conclusions References Appendices... Error! Bookmark not defined. viii

9 List of Figures Page Figure 1: Methodology for obtaining conversion densities used for this study Figure 2: Flowchart showing six steps in climate data acquisition Figure 3: Process for removing outliers from SWE distribution Figure 4: NETCDF to ArcGIS shapefile conversion Figure 5: Satellite sample points used in SWE interpolation Figure 6: A hypothetical data trend line used in kriging calculations Figure 7: Illustration of different kriging models Figure 8: Flowchart showing nine steps used in kriging process Figure 9: ArcGIS model used to generate simple kriging maps Figure 10: Snow load contour lines near Anchorage, AK Figure 11: Snow load contour lines near Barrow, AK Figure 12: Snow load contour lines near Barter Island, AK Figure 13: Snow load contour lines near Fairbanks, AK Figure 14: Snow load contour lines near Homer, AK Figure 15: Snow load contour lines near Kotzebue, AK Figure 16: Snow load contour lines near McGrath, AK Figure 17: Snow load contour lines near Nome, AK Figure 18: Snow load contour lines near Talkeetna, AK Figure 19: Snow load (psf) isolines for central Alaska Figure 20: Snow load (psf) isolines for Arctic and NW Alaska Figure 21: Work-flow process for geospatially estimating SWE Figure A-1: Snow load box plot for Anchorage Figure A-2:Empirical distribution for Anchorage snow load Figure A-3: Snow load box plot for Barrow Figure A-4: Empirical distribution for Barrow snow load Figure A-5: Snow load box plot for Barter Island Figure A-6: Empirical distribution for Barter Island snow load Figure A-7: Snow load box plot for Fairbanks AP ix

10 Page Figure A-8: Empirical distribution for Fairbanks AP snow load Figure A-9: Snow load box plot for Homer AP Figure A-10: Empirical distribution for Homer AP snow load Figure A-11: Snow load box plot for Kotzebue AP Figure A-12: Empirical distribution for Kotzebue AP snow load Figure A-13: Snow load box plot for McGrath AP Figure A-14: Empirical distribution for McGrath AP snow load Figure A-15: Snow load box plot for Nome AP Figure A-16: Empirical distribution for Nome AP snow load Figure A-17: Snow load box plot for Talkeetna Figure A-18: Empirical distribution for Talkeetna snow load Figure A-19: Snow load box plot for King Salmon Figure A-20: Empirical distribution for King Salmon Figure A-21: Snow load box plot for Cordova AP Figure A-22: Empirical distribution for Cordova AP Figure A-23: Snow load box plot for Yakutat Figure A-24: Empirical Distribution for Yakutat Figure A-25: Snow load box plot for Kodiak AP Figure A-26: Empirical distribution for Kodiak AP Figure A-27: Snow load box plot for Valdez WSO Figure A-28: Empirical distribution for Valdez WSO Figure B-1: An example of a linear variogram graph and its equation [2] Figure B-2: An example of a spherical variogram graph and its equation [2] Figure B-3: An example of an exponential variogram graph and its equation [2] Figure B-4: Anisotropy example graph showing directional variation Figure B-5: Prediction grid example where value at point zero is sought Figure B-6: Spherical variogram for given examples and covariance equation Figure B-7: Kriging weights linear algebra equation for given example x

11 Page Figure B-8: Math used for computation of the kriging weight vector Figure B-9: Formula used for the estimate of an unknown value xi

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13 List of Tables Page Table 1 : Fifty year empirical snow load estimates Table 2 : Divergence of satellite predictions from GHCN predictions Table 3 : SWE prediction sensitivity to number of nearest neighbors Table 4 : GHCN vs. satellite comparison of SWE with small differences Table 5 : GHCN and satellite load comparisons near Bethel Table A-1 : Outlier removal sensitivity on snow loads xiii

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15 List of Appendices Appendix A : SWE Data Series Appendix B : Kriging Math Structures Appendix C : R Scripts Used to Manipulate Climate Databases xv

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17 Acknowledgments I would like to thank Professors Lang, Hamel, and Gienko for their patience and whose tutelage has immeasurably improved my value as an engineer. I would also like to thank the University of Alaska Anchorage for its continued employment of my services as an adjunct mathematics instructor from which I received financial support in the development of this thesis. I would also like to thank the University Information Technology Department for its quality technical services and support in standing up key data repositories and application frameworks on which this thesis was developed. xvii

