GEOG352 FINAL REVIEW

main goal of spatial interpolation

estimate the values for the entire study area based on the measured attribute values of existing sample points

analysis of spatially continuous data

focus on understanding the spatial distribution of values of one single attribute over an entire study region

how is the main goal of spatial interpolation achieved?

• by finding a function or a model that approximate the missing values

• predicting/estimating the values at points where the attribute has not been sampled

the spatial interpolation problem

given a set of spatial data in the form of points (or areas), finding the function that will estimate or predict values for any other points (or areas) and will best represent the whole surface

spatial interpolation

mathematical process of finding a function that will estimate/predict unknown attribute values at any point from a given set of sample point locations

the continuous surface

defined as a feature which contains continuous information about estimated z attribute values across a given study area

when should spatial interpolation methods not be applied?

if the attributes of sample points represent the presence of an event (crime/disease), people, or some physical phenomenon (volcanoes, buildings), as the created continuous surface will be meaningless

types of interpolation surfaces

continuous: statistical surfaces where data occurs at every possible location in the study area / discrete: statistical surfaces where data is limited to selected areas are are mostly suitable for qualitative (nominal) data

methods of spatial interpolation for discrete surfaces

spatial tessellation: vonoi diagrams and delaunay triangulation

methods of spatial interpolation for continuous surfaces

approximate spatial interpolation: trend surface analysis

exact: spline, inverse distance weighted (idw) and kriging

spatial tessellation method objective

divide and delineate landscape/study area by creating boundaries of new polygons - discrete surfaces

advantages and disadvantages of spatial tessellation methods

• the computation of irregular tessellation methods are well known algorithms and there are mathematical discussions

• the interpolated surface within each diagram/tessellation will have the same value, so errors cannot be calculated and the computation of a value at an unsampeld point becomes a problem

• adding or removing a point will require redesigning the entire tessellation structure

  • the edges of the study area tessellation structures have weird shapes

trend surface analysis

approximative method to derive continuous surface by using a low order polynomial or trigonometric function

• fitted between the sample data points

• parameters are estimated using the least squares method - minimize the total difference b/w the original surface and the polynomial function

advantages and disadvantages if trend surface interpolation

• the trend surface method generates the broad range models of low-order surfaces

• the statistical significance of trend surface interpolation can be tested by using the tecniques of analysis of variance b/w the trend and the residuals from the trend

• this method becomes increasingly difficult to describe a physical meaning of obtained surfaces when higher polynomials are used

• this is a smoothig technique, rarely passing through original sampling data points so it is often used as exploratory method for th

exact spatial interpolation methods

• points that are closer together on the ground are more likely to have similar values of a property than ponts further apart (first law of geog)

• spatial interpolation proceeds by finding the function that will permit fitting a surface model to the measure sample data points, and then the values at any desired locations can be estimated/predicted using the spatial interpolation method

spline interpolation method purpose

fits a minimum curvature through input points (sample points)

linear, quadratic, cubic spline

• • linear spline: uses linear functions

    • n=1, degree of freedom=0

    • a set of line segments that simply connect the known values of the function at sample points

  • quadratic spline uses quadratic functions

    • n=2, degree of freedom=1

    • sensitivity: if one point is slightly moved, the curve changes in four intervals — moving one point affects a large portion of the spline, not entirely local

  • cubic spline: cubic functions

    • n=3, degree of freedom=2

advtanges and disadvantages of spline interpolation

• the interpolation can be quickly calculated

• splines retain small scale features, are aesthetically pleasing and can produce quick and clear spatial overview of the data

• cubic splines provide the most natural and smooth surface found in the real world while linear or quadratic splines do not generate smooth surfaces

• problem with using splines is that different results will be obtained with choosing different break/sample points or when using a different number of sample points

• there is no direct estimates of the errors assocaited with spline interpolation

trend surface vs spline interpolation

both methods aim to predict values between known points, but use different strategies

trend surface

• fit a general mathematical function (polynomial or regression plane)

