Modelling spatial uncertainty of soil erodibility ... - Wiley Online Library

land degradation & development Land Degrad. Develop. 19: 198–213 (2008) Published online 23 August 2007 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/ldr.836

MODELLING SPATIAL UNCERTAINTY OF SOIL ERODIBILITY FACTOR USING JOINT STOCHASTIC SIMULATION ` 1, G. BUTTAFUOCO2*, A. CANU3, C. ZUCCA4 AND S. MADRAU4 A. CASTRIGNANO 1 Agronomic Research Institute, CRA, Bari, Italy Institute for Agricultural and Forest Systems in the Mediterranean, National Research Council, Rende (CS), Italy 3 Institute of Biometeorology, National Research Council, Sassari, Italy 4 NRD, University of Sassari, Italy

2

Received 19 December 2006; Revised 5 July 2007; Accepted 6 July 2007

ABSTRACT Soil erosion varies greatly over space and is commonly estimated using the revised universal soil loss equation (RUSLE). Neglecting information about estimation uncertainty, however, may lead to improper decision-making. One geostatistical approach to spatial analysis is joint stochastic simulation, which draws alternative, equally probable, joint realizations of a regionalized variable. Differences between the realizations provide a measure of spatial uncertainty and allow us to carry out an error propagation analysis. The objective of this paper was to assess spatial uncertainty of a soil erodibility factor (K) model resulting from the uncertainties in the input parameters (texture and organic matter). The 500 km2 study area was located in central-eastern Sardinia (Italy) and 152 samples were collected. A Monte Carlo analysis was performed where spatial cross-correlation information through joint turning bands simulation was incorporated. A linear coregionalization model was fitted to all direct and cross-variograms of the input variables, which included three different structures: a nugget effect, a spherical structure with a shorter range (3500 m) and a spherical structure with a longer range (10 000 m). The K factor was then estimated for each set of the 500 joint realizations of the input variables, and the ensemble of the model outputs was used to infer the soil erodibility probability distribution function. This approach permitted delineation of the areas characterized by greater uncertainty, to improve supplementary sampling strategies and K value predictions. Copyright # 2007 John Wiley & Sons, Ltd. key words: Sardinia; spatial uncertainty; revised universal soil loss equation (RUSLE); turning bands simulation; soil erosion

INTRODUCTION Soil erosion is a major, worldwide environmental threat to the sustainability and productive capacity of conventional agriculture. In Europe, soil erosion has become a serious problem in many areas and particularly in the Mediterranean region. Therefore, assessing and mapping soil erosion are important for management and conservation of natural resources (Morgan, 1995; Agassi, 1996). One of the most widely applied soil erosion models is the revised universal soil loss equation (RUSLE) (Renard et al., 1991), where long-term average annual soil loss is predicted as a product of rainfall-runoff erosivity factor (R), soil erodibility factor (K), slope length factor (L), slope steepness factor (S), cover management factor (C) and support practice (P) (Renard et al., 1991). Soil erodibility is the inherent susceptibility of soil to be lost to erosion, and it is one factor that affects the likelihood and severity of soil erosion (Morgan, 1995). In soil erosion models like the RUSLE, the soil erodibility factor is defined as the rate of soil loss per erosivity index unit as measured on a standard plot, 22.1 m long with a 9% slope, in a continuously clean-tilled fallow condition with tillage performed up and downslope (Wischmeier and * Correspondence to: G. Buttafuoco, CNR – ISAFOM, Via Cavour 4/6 - 87030 Rende (CS), Italy. E-mail: [email protected]

Copyright # 2007 John Wiley & Sons, Ltd.