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19 Section 1 Introduction Snow loads are an important aspect of structural design in cold climates. In some areas, the weight of accumulated snow on a structure represents the most significant load that the structure will endure in its lifetime. In such areas, accurate estimates of potential snow loads directly affect the safety, reliability, and economy of that structure. The 2012 International Building Code (IBC) [10] defers the determination of snow loads for structural design to the Minimum Design Loads for Buildings and Other Structures [1], commonly referred to by its designation, ASCE 1 7. Ground snow load information with 50 year mean reoccurrence intervals (MRI) is readily accessible for the contiguous states in the forms of ASCE publications and bulletins. Such knowledge base generally allows structural engineers to estimate an appropriate snow load at any location therein. There are no equivalent estimates for Alaska, however, and snow loads are expressed in ASCE tables for a scant 30 urban locations in the state. Given the size of the state, the distance between municipalities and the sparseness of observed snow load measurements, interpolation between those locations does not yield meaningful results. In addition, the Alaska ground snow load design values have remained generally unchanged since the document s inception from the early 1980 s whereas the foundational climate data set has evolved substantially. In the absence of prescribed snow loads that are officially authorized by the IBC and ASCE 7, the Alaskan engineer may turn to a number of additional references when working on a project outside one of the 30 municipal boundaries. The most recent of these publications is a 1994 thesis written by Andy Stember at the University of Alaska Anchorage [21]. Stember presents a history of Alaskan snow load publications, summarizes the application of snow loads, and corrects a calculation error and reprints the data from a previous study from 1987: Snow Loads in Alaska [12] published by the Arctic Environmental Information Data Center (now ENRI 2 ). The 1987 report presents 1 ASCE is the American Society of Civil Engineers. 2 ENRI is the Environmental Natural Resources Institute. 1

20 data from 323 sites collected by the National Weather Service (NWS) and through its affiliate the Soil Conservation Service (SCS). Unfortunately, the data used by the 1987 report did not represent a significant time span for predicting an event with a 50 or 100 year MRI. On average, less than 20 years of data had been accumulated for each site, with one-fourth of the results based on 12 years of data or less. The minimum number of years of data was six with a maximum of 37 years. Since then, the NWS and SCS have continued to collect data at these and other sites. In addition, new data are available from the Natural Resources Conservation Service (NRCS), the so-called SNOTEL sites. There are now over 50 of these sites in Alaska, only a few which existed in the 1980s. Due to the scarcity of reliable, ground-based observational snow load data prior to 1990, there has been considerable disagreement about the density of accumulated snow in Alaska. Since much of the data available has been recorded snow depth, those depths must be multiplied by their corresponding snow densities in order to determine snow weight. Mathematical representations of snow density have been formulated using data from many locations at which both the depth and volume were available. Volume of snow has generally been expressed as Snow-Water Equivalent, or SWE. SWE is the depth of water, usually measured in millimeters (mm), which would result from melting a column of the snow-pack. Historically, snow volume was measured in this manner, though there are other methods available to supplement this practice. The majority of depth and volume data points prior to 1987 were in the contiguous states where observation sample sizes were large enough to afford reasonable statistical modeling. Efforts were made to create an equation using 25 Alaskan sites and the results were published in 1992 by Tobiasson et al [27] using an exponential model. The Alaska version of the equation resulted in seemingly high density predictions which are typically have not been found under natural circumstances. Aside from anecdotal judgment, the main issue regarding predictions of snow density rests in their verification. In the Tobiasson studies, snow density predictions were generally based on simplified, linear or exponential approximations using wind and temperature gradients as input parameters [28]. Such 2

21 snow density predictions were generally not validated through sampling of direct measurements. This study has taken an alternative approach which avoided the use of snow-density predictions. Snow loads were estimated from satellite-based microwave radiometer data. The data set extracted from satellite-based microwave radiometer data provided a larger sample size for which snow loads could be estimated on a finer geographic scale [18]. Those estimates were verified with ground-based snow load measurements at discrete locations. Information gathered from decades of space-borne microwave radiometer data was used to estimate 50 year snow water equivalent returns on a 25 kilometer grid that spans most of the state s non-mountainous terrain. Snow water equivalent (SWE) returns were estimated at the centroids of each cell in the grid. Those estimates were subsequently interpolated using simple kriging to estimate snow water equivalency at locations where it was physically measured. Although there was little physical verification of snow water equivalent measurements at those centroids, it was shown that those estimates that formed neighborhoods around locations where physical measurements were made predicted, within 11% or better, actual snow water equivalent measurements. The process of using satellite point estimates to predict within 11% or better measured snow loads suggested that those satellite estimates for snow water equivalence at the grid centroids may be used as 50 year mean recurrence interval predictions of snow water equivalence at those locations. 3

