Definition
The semivariogram was first defined by Matheron (1963) as half the average squared difference between the values at points ( and ) separated at distance . Formally : where is a point in the geometric field , and is the value at that point. The triple integral is over 3 dimensions. is the separation distance (e.g., in meters or km) of interest. For example, the value could represent the iron content in soil, at some location (withProperties
According to (Cressie 1993, Chiles and Delfiner 1999, Wackernagel 2003) the theoretical variogram has the following properties: * The semivariogram is nonnegative , since it is the expectation of a square. * The semivariogram at distance 0 is always 0, since . * A function is a semivariogram if and only if it is a conditionally negative definite function, i.e. for all weights subject to and locations it holds: :: : which corresponds to the fact that the variance of is given by the negative of this double sum and must be nonnegative. * If the_Parameters
In_summary,_the_following_parameters_are_often_used_to_describe_variograms: *_''nugget''_:_The_height_of_the_jump_of_the_semivariogram_at_the_discontinuity_at_the_origin._ *_''sill''_:_Limit_of_the_variogram_tending_to_infinity_lag_distances.__ *_''range''_:_The_distance_in_which_the_difference_of_the_variogram_from_the_sill_becomes_negligible._In_models_with_a_fixed_sill,_it_is_the_distance_at_which_this_is_first_reached;_for_models_with_an_asymptotic_sill,_it_is_conventionally_taken_to_be_the_distance_when_the_semivariance_first_reaches_95%_of_the_sill._Empirical_variogram
Generally,_an_empirical_variogram_is_needed_for_measured_data,_because_sample_information__is_not_available_for_every_location._The_sample_information_for_example_could_be_concentration_of_iron_in_soil_samples,_or_pixel_intensity_on_a_camera._Each_piece_of_sample_information_has_coordinates__for_a_2D_sample_space_where__and__are_geographical_coordinates._In_the_case_of_the_iron_in_soil,_the_sample_space_could_be_3_dimensional._If_there_is_temporal_variability_as_well_(e.g.,_phosphorus_content_in_a_lake)_then__could_be_a_4_dimensional_vector_._For_the_case_where_dimensions_have_different_units_(e.g.,_distance_and_time)_then_a_scaling_factor__can_be_applied_to_each_to_obtain_a_modified_Euclidean_distance._Variogram_models
The_empirical_variogram_cannot_be_computed_at_every_lag_distance__and_due_to_variation_in_the_estimation_it_is_not_ensured_that_it_is_a_valid_variogram,_as_defined_above._However_some__Discussion
Three_functions_are_used_in__Applications
The_empirical_variogram_is_used_in__Related_concepts
The_squared_term_in_the_variogram,_for_instance_,_can_be_replaced_with_different_powers:_A_''madogram''_is_defined_with_the_Parameters
In summary, the following parameters are often used to describe variograms: * ''nugget'' : The height of the jump of the semivariogram at the discontinuity at the origin. * ''sill'' : Limit of the variogram tending to infinity lag distances. * ''range'' : The distance in which the difference of the variogram from the sill becomes negligible. In models with a fixed sill, it is the distance at which this is first reached; for models with an asymptotic sill, it is conventionally taken to be the distance when the semivariance first reaches 95% of the sill.Empirical variogram
Generally, an empirical variogram is needed for measured data, because sample information is not available for every location. The sample information for example could be concentration of iron in soil samples, or pixel intensity on a camera. Each piece of sample information has coordinates for a 2D sample space where and are geographical coordinates. In the case of the iron in soil, the sample space could be 3 dimensional. If there is temporal variability as well (e.g., phosphorus content in a lake) then could be a 4 dimensional vector . For the case where dimensions have different units (e.g., distance and time) then a scaling factor can be applied to each to obtain a modified Euclidean distance. Sample observations are denoted . Samples may be taken at total different locations. This would provide as set of samples at locations . Generally, plots show the semivariogram values as a function of sample point separation . In the case of empirical semivariogram, separation distance bins are used rather than exact distances, and usually isotropic conditions are assumed (i.e., that is only a function of and does not depend on other variables