UNCERTAINTY OF SOIL ERODIBILITY

199

Smith, 1978). Soil erosion models like the RUSLE are site-specific point models, in the sense that the computations are made on independent entities of landscape (polygons), neglecting all spatial relationships between the polygons. This approach to soil erosion, however, requires many Wischmeier and Smith type plot experiments (Wischmeier and Smith, 1978) to determine soil erodibility. Since soil erodibility is a function of many soil properties (Morgan, 1995), it is possible to model K using continuously varying spatial data of soil properties. Moreover, when assessing erosion risk for large areas, spatial data are required because the variability may better reflects the nature of the erosion process in comparison to measurements made only at a few points within ‘a priori’ soil units. It is important modelling the process but the point models must be extended to a regional scale, because erosion is not only a point process. To improve the prediction capacity of erosion models, a continuous spatial function of input data is needed. In most of quantitative erosion modelling with Geographical Information Systems (GIS), the calculation is often assumed to produce an exact result because GIS is intrinsically deterministic and lack information on the impact of errors in input and output data. Knowing the quality of the model results is fundamental, especially when they are used in spatial decision-making and in planning restoration actions. The quality of model predictions essentially depends on three main factors: (1) quality of data, (2) quality of the model and (3) the way data and model interact (Burrough, 2001). Uncertainty analysis (also called error propagation analysis) is an important tool to know how uncertainties in both model parameters and data propagate through the model (Heuvelink et al., 1989. The objective of error propagation analysis is to determine how large the error in the output (u) is, given errors in the inputs (ai) (Heuvelink, 1998). Usually, this is done by defining a joint probability distribution for the inputs. Then, the standard deviation of an individual input is interpreted as the main parameter representing its uncertainty model (Heuvelink, 1998). Error propagation analysis should include the model error caused by the limitations in the mathematical equations, used to simulate the physical system that result from the simplifying assumptions and/or parameter errors in regression equations. Because the model is a simplified representation of reality, errors in model output will occur even if the inputs were exactly known (Heuvelink, 1998). Model uncertainty is generally difficult to quantify, but one method of estimating the errors is to use a validation test with independent data. Model error can be included by making model parameters randomly distributed or adding a residual noise term to the model (Heuvelink, 1998). Data uncertainty results from measurement error, incomplete knowledge of spatial and temporal variation and heterogeneities at a spatial scale smaller than the sampling scale. Analytical solutions to the error propagation problem exist only in a few special cases, such as when a model that operates on m inputs ai (i ¼ 1, . . ., m) is linear (Heuvelink, 1998). The Monte Carlo method (Lewis and Orav, 1989) is often used in error propagation analysis. In the environmental sciences, Monte Carlo simulation is by far the most popular tool for tracing the propagation of errors because it is transparent, easily implemented and generally applicable (Jansen, 1998). The method generates a random data set of input realizations; each describing the joint distribution of all input variables. The model is then run for each set of realizations of the input variables, and the ensemble of model outputs is used to infer the output probability function. A single Monte Carlo simulation consists of a model run for all locations of a fine grid covering the region of interest. The simplest way to obtain error surfaces of interpolated input data in a GIS is to assume that all data are normally distributed and to express the error as a standard deviation. For each soil attribute, there is a mean value and a standard deviation for each cell. A criticism to this approach is that spatial correlation, is neglected, and the spatially uncorrelated error is used for each realization. Application of the Monte Carlo method to spatially distributed inputs requires the simultaneous generation of realizations from random fields. The joint multivariate stochastic simulation (Gomez-Hernandez and Journel, 1992; Goovaerts, 1997) is aimed at making predictions of cross-correlated variables. Predictions are accomplished using a variogram matrix, which includes spatial autocorrelation, and spatial cross-correlation between variables. The latter information improves spatial prediction of soil erosion by reducing its uncertainty when compared with traditional Monte Carlo simulations. Stochastic simulation is used to estimate cell-specific probability distribution functions that reflect the location of known data points and the spatial correlation structures of the variables. Although the Monte Carlo method is Copyright # 2007 John Wiley & Sons, Ltd.

LAND DEGRADATION & DEVELOPMENT, 19: 198–213 (2008) DOI: 10.1002/ldr

` ET AL. A. CASTRIGNANO

200

computer intensive, it produces unbiased predictions of model outputs, estimates of output uncertainties, assessment of error propagation in non-linear and complex models and estimation of probability of exceeding a critical threshold. In the light of the above considerations, the objectives of this research were to quantify the output error of a soil erodibility factor model, resulting from uncertainties in the model inputs and to show the benefits derived from the application of an error propagation analysis for operational decision-making.