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23 Section 2 Literature Review The main technical hindrance to providing reliable locational snow load estimates in Alaska was the lack of measured snow water equivalent (or SWE) data where SWE is the depth of liquid water that is equivalent to the snow on the ground. Since water has relatively constant density, the depth of SWE corresponds linearly to its weight, and thus snow load and SWE can be used interchangeably. In Alaska, no more than 20 geographically dispersed monitoring stations measured SWE with sufficient temporal duration to allow computation of a 50 year return estimate with a high degree of confidence. Works from Tobiasson and others [27], [28] provided rough approximations for estimating SWE at non-monitored sites, but validation of estimate samples were not generally available in ASCE publications. Since the 1990 s, the latter techniques have remained the de-facto standard for estimating Alaska snow loads. The protocol for previously calculating SWE [28] is summarized in five steps following this passage. This procedure represented an evolution of SWE modeling originating from linear models of the 1950 s to an exponential model derived in the 1990 s, where the later has since remained unmodified [21], [27], [28]. The estimation procedure from [27] and [28] did not recommend a data source or a method to determine the quality of the data that those models treated. Consequently, characterizations of model robustness to data errors and varying data sources were not available. The estimators from [27] and [28] were derived from two climate data sources, one being served by the National Resources Conservation Service and the other from records obtained from field monitoring stations operated by defense agencies. Since the inception of that estimating protocol, the original data sets have accumulated considerably more data, but the accompanying metadata has remained unchanged. As a result, there has been no need to revise estimating models in order to accommodate new data standards. This implies that today s monitoring data can be used as inputs to the SWE models developed over 20 years ago. The description below of the last developed SWE estimation protocol from the 1990 s incorporates a hypothetical consumption by those models using today s 5

24 climate data without regard to any modeling errors. The SWE estimation protocol is summarized as follows: 1. Twenty-five geographically dispersed monitoring stations were identified as ones which reliably recorded SWE data. The maximum annual SWE records were presumed to be log-normally distributed. 2. Conversion density factors were calculated at each of the SWE monitoring sites. Conversion density was the ratio of the maximum annual depth to the maximum annual SWE, regardless of the date of those measurements. 3. Those conversion densities were combined with snow survey data from the National Resource Conservation Service (NRCS), U.S. Geological Survey physiographic maps, and snow-cover density maps. Snow density demarcations were assigned to regions in the state. Regionalized densities were further stratified based on elevation, wind characteristics, and proximity to marine environments [28]. 4. Approximately 450 monitoring stations reliably recorded depth data, but not volume (SWE) data. In general, the periods of those depth recordings at those monitoring sites showed that snow depth was log-normally distributed and, subsequently, 50 year maximum snow depths were reliably estimated from those sites. Depth data was used along with regionalized conversion densities to estimate snow loads at those sites. 5. Snow loads where no SWE were measured or approximated by the estimation process of step four were interpolated by averaging observed or estimated snow loads in proximity at those sites [21]. From snow density estimates, structural engineers obtained snow loads at select locations where none was measured. Since the 1980 s, a considerable amount of research has been conducted on estimating high-latitude SWE by assimilating space-borne microwave radiometer data and synoptic snow-depth observations [3],[4],[5], [18],[19],[26]. The origin of this research took place in the early 1960 s when passive microwave snow monitoring equipment using ground- based microwave radiometers showed the potential of inferring snow 6

25 parameters. Microwave brightness temperature measured by space-borne sensors originated from (1) radiation from the underlying surface, (2) the snowpack, and (3) the atmosphere. The atmospheric contribution usually was small and could be neglected over snow covered areas. Under this condition, microwave measurements could be used to extract snowpack parameters. Snow crystals were effective scatterers of microwave radiation. The deeper the snowpack, the more snow crystals there were available to scatter microwave energy away from the sensor. Hence, microwave brightness temperatures were generally lower for deep snowpacks (more scatterers) than they were for shallow snowpacks (fewer scatterers). The physical properties of ground snow emitted characteristic microwave signals from which snow water equivalent retrieval algorithms were developed [18]. The culmination of intensive satellite monitoring and this research resulted in the publication of a SWE dataset for a period from 1980 to the present and which covers an extensive portion of Alaska surface area on a 25-kilometer grid. The dataset was released through the National Snow and Ice Data Center (NSIDC) and is maintained and quality checked by the Global Snow Consortium under the European Space Agency Study Contract 21703/08/I-EC [26]. This data set relies on spectral properties of satellite radiometer (passive microwave) data in order to discern snow density. The satellite produces a large data set over a relatively long temporal scale for which snow density can be interpolated. Like the Tobiasson methodology, however, satellite estimates in Alaska lack rigorous validation through direct measurement of estimated loads. In this study, SWE estimates at satellite sample points were used to predict SWE at locations where SWE were physically measured. This research was the first known attempt at utilizing satellite observations as a means of predicting local SWE (snow loads) for Alaska. The approach departed from historical techniques of estimating snow load measurements based on ground-based observations. Snow load researchers attempted to derive prediction formulae for snow loads with limited to moderate success [11], [12], [22], [23], [24], [25], [28]. This limited success was due principally to 7

26 the geospatial sparseness of SWE observations and to limited temporal observations of climate records. The researchers who today maintain the satellite observation data used to estimate SWE have incorporated physiographic and climate characteristics into models which accurately predict SWE [18]. In addition, the temporal record of those observations spans 30 years, which is sufficient to estimate 50 year mean recurrence intervals of SWE. The use of kriging as an interpolation means for discovery of distributions of snow depth and snow water equivalency was the method of choice for research in SWE distribution using satellite radiometry [24], [25], [26] and was shown to improve estimation accuracy [26]. 8