such as center position). Then, the empirical semivariogram can be calculated for each bin: : Or in other words, each pair of points separated by (plus or minus some bin width tolerance range ) are found. These form the set of points . The number of these points in this bin is . Then for each pair of points , the square of the difference in the observation (e.g., soil sample content or pixel intensity) is found (). These squared differences are added together and normalized by the natural number . By definition the result is divided by 2 for the semivariogram at this separation. For computational speed, only the unique pairs of points are needed. For example, for 2 observations pairs (z_a,z_b),(z_c,z_d)">math>(z_a,z_b),(z_c,z_d)taken from locations with separation only (z_a,z_b),(z_c,z_d)">math>(z_a,z_b),(z_c,z_d)need to be considered, as the pairs (z_b,z_a),(z_d,z_c)">math>(z_b,z_a),(z_d,z_c)do not provide any additional information.Variogram models
The empirical variogram cannot be computed at every lag distance and due to variation in the estimation it is not ensured that it is a valid variogram, as defined above. However someDiscussion
Three functions are used inApplications
The empirical variogram is used inRelated concepts
The squared term in the variogram, for instance , can be replaced with different powers: A ''madogram'' is defined with theParameters
In summary, the following parameters are often used to describe variograms: * ''nugget'' : The height of the jump of the semivariogram at the discontinuity at the origin. * ''sill'' : Limit of the variogram tending to infinity lag distances. * ''range'' : The distance in which the difference of the variogram from the sill becomes negligible. In models with a fixed sill, it is the distance at which this is first reached; for models with an asymptotic sill, it is conventionally taken to be the distance when the semivariance first reaches 95% of the sill.Empirical variogram
Generally, an empirical variogram is needed for measured data, because sample information is not available for every location. The sample information for example could be concentration of iron in soil samples, or pixel intensity on a camera. Each piece of sample information has coordinates for a 2D sample space where and are geographical coordinates. In the case of the iron in soil, the sample space could be 3 dimensional. If there is temporal variability as well (e.g., phosphorus content in a lake) then could be a 4 dimensional vector . For the case where dimensions have different units (e.g., distance and time) then a scaling factor can be applied to each to obtain a modified Euclidean distance. Sample observations are denoted . Samples may be taken at total different locations. This would provide as set of samples at locations . Generally, plots show the semivariogram values as a function of sample point separation . In the case of empirical semivariogram, separation distance bins are used rather than exact distances, and usually isotropic conditions are assumed (i.e., that is only a function of and does not depend on other variables such as center position). Then, the empirical semivariogram can be calculated for each bin: : Or in other words, each pair of points separated by (plus or minus some bin width tolerance range ) are found. These form the set of points . The number of these points in this bin is . Then for each pair of points , the square of the difference in the observation (e.g., soil sample content or pixel intensity) is found (). These squared differences are added together and normalized by the natural number . By definition the result is divided by 2 for the semivariogram at this separation. For computational speed, only the unique pairs of points are needed. For example, for 2 observations pairs (z_a,z_b),(z_c,z_d)">math>(z_a,z_b),(z_c,z_d)taken from locations with separation only (z_a,z_b),(z_c,z_d)">math>(z_a,z_b),(z_c,z_d)need to be considered, as the pairs (z_b,z_a),(z_d,z_c)">math>(z_b,z_a),(z_d,z_c)do not provide any additional information.Variogram models
The empirical variogram cannot be computed at every lag distance and due to variation in the estimation it is not ensured that it is a valid variogram, as defined above. However someDiscussion
Three functions are used inApplications
The empirical variogram is used inRelated concepts
The squared term in the variogram, for instance , can be replaced with different powers: A ''madogram'' is defined with the