MATERIALS AND METHODS The Study Area The area covers about 500 km2 (Figure 1) and is located in central-eastern Sardinia (Italy), between 408 20’ and 408 400 north latitude and between 98 300 and 98 500 east longitude. The climate is dry sub-humid, with a mean annual temperature of about 178C and mean annual precipitation ranging from 500 to 700 mm. Temporal rainfall distribution is irregular and mainly concentrated in the autumn and is characterized by high erosivity. The early autumn events are intense storms that have a serious erosive impact on pastures deprived of the vegetation cover due to aridity and grazing. The geological substratum is highly heterogeneous with Palaeozoic granites and metamorphic rocks, Jurassic crystalline limestone, Plio-Pleistocene basalt and ancient and recent alluvial deposits. The landscape is complex (Figure 2). The southeastern part of the study area is a coastal plain characterized by alluvial deposits. This plain separates the northern basaltic plateau from the limestone relieves of Tuttavista Mountain, which characterizes the southern area. On the central and northern areas, granite outcrops show steep and irregular landforms, which are deeply dissected; and metamorphic formations show undulating relief. Most of the area, especially on hilly slopes, is characterized by shallow soils that are degraded due to over-cultivation and overgrazing. Erosion processes are widespread.

Figure 1. Location of study area and sampling points. Copyright # 2007 John Wiley & Sons, Ltd.



201

Figure 2. Lithologic map showing the distribution of major rock types in the study area.

Soil Sampling The soil was sampled at 152 points according to a classical soil survey and following an irregular scheme (Figure 1). In this paper, we report on the analysis of the particle size fractions, sand (2–0.02 mm), silt (0.02–0.002 mm) and clay ( 0.10. The data distributions showed long positive tails, with the exception of sand (Figure 3). Therefore, before conducting joint Gaussian simulation, we applied a Gaussian Copyright # 2007 John Wiley & Sons, Ltd.



204

Table I. Basic statistics of clay, silt, sand and organic matter (OM) Variable Clay Silt Sand OM

Minimum (%)

Maximum (%)

Mean (%)

Median (%)

St. Dev. (%)

Variance (%)

Skewness

Kurtosis

C.V.

0.50 0.80 38.80 0.30

33.00 45.40 98.20 13.40

11.43 15.63 72.93 4.52

10.80 14.00 74.45 3.40

6.17 8.73 11.95 3.17

38.00 76.00 143.00 10.07

1.23 0.96 0.48 1.22

4.84 3.96 3.04 3.92

0.54 0.56 0.16 0.70

transformation to all three variables (clay, silt and organic matter contents) and a LMC (Table II) was fitted to all direct and cross-variograms of the Gaussian transformed variables (Goovaerts, 1997). The LMC included three different spatial structures: a nugget effect, a spherical structure with a shorter range (3500 m) and a spherical structure with a longer range (10 000 m). The goodness of fit was evaluated by a cross-validation test, whose results

Figure 3. Histograms of sand, silt, clay and organic matter contents. Copyright # 2007 John Wiley & Sons, Ltd.


205


Table II. Fitted linear model of coregionalization of anamorphosed clay (G Clay), silt (G Silt) and organic matter (G OM) Variable (1) Nugget effect G Clay 0.5567 G Silt 0.1045 G OM 0.1790 Eigenvalue 0.7077 (55.8%) (2) Spherical model (range ¼ 3500 m) G Clay 0.1115 G Silt 0.0762 G OM 0.0095 Eigenvalue 0.7485 (83.7%) (3) Spherical model (range ¼ 10 000 m) G Clay 0.3754 G Silt 0.1493 G OM 0.0357 Eigenvalue: 0.6104 (68.0%)

0.4475 0.0846 0.3856 (30.4%)

0.2641 0.1750 (13.8%)

0.1785 0.2828 0.1461 (16. 3%)

0.6046 0

0.3896 0.2211 0.2876 (32.0%)