27 Section 3 Scope This project will assemble and review substantive documentation on snow loads in Alaska. Recorded precipitation and snow-water equivalent data will be extracted and stored in a relational database management server that is shared to snow load computation methods. The data will be evaluated for 1) the appropriate statistical distribution for most sites (e.g. Pearson distribution), 2) the calculated 50 year ground snow load mean recurrence interval (MRI) at each site, and 3) the snow density at satellite sample points where this evaluation is possible. From these evaluations, ground snow load values will be determined for as many Alaskan sites as is practical. Where necessary, these determinations will be generated from weighted averages that account for the geographic variability and reliability of the data. The weights will be calculated using simple kriging. See Appendix B for an explanation of weights calculation. Using the data gathered, an attempt will be made to generate contour maps and ground snow load values for some or all of Alaska. 9

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29 Section 4 Methodology The procedure for calculating snow loads at selected locations in Alaska consisted of four essential steps. These steps were (1) data management, (2) data exploration, (3) SWE measurement prediction using kriging over the Alaska spatial extent, and (4) prediction validation. Each step is explained in detail in a separate subsection. A flowchart outlining the procedure is shown in Figure 1. Figure 1: Methodology for obtaining conversion densities used for this study A justification for using kriging in order to estimate snow density (load) was warranted. All interpolation algorithms, whether they are inverse distance weighted, splines, radial basis functions, or triangulation, estimate a physical value at a given location as a weighted sum of data values at surrounding locations. Almost all assign weights with increasing separation distance. Kriging assigns weights according to a data-driven weighting function, rather than an arbitrary function. Nevertheless, kriging is still just an interpolation algorithm and will give very similar results to other in many cases [2]. In particular: 11

30 If the data locations are fairly dense and uniformly distributed throughout the study area, as is the case with satellite data, one can get fairly good estimates regardless of the interpolation algorithm. If the data locations fall in a few clusters with large gaps in between, then one can get unreliable estimates regardless of the interpolation algorithm. Almost all interpolation algorithms underestimate the highs and overestimate the lows; this is inherent to averaging and if an interpolation algorithm did not average, then it would not be considered reasonable. Some advantages of kriging [25], [26]: It helps to compensate for the effects of data clustering, assigning individual points within a cluster less weight than isolated data points (or, treating clusters more like single points). It offers an estimate of estimation error (kriging variance) along with an estimate of the variable itself. The availability of estimation error provides a basis for stochastic simulation of possible realizations of the variable at ad-hoc locations. In this study, the following interpolation methods used to estimate snow density at ad-hoc locations were tried using ArcGIS 3 geospatial analyst tool: inverse distance weighting, global and local polynomial interpolations, radial basis functions, kriging and cokriging, and empirical Bayesian kriging. 3 ArcGIS is a proprietary geographical information science tool developed by Environmental Sciences Research Institute (ESRI) 12

31 The one interpolation method that presented the more accurate snow predictions was that of simple kriging. All others tended to overestimate snow density. See for instance [23]. 4.1 Data Management In order to begin snow load estimation, a vetted climate data source was selected [13],[14]. In a personal communication with Dr. Peter Olson, the Alaska state climatologist, a determination was made that the most reliable data set at the time this paper was authored was the Global Historical Climatology Network (GHCN) data store maintained by the National Climatic Data Center (NCDC), which URL is The flowchart in Figure 2 shows the process by which data was assembled for research. R codes were written for data extraction from GHCN raw files into a SQL Server 4 database. See Appendix C for details. The SQL Server tables were shared out via a web service which presented queries to the database engine and which relayed data result sets back to client methods. Figure 2: Flowchart showing six steps in climate data acquisition The GHCN extracts served principally to provide estimates of 50 year return SWE measurements at those monitoring stations which measure SWE. The second phase of data management was to assemble a SWE dataset from the Global Snow Monitoring Network 5. The later dataset provided daily SWE estimates on a 25-kilometer grid which 4 SQL Server is a relational database management system developed by Microsoft Corporation 5 European Space Agency Study Contract, ESRIN Contract 21703/08/I-EC 13

32 covered portions of the Alaska spatial extent. This dataset was aggregated to provide 50 year SWE point estimates from satellite sample points. 4.2 Data Exploration Data exploration consisted of two essential steps. These steps included: (1) estimation of SWE at locations where SWE was not directly measured, and (2) generation of a prediction map for SWE. Data exploration was aimed at developing hypotheses and made extensive use of graphical views of the data such as maps or scatter plots. Modeling of SWE spatial phenomena has to incorporate the possibility of spatial dependence in order to provide a true representation of the existing effects [2]. Such spatial effects can be either large scale trends or local effects. Large scale trends are first order effects and describe overall variation in the mean value of SWE. An exploration of first order effects typically involves studying spatial moving averages. Local variations in SWE are often labeled as second order effects. Second order effects represent the tendency of neighboring values to follow each other in terms of deviation from the mean. The presence of second order effects result in positive covariance between observations a small distance apart and lower covariance or correlation if they are further apart Data Distribution of SWE SWE-monitoring stations are those which have climate records containing snow water equivalent measurements for a period of sufficient duration which allows the computation of an empirical cumulative distribution function. A database query was authored to filter SWE-monitoring stations located in Alaska and their corresponding observations from GHCN extracts. Annual maximums of those measurements were extracted based on water year, which is the year beginning on September 1 and ending on August 31 of the next calendar year. Note that the start of a water year is specific for Alaska whereas the start of the water year in the contiguous states is October 1. Once those values were selected, an assessment was made as to the type of distribution the data settled into. In general, those data generally showed log-normal distributions, but 14