  • resulting curve does not necessarily pass through the sample points - approximates the trend

  • this is useful when the data has noise or youre more interested in the general pattern than exact values

    • = approximate interpolation

exact interpolation (spline)

• creates a curve that passes through all control points

  • uses a piecewise polynomial function to create a smooth and continuous path through the data

  • makes it more accurate when exact values matter — (modelling terrain or temp)

    • = exact interpolation

IDW

weights the points closer to the data source greaterr than those further way - incorporating first law of geog

procedure of IDW

• • superimposes an equally spaced grid of points onto the control/sample points

  • estimates values at each grid points as a function of their distance from the control/sample points

  • interpolates between the grid points

what does k control in IDW

significance of the surrounding points upon the interpolated values

• a higher power results in less influence from distant points and typically the generated surface will be smoother

advantages and disadvantages of the IDW method

• most commonly used spatial interpolation method that can applied on large datasets and study areas

• the surface resulting from IDW depends on the power exponent k and on the size of the window/radius

• the size of the window considers certain number of neighbour points, which affects the average values on the estiamted control/sample points and the computational time

• the choices of location and the number of control/sample points also have a direct influence on the outcome of the IDW method

what do neighbours control in idw

fewer neighbours increases detail, while more neighbours smooth the surface

kriging is used when

• the spatial variation of any spatial geogrpahical property is too irregular (heteogen) to be represented by a smooth mathematical function

kriging procedure

• • kirging superimposes an equally spaced grid of points onto the measured points

  • interpolates between the grid points giving consideration tos patial autocorrelation (the statistical variation of attribute z values of surrounding points)

  • estimate the predicted values for each point at location s

semi variance

half of the variance in the data points over a distance h

gives an indication of the variation b/w the atrribute values of z of sample/control points

semivarigoram

graph that relates each sample/control point to all other sample/control points with repect to the values of attribute z and distance h (lag intervals) between points

• semivariance used to make graph is

sill

• particular value of semivariance at which there is no more spatial dependency between the data points

range

gives an indication of distance h (lag intervals) over which spatial dependency occurs

nugget

spatially uncorrelated noise in the data set

advantages and disadvantages of kriging

• kirging is an advaned and complex technique that relies heavily on statistical theory and on computing abilities

• it is the most useful method when applied on data that contain well defined local trends of the attribute value

• the form of semivarigoram is central. to kriging, but it is uncertain if particular functions that estimate semivariogram is a fact the true estimator of the spatial variation in the study area

• scale free interpolation technique, can be applied to smaller study areas

application of interpolation methods

DTM - vector (TIN) and raster (DEM)

geodemographics

• profiling people based on the location where they live

• understanding the composition of human settlements in different kind of neighbourhoods

  • creation of market segmentation systems

geodemographics require

• various spatial data analysis methods

• census and postal geography data

• large geospatial and socio-economic datasets

• GIS software for analysis, mapping and display

goal of segmentation

• classifying neighbourhoods into homogeneous types of clusters

  • find groups or clusters of neighbourhoods that are similar to eachother, demographically and in terms of lifestyle

space time analysis

• study of things change across both space and time simultaneously

  • combines geography with temporal pattens to help us understand complex processes that unfold dynamically

    • ex: covid 19 - spatial element: cities/countries affected, temporal elemts: weekly case data

reaction

• complex adaptive systems theory suggest that near can be sufficient

  • simple and local interactions among entities of the system at local level can produce complex behaviour and patterns at global level, that are not completely predictable or controllable

    • in the real world - everything is process

    • space and time need to be considered together = spatio-temporal analysis and modelling

complex system theory

permits modelling of wide range of spatial and geographical dynamic phenomena especially when integrated with GIS and geospatial data

cellular automata

geosimulation models: modeling regional urbanization process

sagregation model

people tend to linein neighbourhoods where 50% or more are like themselves