0.1331 0

The coregionalization matrices, the eigenvalues and the corresponding percentage of variance explained by each eigenvector for the three basic structures are reported.

in terms of mean experimental error and MSDR are reported in Table III. Having 0 for the mean experimental error and 1 for the MSDR, which are the optimal values for the two statistics, the multivariate model of spatial correlation was unbiased and reproduced the experimental variance adequately. The two ranges represent a double spatial variability structure, which is likely affected by two different factors: (1) a shorter scale related to local morphology and land use and (2) a longer scale related to the main soil forming factors. The coregionalization matrices, consisting of the sills of direct and cross-variograms, are reported in Table II for each basic structure together with the corresponding eigenvalues and the proportion of spatial variance at that scale explained by each of them. The first eigenvalue of the shorter-range component accounted for more than 80% of the variance, about 68% for the one of the longer-range component and 56% for the one of the nugget effect. Moreover, all the three variables were highly spatially correlated at any scale, because the first two eigenvectors explained more than 90% of the variance at that scale. Figures 4(a–d) and 5(a–d) present a way to treat the co-simulated images of the four variables jointly, by calculating the mean and the standard deviation of the 500 simulations at each grid node and then mapping the results for each variable. The mean maps (Figure 4) show the complexity in spatial distribution of the textural components. Sandy soils (Figure 4a) prevail on the intrusive rocks and along the north-east border of the area, characterized by littoral sand dunes and sandy sediments. High contents in silt (Figure 4b) are located in the south on the basalt plateau and in part of the metamorphic region in the north; low contents in silt are located along the north-east border of the area and in other spots along the intrusive central large outcrop. High contents in clay (Figure 4c) are mostly located in the south-east corner, on the basalt and limestone outcrops as well on the other limestone formations located in the north-western sector.

Table III. Results of cross-validation Variable

Mean error (%)

Mean squared deviation ratio (MSDR)

G Clay G Silt G OM

0.01124 0.00471 0.01256

1.05 0.98 1.10




206

Figure 4. Mean maps of simulations for sand (a), silt (b), clay (c) and organic matter (d).




207

Figure 4. (Continued).




208

Figure 5. Standard deviation maps of simulations for sand (a), silt (b), clay (c) and organic matter (d).




209

Figure 5. (Continued).