33 with few exceptions, some record sets were of type Pearson I-VII distribution. The choice of distribution was made on a best-fit approach for which the SWE conformed to. R software numerical routines made the determinations base on the data moments (mean, variance, skewness, and kurtosis). Once the empirical cumulative distribution functions were plotted for each SWE-monitoring station, then the 50 year return periods were derived for SWE. The derivation of the 50 year return periods often involved removing extreme data outliers of snow density observations and then recalculating the empirical distribution function. The procedure for selecting which outliers to remove from the SWE distribution is shown in Figure 3. 15

34 Figure 3: Process for removing outliers from SWE distribution 16

35 The process of removing snow load outliers in this study is summarized as follows: 1. If the 50 year satellite mean recurrence interval (MRI) for snow load approximated the GHCN 50 year snow load at a given monitoring site within a given tolerance then the decision was one to use the GHCN 50 year value. 2. If the response to the previous inquiry was negative, then an examination of the boxplot outliers of the GHCN distribution for the site under investigation was conducted. If the data did not contain outliers, then the decision was to use engineering judgements to make the SWE estimate. If the data did contain outliers, then the outliers were ranked in descending order and the process proceeded to step three. 3. An engineering judgement was made as to whether the largest outlier should have been retained for load estimation or whether it should have been discarded. In the former case, when the GHCN 50 year MRI and satellite 50 year MRI were suitably close then the satellite data were regarded as sufficient for interpolating loads in proximity to the GHCN monitoring site. If the two values had an unacceptable difference then it was left to engineering judgement as to whether to use the satellite data for load interpolation and the procedure concluded. In the later case, the procedure advanced to step four. 4. Step three was repeated until there were no more outliers. If all outliers were consumed and the satellite value did not approximate the GHCN 50 year MRI, then the choice was left to use engineering judgement to make the SWE estimate. See Appendix A for a summary of the findings. See also Table A-1 in Appendix A for sensitivities of load estimates to outlier removals. With 50 year snow water equivalent in hand, the corresponding snow load estimates were calculated. 17

36 Load values in pounds per square foot were calculated from GHCN SWE measurements (in tenths of millimeters) using the following formula: L = SWE 0.1mm 1in 25.4 mm 5.2 psf/in ( ) Satellite SWE Data Extraction Satellite measurements from the Global Snow Monitoring Network provided observations which could be used to estimate SWE over Alaska. The observations were bundled into NETCDF 6 files which were distributed by NSIDC 7. An R script was written to load daily SWE data from that data repository into a SQL Server database and which then generated an ArcGIS point shapefile. The code is included in Appendix C and is Figure 4: NETCDF to ArcGIS shapefile conversion 6 NETCDF (Network Common Data Form) is a product of Unidata Corporation which is a set of software libraries and self-describing, machine-independent data formats that support the creation, access, and sharing of array-oriented scientific data. 7 National Snow and Ice Data Center, 18

37 flowcharted in Figure 4. The daily temporal extent covered a period from years 1980 to A database query was authored to filter satellite sample points located in Alaska and their corresponding estimates of SWE. Annual maximums of SWE measurement estimates were extracted based on water year, which is the year beginning on September 1 and ending on August 31 of the next calendar year. Once those values were selected, an assessment was made as to the type of distribution the SWE data settled into. In general, this data generally showed log-normal distributions. Once the log-normal cumulative distribution functions were plotted for each SWE satellite sample point, then the 50 year return periods (mean recurrence interval or MRI) were estimated for SWE at those points. Figure 5 shows the satellite sample points used for estimating SWE depicted as small green dots. GHCN SWE monitoring stations are shown as large blue dots. The gaps in the coverage are due to mountainous areas which are not treated by satellite observations. Figure 5: Satellite sample points used in SWE interpolation 4.3 Predicting SWE Measurements Using Kriging The technique for predicting under-sampled physical phenomena using kriging is well established and the literature on its implementation is extensive [2],[11],[15],[16],[17],[20],[26]. The application of simple kriging as a means of estimating Alaska snow loads is unprecedented and is one of the few statistical methods 19

