Multidimensional Analysis of Snow Cover Data in South Dakota Diversity of Landscapes

Snow distribution in SD was studied with Factor Analysis (FA) of monthly total snowfall [in]. The long-term data obtained from the High Plains Regional Climate Center were used for the territory of South Dakota. The perspective for creating an Atlas of Climate and Water Resources for SD directed this study of total monthly snowfall with connection to landscape diversity.

The initial matrix {Xn*p} where n is number of stations and p is number of variables of monthly average for period of observations. The maximum number of stations n with mutual interval of observations for SD is equal 93 (n=93). These stations have mutual time interval of 18 years observations (1952-53 – 1969-70). Total monthly snowfall has data for p=10 is number of months with observation and p=11 is number of months with observations with total snowfall for the winter season. The second matrix contains proportions of total monthly snowfall to total annual (proportion is the monthly total snowfall divided on total seasonal snowfall); the number of rows n and the number of variable are the same as in the first case: n=93 and p=10, 11.

The average annual sum of total monthly snowfall (September-June) for SD 34.64 in obtained on 93 stations for 1952-1970, the median is 31.84 from the same data; ranged from 10.21 to 152.27 from the same data. The most variable month is April with average
4.89 [in], median 3.67, min 0.75 and max 33.18; the average proportion for April is 0.13, min 0.03 and max 0.24. The averages for November to April grow as sequence: 3.60, 5.70, 4.88, 7.25, 7.46 and 4.89 the variability of those months has sequence: 2.49 2.52 2.25 3.33 3.53 and 4.87. The Pearson coefficient of correlation for the monthly snowfall averages from September to June with annual sum has sequence: 0.83, 0.92, 0.96, 0.92, 0.93, 0.93, 0.94, 0.92, 0.90, and 0.82; the correlation for the monthly proportions from September to June with annual sum has sequence: 0.38, 0.45, -0.01, -0.32, -0.33, -0.21, -0.15, 0.27, 0.36, and 0.35.

The FA of both initial matrixes had extracted two factors for monthly observations with incorporation of 93 % of total variability of the data in the model; and three factors model with 70% variability of data for monthly proportions. The two factor groups in model of observed snowfall contain winter months (Dec – Mar) and all other. The model of monthly proportions with tree factor groups include winter months in two group and combination of spring and fall months in third. The factor scores presented as maps for SD allow trace distribution of factor groups of monthly snowfall patterns for two models through the territory of SD. The combination of factor scores distribution for two models is in a good agreement with main landscape regions and subregions of SD.
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South Dakota Diversity of Temperature: Pictures from Statistical Analysis

The regional diversity of monthly temperature was analyzed based on long-term data obtained for South Dakota (SD) from the High Plains Regional Climate Center. Multidimensional statistical methods were used and the principal results presented as a sequence of 2- and 3-dimensional scatterplot pictures depicting the quantitative results. 
System hierarchical model of landscape was used for research tasks formulation. Initial matrixes for three research tasks were compiled for the state. The first set of initial matrices of time series {Xt*n}, where t = number of years and n = number of meteorological stations, contains two matrixes: X1(67*29) and X2(33*94). The second set -{Xt*m}, where t = number of years and m = number of months in a year: X3(113*12), X4(110*12), and X5(102*12). The third set - {Xn*m}, where n = number of meteorological stations and m = number of months in a year, contains two matrixes: X6(29*12) and X7(94*12). 
Statistical analysis allowed us to differentiate weather stations by temporal trends and spatial distribution for the time interval 1932-1998. The most variable stations (Brookings, Camp Crook, and Highmore) were determined; their seasonality was described (the most variable months and correlation among months during the year) and their seasonal regime determined. The average annual and monthly temperature distributions were presented for South Dakota based on 29 and 94 stations for the time intervals 1932-1998 and 1963-1995.
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Understanding and mapping water resources by multidimensional statistics and fuzzy logic: Missouri River basin case