210


The organic matter shows a mostly random distribution; however, the higher contents are observed in clay soils and in areas characterized by thick vegetation cover. The maps of the standard deviation (Figure 5), obtained by post-processing of simulations, assess the uncertainties of non-gaussian variables and overcome the drawback of kriging variance of its independence from the actual sample values. From visual inspection of these maps, the uncertainty distributions of clay, silt and organic matter seem to depend not only on the spatial distribution of the samples but also on the size of the estimates. It is evident that the standard deviation tends to increase jointly with the size of the estimate. A quite different behaviour can be observed for the sand, due to its near-normal properties. As expected for a Gaussian distribution, the standard deviation is mostly related to the density of the sample data (Figure 1). The Monte Carlo simulation has shown how the previous uncertainties in input variables can affect the predictions of the K model. Figure 6 shows the maps of the mean and standard deviation of K, which were obtained in a similar manner to the maps of Figures 4 and 5. The predicted K ranges from near to 0 to 0.04 Mg h MJ- 1 mm- 1 and reveals a very complex spatial distribution, due to the several factors affecting soil erodibility. Areas with low K values were located on the limestone relief in the southern part, where several samples had high organic matter content and in the central parts of both the granite and metamorphic reliefs, where medium organic matter content and light texture coexist. On the contrary, the highest values were concentrated along the south-east border including part of the basalt plateau, the limestone colluvia and the coastal plains, characterized by heavier texture (relatively rich in clay). The remaining high values were located mostly on alluvial soils. The model should be locally calibrated for producing reliable regional maps of erosion hazard. Looking at the standard deviation map of K factor (Figure 6b), only a weak spatial pattern was distinguished. Errors were uncorrelated with the estimates of K, but were correlated with the density of sampling, thus, hinting properties of variance homoscedasticity. Several factors, contributed to spatial variation of K with none dominating. Owing to the extremely varying morphology and genesis of the subsoil, however, some spatial structure of K was expected. The randomness might be a consequence of the variability of the input variables and also of the poor performance of the Torri et al. model, which needs a local calibration of a regression. In addition, the model uses only texture and organic matter as regressors, whereas the complexity of the landscape under study would require a non-linear model of other variables. The objective, however, was not to validate the model or assess the errors associated with the model type and coefficients but rather to evaluate how the variability of inputs affects uncertainty of model prediction. By definition, a model is an approximation of reality and some models describe reality better than others. Therefore, the choice of model plays an important role in the prediction error. In this paper, however, it was assumed that an appropriate model was selected and that the model errors were associated only with the spatial variation of the input attributes. This approach has demonstrated that is possible to produce regional maps of erosion hazard, which are more useful than the simple extrapolations of the plot experiments. After local calibration and validation, the model provides a method for exploring the effectiveness of the measures designed to reduce erosion hazards or for comparing alternative policy scenarios. Of course, one must evaluate how well the model approximates reality, that is, model uncertainty. If the model has high uncertainty, a difference in model output may not indicate a real change and thus could be meaningless. Therefore, it is important to know model uncertainty for operational decision-making. In order to obtain realistic values of the model output uncertainty, when the model outputs are supposed to be spatially correlated, it is critically important to model and assess spatial correlations of input variables (Heuvelink and Pebesma, 1999). Ignoring spatial correlation between input variables, as in the traditional Monte Carlo approach, means modelling input variables as white noise. In this case, all uncertainty in K predictions might vanish after mapping with a dense point grid. The required density depends on the estimate precision level, and it is critical to model spatial correlation correctly to separate input error from model uncertainty. The two types of variation are quite different. Spatial variation refers to the deterministic variation of K for a single realization of the input parameters, whereas uncertainty refers to K distribution for a single point obtained from the ensemble of Monte Carlo simulations. Copyright # 2007 John Wiley & Sons, Ltd.



211

Figure 6. K mean (a) and standard deviation (b) maps calculated from joint simulations of sand, silt, clay and organic matter.




212

Figure 7. Probability map of K factor greater than 0.03 Mg h MJ1 mm1.

One approach to assess spatial correlations is to delineate all locations where erosion is greater than a critical threshold value. Such an approach requires estimation or simulation of K, but due to uncertainty, there is a risk of incorrectly declaring as save a location ‘at erosion hazard’ or, conversely ‘at erosion hazard’ a ‘save’ location. Therefore, it is critical to include uncertainty in the classification process. One method is to express the erosion hazard in probabilistic terms of exceeding a given level of erosion. In Figure 7, a map is shown where the threshold value was arbitrarily set equal to 0.03 Mg h MJ- 1 mm- 1. It is possible to delineate the areas within which the probability of exceeding the threshold value is greater than some probability level. The areas at risk of exceeding a probability of 0.7 are located mostly along the south-east edge and on the northern border of the study area (Figure 7). The main difficulty is to appropriately choose a probability threshold aimed at designing some remediation action. CONCLUSIONS The results of a spatial uncertainty analysis have shown that the prediction quality depends on the uncertainties of the data used in the analysis. Therefore map makers should convey the accuracy of the maps they produce (Heuvelink, 1998). A complete characterization of the accuracy of spatial data should also include the spatial correlation of the attributes used for estimation and stored in a GIS. In the past a single Root Mean Squared Error was considered as sufficient to assess spatial accuracy, but now it is no longer sufficient, and more information is needed to characterize the quality of a map. The proposed methodology is able to highlight areas characterized by higher soil erodibility and erodibility may be compared with other territorial or anthropic information such as crops, agricultural practices, etc. to identify potential erosion risk areas, in extensive contexts where the scarce data availability and the high heterogeneity in data density are important to account for uncertainty and its spatial distribution. Copyright # 2007 John Wiley & Sons, Ltd.