38 available for estimation. The motivation for using this technique was principally due to the absence of climate monitoring infrastructure and to the sparseness of logging those values by existing Alaska climate monitoring stations. In this study, SWE estimates at satellite sample points were used to predict SWE at locations where SWE was physically measured. This approach suggested that if satellite observations were predicting measured SWE values, then those satellite observations presented reasonable estimates of SWE at locations where SWE was not actually measured. This approach historically had little success with ground-based observations [27], but ground-based prediction models have evolved to become more accurate [22]. 4.4 Exploratory Variogram Analysis The following subtopics present steps needed to generate a rudimentary prediction grid for SWE. The prediction techniques selected for this study used simple kriging with snow water equivalent measurements extracted from satellite scans. To reiterate, kriging is a specialized form of spatial interpolation. Kriging is a process of determining the characteristics of objects from those of nearby objects. Kriging fits a function to a specified number of points where those points fall within a specified radius. The use of kriging is appropriate when the presence of spatial correlations are suspected or when there is a directional bias in data. The following three points suggest the rationale for using this interpolation technique. Kriging is based on the idea that one can make inferences regarding a random function Z(x), given data points Z(x 1 ), Z(x 2 ), Z(x n ), The basis of this technique is the rate at which variance between points changes over space, and This is expressed in the semivariogram (or just variogram ) which shows the average difference between values at points changes with distance between points. In kriging, the random function may be written as: Z(x) = mean(x) + spatial signal g(h) + error (4.4.1) 20

39 Kriging data may be described as a sum of a trend plus small scale and microscale variations plus measurement error. An example of a kriging trend line would be shown in the hypothetical data plot in Figure 6. Figure 6: A hypothetical data trend line used in kriging calculations The amount and form of spatial autocorrelation can be described by a variogram that shows how differences in values increase with geographical separation. The variogram is a function that relates dissimilarity of data points to the distance that separates them [6], [8]. A worked numeric example of calculating a variogram is presented in Appendix II. There are three fundamental types of kriging. These are: Simple kriging: the mean value of the data is known, Ordinary kriging: the mean of the data is unknown and is estimated as a constant in the prediction neighborhood, Universal kriging: the mean of the data is unknown, varies locally, and is predicted as a trend. Figure 7 shows the difference between the types of kriging using a hypothetical data set. 21

40 Figure 7: Illustration of different kriging models In this study, simple kriging was used because the mean of the satellite observations for SWE could be found Task 1: Construct Sample Variogram The first task for snow water equivalent data exploration was to produce a sample variogram from which the sill, range, and nugget could be extracted as parameters for kriging processing. The precise mathematical structure of the kriging variogram is presented in Appendix B. In this study, a sample variogram consisted of at most 250 SWE estimates from satellite observations in the neighborhood of a monitoring station. The choice of 250 nearest neighbors for SWE kriging is elaborated on in a sensitivity analysis in section 5.4 of this document. Once a sample variogram was constructed, that variogram was fitted with a model variogram. A number of choices arose for fitting the sample variogram. Variogram models must be mathematically positive definite matrices so that the covariance matrix based on it can be inverted during the kriging process. This 22

41 restriction limits the types of variograms that can be used. The reader is referred to Appendix B for a detailed explanation of this restriction Task 2: Generate Prediction Grid The second task for snow load estimation was to create a prediction grid covering Alaska. The geographic bounding box for the State of Alaska consists of a latitude range from N to N and a longitude range from W to W. The prediction grid consisted of sample points for which the kriging estimates were calculated. The grid had to be fine enough so that subsequent rasterization of the predicted values gave the appearance of smooth serpentine contour lines. To further enhance the prediction grid, prediction points lying beyond the nearest 250 neighbors were removed (clipped). The choice of 250 nearest neighbors was compatible with the extent of their coverage overlap of larger Alaska municipal boundaries where building construction was more prevalent. In addition, simple kriging showed increasing sensitivity to choices of larger neighborhood sizes as well as to smaller neighborhood sizes of twenty or fewer points as shown in section Task 3: Generate Map Products The following flowchart in Figure 8 outlines the process by which the snow water equivalent prediction map was produced. 23

42 Figure 8: Flowchart showing nine steps used in kriging process Task 4: Validate Kriging Results Validation is a function which interprets the goodness-of-fit of the kriging estimates. Validation is used to check for correctness in the application of kriging. In this study, a criteria was not presented for validating kriging models as a means of predicting snow density at ad-hoc locations. This remains a topic for future study. Cross validation is a technique that allows the structural engineer to compare predicted values with true values. In spatial data this technique assists in deciding which variogram model to choose or which prediction method gives better results. The essential idea of cross validation is the following. Omit point i from the data set and predict its value using the remaining n 1 data points. This procedure provides a comparison of the predicted value with the true value at location s i. This is essentially the process described in section of this document. Additional validation analysis is presented in section 5.4 where a treatment of model sensitivity to satellite data sample sizes is presented. 24