Time series from 46 gauging station with drainage areas from 113 to 398 sq mi in the Upper Missouri River basin with mutual period of observation from 1963 to 1991 were used for analysis. Factor analysis of average annual flow revealed five patterns of river runoff within four distinct subregions of the territory (east, two carbonate karsts areas, uplands). This factor model reflected 62% variance of initial matrix. Each of four groups of watersheds obtained as a factor was presented by one gauging station with time series of annual and monthly discharges (I- 06218500, II- 06478690, III- 06412500, and IV- 06323000). Streams represented by patterns I, II and IV have increase of values and those represented by III have a decrease. The positive trend for pattern II is statistically significant. For four typical flow records, monthly average values were obtained from three to four seasons composed of different ensembles of months. The trends for seasonal components were analyzed for four typical watersheds and a significant increase was obtained for fall-winter season for type IV. Stream runoff is the most appropriate regional indicator for hydroclimatological processes. With multidimensional statistics this process can be considered as spatiotemporal structure of different scale of landscape properties and dynamics. Uncertainties of process originating stream runoff based on dynamic of regional meteorological system and diversity of local landscapes. Boundaries for domains with different annul and seasonal regimes of stream runoff were defined with factor loadings and fuzzy logic rules. With case of Missouri River basin presented that more complete decryption of real events in nature requires use probability and fuzzy logic together

Missouri River Watershed: the Object for Hydrological Study and Uncertainty of Models

The longest river (2540 mi) and the second drainage area (529 346 mi2) in the North America the Missouri River is one from the most interesting hydrological objects to study. The issue of uncertainty is the basis for any application of knowledge (“Uncertainty is an attribute of information.” From Zadeh, 2005) and has to be one of the main tasks in Earth’s systems study. Knowledge about natural systems (watershed in our case) may be only obtained by the analysis of the empirical (instrumental) data (observations). Principle of Uncertainty is the basic low in Physic. In Hydrology, the Uncertainty starts from the unveiling of the research task by the researcher. The main source of the uncertainty comes from the natural system “extraction” (unit’s boundaries) for modeling and from the limitations of data representing both time and space variability. The watershed has the formal determined boundary and this property places hydrology in the center of regional climate research. The uncertainty is considered in context of time and space with use of cybernetic model of the watershed and described on base of specification of the system in the coordinates on the Earth. The math model does not have criteria to verify itself (Gödel's incompleteness theorems) – multitasks & multiscales studies have to be completed. The data analysis for Upper Missouri River provide a base for regionalization, a multi-scaled description of the structure of river watersheds and their interaction with climate characteristics, and uncertainty of the obtained knowledge. The formulation of the uncertainty for watershed helps to explain the scope of practical applications to be developed, and the tasks to study, communicate and educate the public/communities about water resources and environmental issues, including extreme events like drought/flooding. Science is one of many components of life but the scientist is the only holder of the “truth” and a creator of formal knowledge of the nature. “… data analysis assists in the formulation of a model … A model is merely your reflection of reality and, like probability, it describes neither you nor the world, but only a relationship between you and that world” (From Lindley, “Principles of Statistics”, 2006)

The spatial temporal regime of stream flow of the conterminous U.S. in connection with indices of global atmospheric circulation

Long-term stream flow records (1929-1988) from seventy one U.S. Geological Survey gauging stations with drainage area in range 1000-10000 sq mi were analyzed using multivariate statistics. Factor analysis of average annual flow revealed seven patterns of river runoff within seven distinct regions of the territory. This factor model reflected 69% variance of the initial matrix. The second set of stream flow records (1939-1972) from ninety-seven gauging stations was used as control. This set contains all seventy one from first one and additional stations with shorter observation period. Factor analysis of this expended set again yielded seven factors (69% variance of the initial matrix) with very similar spatial distribution of gauging stations.

Every group of watersheds obtained as a factor was presented by one gauging station with time series of annual discharges (1- 05474000, 2- 14321000, 3- 07019000, 4- 0815000, 5- 11186001, 6- 01666000, 7- 06800500) as the most typical for group. For the same time interval, streams represented by all patterns have increasing values (i. e. the positive difference between two time subintervals); but only the positive linear trend for patterns 1 and 7 are statistically significant. 

For the seven typical flow records, monthly average values were obtained from three to five seasons composed from different ensembles of months. 

For each annual time series of the typical seven stream flow patterns, regression equations were obtained from indices of global atmospheric circulation (AO, NAO, NPO and AAO). The equations contain from one to five variables (predictors) and have coefficients of correlation from 32% to 73%. 
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Precipitation in Aberdeen, SD: data analysis approach