213

At the present time, there is reluctance to perform error recognition, quite probably because of the increased analysis required. Studying uncertainty, however, leads to increased understanding of the roles played by the different environmental parameters, and it helps to evaluate the relative costs and benefits of using different scenarios. The complexity of the concepts and the software tools involved by the proposed approach is greater than for the traditional analysis and thus constitutes a practical difficulty. The introduction of such an approach in the public administration would require adequate skill and facilities. The current tendency is to use more and more ready to use, user-friendly spatial prediction software, which could introduce unpredictable errors when used to predict single variables that are then integrated into distributed deterministic models. acknowledgements The authors thank the reviewers of this paper for providing constructive comments which have contributed to improve the published version. references Agassi M. 1996. Soil Erosion, Conservation, and Rehabilitation. Marcel Dekker, Inc.: New York, NY. Burrough PA. 2001. GIS and geostatistics: Essential partners for spatial analysis. Environmental and Ecological Statistics 8: 361–377. Castrignano` A, Buttafuoco G. 2004. Geostatistical stochastic simulation of soil water content in a forested area of south Italy. Biosystems Engineering 87: 257–266. Chile`s JP, Delfiner P. 1999. Geostatistics: Modelling Spatial Uncertainty. Wiley: New York, NY. Christakos G. 1987. The space transformations in the simulation of multidimensional random fields. Journal of Mathematics & Computers in Simulation 29: 13–319. Christakos G. 1992. Random Field Models in Earth Sciences. Academic Press: San Diego, CA. Geovariances. 2006. ISATIS: Software Manual Release 6.0.6. Geovariances & Ecole des Mines de Paris: France. Gomez-Hernandez JJ, Journel AG. 1992. Joint Sequential Simulation of Multigaussian Fields. In Geostatistics Troia, 1992, Vol. I, Soares A (ed.). Kluwer Academic Publishers: Dordrecht; 85–94. Goovaerts P. 1997. Geostatistics for Natural Resources Evaluation. Oxford University Press: New York, NY. Heuvelink GBM. 1998. Error Propagation in Environmental Modelling With GIS. Taylor, Francis: London. Heuvelink GBM, Burrough PA, Stein A. 1989. Propagation of errors in spatial modelling with GIS. International Journal of Geographical Information Systems 3: 303–322. Heuvelink GBM, Pebesma EJ. 1999. Spatial aggregation and soil process modelling. Geoderma 89: 47–65. Jansen MJW. 1998. Prediction error through modelling concepts and uncertainty from basic data. Nutrient Cycling in Agroecosystems 50: 247–253. Journel AG. 1983. Non-parametric estimation of spatial distributions. Mathematical Geology 15: 445–468. Journel AG, Huijbregts CJ. 1978. Mining Geostatistics. Academic Press: New York, NY. Lewis PAW, Orav EJ. 1989. Simulation Methodology for Statisticians, Operations Analysts, and Engineers, Vol. 1. Wadsworth Publ. Co.: Belmont, CA. Morgan RPC. 1995. Soil Erosion and Conservation, 2nd ed. John Wiley & Sons: New York, NY. Renard KG, Foster GR, Weesies GA, Porter JP. 1991. Rusle—revised universal soil loss equation. Journal of Soil and Water Conservation 46: 30–33. Shirazi MA, Boersma L, Hart JW. 1988. A unifying quantitative analysis of soil texture: Improvement of precision and extension of scale. Soil Science Society of America Journal 52: 181–190. Torri D, Poesen J, Borselli L. 1997. Predictability and uncertainty of the soil erodibility factor using a global dataset. Catena 31: 1–22. Torri D, Poesen J, Borselli L. 2002. Corrigendum to ‘Predictability and uncertainty of the soil erodibility factor using a global dataset’ [Catena 31 (1997) 1–22] and to ‘‘Erratum to Predictability and uncertainty of the soil erodibility factor using a global dataset’’ [Catena 32 (1998) 307–308]. Catena 46: 309–310. Wischmeier WH, Smith DD. 1978. Predicting rainfall erosion losses. A guide to Conservation Planning. U.S. Department of Agriculture, Agriculture Handbook No 537, 58 pp.