43 Section 5 Results 5.1 Physically Measured SWE Distributions Table 1 shows 50 year MRI snow load estimates from GHCN SWE monitoring sites. The load estimates were extracted from Pearson distributions. The third column shows the 50 year log normal distribution of snow depth. The fourth column shows a conversion density factor which, when mulitplied by the 50 year depth return gives an estimate for the 50 year load. This parameter is used in ground-based estimates of snow loads described in [27],[28]. Table 1 : Fifty year empirical snow load estimates Snow load estimates derived from simple kriging of satellite observations were crossvalidated with load observations in Table 1 where the later observations were contained in the satellite scanning grids. For some of the loads presented in Table 1, maximal outliers were removed prior to the computation of the 50 year load MRI. In many cases a judgemental choice based on statistical reasoning was made to remove an outlier that 25

44 was much greater than the third quantile value for a given load distribution. In some cases, no outliers were removed. An inventory of which outliers were removed and which were retained at particular locations is presented in Appendix A. 5.2 Local SWE Variograms The next result was obtained from the plots of the omnidirectional variograms of SWE estimates from satellite observations in the neighborhoods of the ground-based SWE monitoring stations. Those graphs yielded the range estimates for fitting the variograms used in simple kriging for SWE estimation at ground-based monitoring sites. Simple kriging analysis could not be performed in the neighborhoods of Adak, Annette Island, Bethel, Cold Bay, Juneau, St. Paul Island, Shemya, and Unalakleet. Those locations are generally characterized as being coastal villages and/or small towns surrounded by tall mountains. In those locations, satellite signatures needed for SWE estimates were greatly diminished by marine effects contributing to wetter snow [18], [19], [26]. Those omissions correlated with the satellite coverage gaps shown in Figure 5. Geospatial interpolation was generally not possible in the neighborhoods of these communities due to the sparseness of local sample points. Additional investigation is required to interpolate load information in spatial extents surrounding those locations. Agreement in load estimates was observed for Anchorage, Barrow, Barter Island, Fairbanks, Homer, Kotzebue, McGrath, Nome, Talkeetna, and Yakutat. Those communities were generally subjected to drier, lighter snow which was readily detectable by satellite observations [18], [19], [26]. In all cases, a fifth-order polynomial variogram with first order trend removal was used to fit a simple kriging surface to within 11 percent of the measured values. The choice for this variogram model was one of best fit using ArcGIS Geospatial Analyst extension. Between 220 and 250 nearest neighbors were used from satellite data to estimate snow loads at ground-based monitoring sites. Satellite load estimates for Bethel, Unalakleet, and King Salmon did not match groundbased measurements. Those communities are in proximity to Bristol Bay and snow 26

45 water measurements were subjected to the Bay s marine influences. Satellite load estimates were comparable to GHCN estimates for 50 year return periods for Valdez and Cordova only after adjustments were made. In the case of Cordova, an exponential variogram was used for the nearest 250 satellite observations. For Valdez, the Global Snow data set was too sparse to perform interpolation. When the Valdez data set was augmented with 50 SWE satellite observations from the Canadian Meteorological Center, then the kriging interpolation model was near the GHCN measured value. See Appendix A. There were no satellite observations for Juneau. Table 2 : Divergence of satellite predictions from GHCN predictions 5.3 Simple Kriging Predictions Figure 9 represents an ArcGIS model builder solution for generating kriging prediction maps. Figure 9: ArcGIS model used to generate simple kriging maps 27

46 The GA Layer methods as shown in Figure 9 were used to calculate kriging values at satellite sample points. The GA Layer to Points method was used to calculate kriging values of satellite sample points. The GA Layer to Contour method was used to generate SWE contour lines as shown in the following maps. The AKBoroughsMerge object is a borough map of Alaska with the borough boundaries dissolved. This is the clipping mask needed to render the contour grid in the shape of the state s outline. The shapefile akk_simp.shp was a point shapefile containing the results of kriging estimates at the satellite sample points. It was the output of the GA Layer to Points method. This shapefile was used to explore the errors in the kriging estimates at each satellite sample point. The following plates (Figures 6 19) exhibit simple kriging predictions of ground snow load for Alaska near ground-based monitoring stations. Sample points from satellite observations were used to derive the ground snow load contour plots in the neighborhoods of density monitoring stations. The sizes of these sample point sets ranged from observations. In some cases, sample points that resided on ocean surfaces were removed. These plates may be considered tabular functions of ground snow load and position. One observation must be made regarding tiling adjacent contour maps. When tiling adjacent contour maps, the reader may notice that contour lines are broken at the borders where two adjacent contour maps are tiled together. For example, when tiling the Fairbanks contour plot and the Anchorage contour plot, the contour lines do not connect, although subsets of the contour data sets overlap. This lack of serpentine continuity in the tiling is due essentially to the variation in the separate samples used to generate the localized contours. Although the same kriging models were used to generate the contour graphs, the localized variations in the samples were sufficiently different which resulted in distinctive localized contour patterns which were not dependent on the properties of other contour models. In addition to statistical variations of the localized SWE data sets, the map scales for the monitoring sites are not uniformly the same. Such different scales contribute to 28