The daily, monthly and annual sum of precipitation was analyzed for station Aberdeen-Airport, South Dakota (NOAA COOP #390020) with traditional normative statistical descriptions of precipitation variability and with methods of data analysis. The goal for research was formulated as finding a way to forecast extremes, such as the flooding of Aberdeen in May 2007 after a 30-hour, 8.42 inch rain event. Daily data were available, analyzed and cannot be considered as having any use for the forecast. The observations of 1423 monthly sums of precipitation have the following characteristics: mean = 1.802 in.; geometric mean = 0.856 in.; median = 1.26 in.; minimum = 0.0; maximum = 12.39 in. Simplified Fourier analysis was performed for monthly data, and 31 cyclic components were determined and used in a model to represent the data with periods from 4 to 379 months. The biggest amplitude from those components belongs to period of 12 months (1.396), and the next largest amplitude (0.261) had a period length of 6 months. The lowest amplitude (0.033) has a period of 75 months; the longest period 379 months has an amplitude of approximately twice that height (0.066). Fourteen periods have amplitudes higher that 0.1. The same analysis was performed on data obtained from the High Plains Regional Climate Center that was collected between January 1932 and April 2006. The model has mean equal 1.623 in. and the same two periods: 12 and 6 months, - with the biggest amplitudes: 1.315 and 0.258, and the model showed the increase in precipitation during May 2007 (3.494 in) that bigger that in April (2.965 in), June (3.115 in) 2007, and May (2.246 in) 2006. Mean annual precipitation for the time interval 1891- 2007 was 21.627 in. Simplified Fourier analysis presented a model of annual variability with eight cyclic components. The biggest amplitude (2.26) has the longest period of 46 years; the smallest (0.336) has a period of 31 years. Five components (46, 3, 10, 24, and 8 years) have amplitudes larger that 1.0 The same analysis for annual sum for time interval 1932-2006 presents mean = 19.514, and a model with five components of 3, 7, 14, 18, and 26 years with sequential altitudes 1.525, 1.451, 1.084, 1.066, and 1.451. The model shows that 2007 annual precipitation is moderately greater than the long-term mean value. During 118 hydrological years, the mean of annual sum for hydrologic year is 21.617 in., the minimum is equal 9.07 and maximum is equal 41.44. The highest mean of 3.704 in. for monthly sum during this time interval has June, the second of 2.948 in. has May; but May has the highest maximum equal 12.39 and June only has 10.91. The factor analysis on the matrix of monthly sum of precipitations {P118*12,13} shows that the annual sum mostly associated with May and August; the seasonal variability may be described by five factors (cumulative variance reflected by the model is equal 57.4%). The seasonal regime may be well represented by the factor scores; four factors charts (for factors 1, 3, 4, and 5) show different degrees of decrease in sum of precipitations for analyzed interval. In contrast, for factor 2 slightly increased. The assumption of possible forecast for precipitation and connected flooding was considered based on the obtained results. The highest daily sum of precipitation may be expected in month with biggest monthly sums in May and June. The time series of annual sum and monthly sums of precipitations have cycles with periods of 24, 31 and 46 years, and 286 and 379 months; traditional tools of characterization variability have to be applied to the longest as it is possible time series

Snow Cover in South Dakota: Statistical analysis of spatiotemporal diversity

Snow distribution and accumulation influence many human activities and the dynamic sustainability of ecological systems. Snow cover distribution analysis is a second research step towards creating a Atlas of Climate and Water Resources for SD (temperature analysis was presented last year). We analyzed the regional diversity of monthly total snowfall based on long-term data obtained from the High Plains Regional Climate Center for South Dakota. Multidimensional statistical methods were used, and the results presented for the State of South Dakota. 

The sets of initial matrixes were compiled with snow observations for the state. The first type of initial matrix of time series {Xt*n}, where t = number of years (67) and n = number of meteorological stations (25), contains 25 stations with mutual observational interval (1931-1998) of 67 winters. The second type of initial matrix {Xt*m}, where t = number of years (67) and m = number of months in a year, included seven initial matrixes.

Statistical analysis allowed us to differentiate weather stations by temporal trends and spatial distribution for the time intervals 1931-1998. The average annual sum of total monthly snowfall (October-May) ranged from 25.5 to 53.5 inches for 25 stations trough this time interval. The most variable stations (Bowman Court House, Alexandria, Colony, Gordon, Clark, hot Springs and Eureka) were determined; their seasonality was described (the most variable months and correlation among months during period of snowfall) and their seasonal regime determined. 
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Spatiotemporal regime of climate and streamflow in the U.S. Great Lakes Basin

We analyzed interannual and seasonal regimes of river runoff, precipitation, and air temperature for three nested regions: (1) the Upper Peninsula of Michigan, (2) the U.S. portion of the Great Lakes Basin, and (3) glaciated portions of the northeastern U.S. spanning from the Dakotas to New England. Data sources included historical records of 50 to 105 years duration from 45 USGS gauging stations and 198 U.S. National Climate Network stations, and satellite-derived estimates of global monthly precipitation gridded at 2.5 resolution for 1979-2008 (partly data about precipitations were collected and passed for analysis by Glenn Hodgkins from USGS). We examined the spatiotemporal variability of climate characteristics for these regions as a multidimensional structure obtained from empirical data using factor analysis. The structure consisted of a few (2 to 7) centers of variability for each characteristic, and reflected the diversity of landscapes within the regions examined. Trends and regime shifts for mutual time intervals of river runoff, precipitation and temperature showed different direction of changes. The results obtained at the three scales were generally in agreement

Geological Controls on Water Resource Variability in Minnesota, USA

Sustainable management of water resources requires quantitative description of spatio-temporal variability, and the map is a universal medium to reflect the spatio-temporal distribution of water resources. The long history of cartography and the recent digital revolution have culminated in the Google Earth web portal with unprecedented frequency of daily use. System analysis with combination of a cyber model of landscapes, multidimensional methods of data analysis, and GIS cartography of water resources in Minnesota started in 1996 with support from faculty of Department of Geology University of Minnesota-Duluth and has continued ever since. The “Water Resource Sustainability” project, funded by the Legislative Citizens Commission on Minnesota Resources (2007-2009) was the most resent phase of the research. 

Research using river flow monitoring data available from USGS for Minnesota and bordering areas of North Dakota, South Dakota, Iowa and Wisconsin was completed for the territory. Analysis of landscapes properties for watersheds taken from maps - Bailey’s Ecological Provinces, Soil Taxonomy Order, topographic characteristics (average altitude, average watershed slope, total, intermittent, and perennial drainage density), thickness of quaternary sediments, and Hydrogeological Hierarchical Regionalization - revealed control of geological conditions on water resource variability. The trends of interannual patterns and seasonality of river runoff depend on bedrock type and presence or absence of thick depositions of quaternary sediments in NE and SE of research territory and also on thickness of quaternary sediments in NW. The same parts of territory have main differences in annual and February monthly yields for interval of observations 1955-1978. The numbers of river discharge yield reach difference from 5 to 20 times. 

The control over water resource distribution and variability belongs to geological boundaries for types of bedrocks, lithology, and thickness of quaternary sediments. Groups of watersheds recognized by mutual landscape properties (geological conditions) with statistically proven influence on hydrologic characteristics provide a basis for regionalization and creation of a water resource map. The regionalization on the water resource map opens the way to study and climate change for regional level

Atlases of Minnesota Water Sustainability: Creation from Models, Analytical Methods, and Database of Watershed Characteristics

Confident assessment of the numbers of regional water balance and water resources for Atlases of MN Water Sustainability (AMNWS) has to be based on the best science, advanced knowledge and modern technology. Creating a watershed characteristic database is the first and most significant step in the analysis of time spatial distributions of Minnesota’s water balance components. The database spatially associates hydrologic data (annual discharge, monthly proportion, and yield) with topographical, soil and vadoze zone, ecological and hydrogeological conditions and properties. USGS stream flow data for 129 gauging stations surrounding MN was manipulated within ArcGIS. Characteristics of the 129 watersheds (drainage areas 100-10000 square miles) were summarized to create a matrix database (1947-1971 (79 watersheds), 1955-1979 (93), and 1976-2006 (74). Model as formal integrator of knowledge plays a central role for hydrological process in the landscape environment study. The use of cyber model of landscape (Krcho, 1978) and watershed as part of it (Shmagin, 1997) provides research tasks for data analyses. Six research tasks implied 17 initial matrixes incorporated the existing longtime hydrological data (from 25 to 109 years). Special part of database includes temperatures and precipitation (from 56 to 106 years) for 90 meteorological stations for MN and surrounding states (from US Historical Climatology Network: http://cdiac.ornl.gov/epubs/ndp/ushcn/usa_monthly.html). The data analysis (multivariate exploratory techniques) operates with two kinds of data: time series and maps that are integrated with use of GIS. Statistic was used with this database to transform initial matrixes of watershed characteristics into linear components and residual matrixes, establishing characteristics interconnections. To combine the results of data analysis of layered quantities and qualitative information onto a map of water resources with defined unit boundaries requires the fuzzy logic approach for regionalization (Shmagin and Chen, 2006). For AMNWS future developments, the watershed characteristics database can be upgraded to include monitoring of the surface and ground water regimes