47 misalignment of contours among the different map tiles. The map scales are included in the map errata. 29

48 Figure 10: Snow load contour lines near Anchorage, AK 30

49 Figure 11: Snow load contour lines near Barrow, AK 31

50 Figure 12: Snow load contour lines near Barter Island, AK 32

51 Figure 13: Snow load contour lines near Fairbanks, AK 33

52 Figure 14: Snow load contour lines near Homer, AK 34

53 Figure 15: Snow load contour lines near Kotzebue, AK 35

54 Figure 16: Snow load contour lines near McGrath, AK 36

55 Figure 17: Snow load contour lines near Nome, AK 37

56 Figure 18: Snow load contour lines near Talkeetna, AK 38

57 Figure 19: Snow load (psf) isolines for central Alaska 39

58 Figure 20: Snow load (psf) isolines for Arctic and NW Alaska 40

59 Comparable maps were not produced near GHCN sites where satellite estimates did not agree with ground-based predictions. 5.4 Sensitivity Observations Slightly better kriging results were obtained for Anchorage, Fairbanks, and Talkeetna when the localized satellite sample points were merged into a single point shapefile and the same simple kriging procedure was applied. This is shown in Figure 19. Worse estimate results were obtained when neighborhood sets were merged from western Alaska sites which included Nome, Unalakleet, Bethel, McGrath, and Kotzebue. Those estimates from simple kriging were much higher than those from smaller, neighborhood-based (localized) simple kriging. When satellite estimates for neighborhoods of Barter Island, Barrow, Kotzebue, and Nome were merged, the SWE predictions improved for the Arctic villages but were about 15% higher for villages near the Bering Sea. A nearest-neighbor sensitivity assessment was done to ascertain a reasonable neighborhood size for predicting GHCN measurements from satellite observations. The choice of 250 nearest neighbors produced better agreement between satellite estimates and ground based measurements than larger neighborhood sizes. See results in Table 3. Smaller neighborhood sizes used for SWE estimates yielded smaller coverage areas where estimates could be reported and the improvement in estimate quality was minimal. 41

60 Table 3 : SWE prediction sensitivity to number of nearest neighbors SWE Monitoring Site GHCN Measurement (psf) Satellite Prediction 250 nearest neighbors (psf) Satellite Prediction 300 nearest neighbors (psf) Satellite Prediction 350 nearest neighbors (psf) Satellite Prediction 400 nearest neighbors (psf) Satellite Prediction 450 nearest neighbors (psf) Anchorage Int l AP Barrow Post Rogers AP Barter Island WSO AP Fairbanks Int l AP Homer AP Kotzebue Ralph Wein AP McGrath AP Nome Muni AP Talkeetna AP Model Diagnostics The predominant issue in this study dealt with establishing a means of estimating snow loads in Alaska at locations where snow loads are not directly measured. Using a simple kriging approach from satellite measurements gave good snow load estimates at locations where ground-based measurements are not available. The quality of those estimates was gaged by how close they were to actual measurements extracted from monitoring sites. Table 4 shows comparisons of measured SWE values with kriged satellite estimates of SWE. The results were promising but could have been generally improved with more direct SWE monitoring. 42

61 Table 4 : GHCN vs. satellite comparison of SWE with small differences SWE Monitoring Site Anchorage Int l AP Barrow Post Rogers AP Barter Island WSO AP Fairbanks Int l AP GHCN Measurement (psf) Satellite Prediction (psf) Percent Difference % % % % Homer AP % Kotzebue Ralph % Wein AP McGrath AP % Nome Muni AP % Talkeetna AP % Yakutat AP % In three cases where satellite estimates were available, monitoring station SWE measurements departed significantly from satellite estimates of SWE. This may be explained by peculiar environmental conditions specific to the monitoring stations, such as at an airport for instance. Those sites were Bethel, Unalakleet, and King Salmon. Table 5 shows the differences between measured snow water equivalency at the Bethel monitoring station and satellite estimates of snow water equivalency from the five nearest satellite sample points for years 1984 through

62 Table 5 : GHCN and satellite load comparisons near Bethel Year GHCN ( , ( , ( , ( , ( , ) ) ) ) ) In Table 5, annual maximum loads in pounds per square feet are compared with GHCN observations at Bethel Airport and with annual maximum satellite load estimates at the five nearest neighbors to Bethel Airport for a period of record from 1984 to The satellite estimates consistently showed higher load (SWE) values than those of GHCN measurements for each year. 44

63 Section 6 Summary and Conclusions Climate monitoring stations reporting snow water equivalent measurements are too few in number to interpolate off site values on the spatial extent spanned by Alaska geography. As a result, 50 year predicted snow load measurements at locations separated by distance from snow load measurement sites have to be approximated. The estimation process is summarized in the following work-flow diagram in Figure 21. Figure 21: Work-flow process for geospatially estimating SWE. 45