Proposal and validation of geostatistical-based metrics to quantify within-field variability

Seidel, Enio Júnior; Oliveira, Marcelo Silva de

doi:10.36783/18069657rbcs20250042

Division - Soil In Space and Time | Commission - Pedometrics • Rev. Bras. Ciênc. Solo 49 • 2025 • https://doi.org/10.36783/18069657rbcs20250042 linkcopy

Proposal and validation of geostatistical-based metrics to quantify within-field variability

Authorship SCIMAGO INSTITUTIONS RANKINGS

Metrics are fundamental to quantify and classify the spatial dependence of soil and agricultural attributes. This study aimed to propose and validate metrics based on two distinct approaches, one additive, which considers the arithmetic mean of the vertical and horizontal components, and the other multiplicative, which considers the geometric mean of the vertical and horizontal components of the semivariogram. Furthermore, we intend to propose the classification of spatial dependence based on the categorization of these metrics. Finally, a function in R language is presented to calculate the metrics and classify spatial dependence. The spatial dependence arithmetic index 1 (SDAI1) and spatial dependence arithmetic index 2 (SDAI2) are constructed in a dimensionless way, in the range between 0 and 100 %, considering the sum (arithmetic mean) between the vertical and horizontal components of the semivariogram. The spatial dependence geometric index 1 (SDGI1) and the spatial dependence geometric index 2 (SDGI2) are constructed in a dimensionless way, in the range between 0 and 100 %, considering the multiplication (geometric mean) between the vertical and horizontal components of the semivariogram. The SDAI1, SDAI2, SDGI1, and SDGI2 metrics are compared with other metrics existing in the literature, such as the spatial dependence degree (SPD), the integral scales J1 and J2, the mean correlation distance (MCD), the spatial dependence index (SDI), and the spatial dependence measure (SDM). For different spatial dependence scenarios, correlations are calculated between the geostatistical-based metrics and the performance measures Moran’s I, mean squared error (MSE), and kriging variance (KV). The metrics perform well in describing spatial dependence, with the exception of J1 (or MCD) and J2. However, the SDAI1, SDAI2, and SDGI1 metrics have slightly better correlations with the Moran’s I, MSE, and KV measures, when compared to the SDI, SDM, SDGI2, and SPD metrics. Furthermore, the SDAI1 and SDAI2 metrics show superior performance in capturing the vertical and horizontal effects of the semivariogram. Finally, a function in R language was developed to calculate the metrics and classify spatial dependence.

Keywords
precision agriculture; spatial variability; autocorrelation; semivariograms

Metric	Classification
Metric	Weak	Moderate	Strong
SDAI1	0 to 38	38 to 68	68 to 100
SDAI2	0 to 48	48 to 75	75 to 100
SDGI1	0 to 30	30 to 64	64 to 100
SDGI2	0 to 40	40 to 73	73 to 100

X	SDI	SDM	SDAI1	SDAI2	SDGI1	SDGI2	J1 (or MCD)	J2	SPD
Exponential semivariogram model
SDM	0.985
SDAI1	0.906	0.923
SDAI2	0.909	0.942	0.992
SDGI1	0.966	0.987	0.963	0.976
SDGI2	0.945	0.984	0.943	0.969	0.991
J1 (or MCD)	0.257	0.217	0.004	0.015	0.095	0.126
J2	0.257	0.217	0.003	0.015	0.095	0.126	0.999
SPD	0.332	0.288	0.603	0.525	0.407	0.318	-0.444	-0.444
$	0.332	0.288	0.603	0.525	0.407	0.318	-0.444	-0.444	1.000
£	0.261	0.239	0.564	0.503	0.371	0.291	-0.541	-0.541	0.980
#	0.921	0.969	0.870	0.908	0.945	0.975	0.279	0.279	0.131
I Moran	0.681	0.698	0.793	0.796	0.781	0.732	-0.379	-0.379	0.654
MSE	-0.672	-0.673	-0.770	-0.764	-0.754	-0.694	0.377	0.378	-0.683
KV	-0.709	-0.707	-0.789	-0.784	-0.780	-0.720	0.336	0.336	-0.667
Gaussian semivariogram model
SDM	0.982
SDAI1	0.942	0.938
SDAI2	0.946	0.961	0.992
SDGI1	0.985	0.986	0.968	0.979
SDGI2	0.958	0.992	0.940	0.970	0.981
J1 (or MCD)	0.016	0.089	0.014	0.049	0.013	0.112
J2	0.016	0.091	0.015	0.050	0.015	0.114	0.999
SPD	0.474	0.358	0.619	0.529	0.495	0.341	-0.359	-0.361
$	0.474	0.358	0.619	0.529	0.495	0.341	-0.359	-0.361	1.000
£	0.437	0.329	0.588	0.508	0.472	0.320	-0.397	-0.399	0.989
#	0.864	0.939	0.837	0.889	0.883	0.954	0.268	0.270	0.087
I Moran	0.908	0.849	0.873	0.857	0.911	0.825	-0.193	-0.193	0.684
MSE	-0.838	-0.776	-0.822	-0.802	-0.860	-0.765	0.245	0.246	-0.713
KV	-0.883	-0.822	-0.856	-0.838	-0.894	-0.804	0.221	0.221	-0.702
Spherical semivariogram model
SDM	0.992
SDAI1	0.940	0.929
SDAI2	0.955	0.954	0.994
SDGI1	0.989	0.983	0.970	0.983
SDGI2	0.983	0.996	0.941	0.967	0.986
J1 (or MCD)	0.436	0.510	0.362	0.411	0.388	0.386
J2	0.435	0.508	0.361	0.410	0.510	0.509	0.999
SPD	0.416	0.341	0.642	0.572	0.496	0.364	-0.327	-0.328
$	0.416	0.341	0.642	0.572	0.496	0.364	-0.327	-0.328	1.000
£	0.388	0.318	0.615	0.553	0.478	0.344	-0.369	-0.371	0.989
#	0.925	0.963	0.845	0.887	0.908	0.963	0.696	0.695	0.133
I Moran	0.880	0.838	0.840	0.838	0.893	0.827	-0.024	-0.026	0.598
MSE	-0.825	-0.774	-0.796	-0.787	-0.845	-0.767	0.129	0.131	-0.634
KV	-0.864	-0.816	-0.817	-0.813	-0.989	-0.803	0.058	0.060	-0.595

Data	Partial sill	Sill	Range	MD
soilmoisture	1.62	4.12	78.83	401.62
NVDI	0.0036	0.0065	7.45	24.19
Data	SDAI1	SDAI2	SDGI1	SDGI2
soilmoisture	39.30 (Moderate)	50.99 (Moderate)	39.30 (Moderate)	49.62 (Moderate)
NVDI	58.59 (Moderate)	68.09 (Moderate)	58.51 (Moderate)	67.78 (Moderate)

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Sociedade Brasileira de Ciência do Solo Sociedade Brasileira de Ciência do Solo, Departamento de Solos - Edifício Silvio Brandão, s/n, Caixa Postal 231 - Campus da UFV, CEP 36570-900 - Viçosa-MG, Tel.: (31) 3612-4542 - Viçosa - MG - Brazil
E-mail: sbcs@sbcs.org.br

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[1] MD: maximum distance; SDAI1: spatial dependence arithmetic index 1; SDAI2: spatial dependence arithmetic index 2; SDGI1: spatial dependence geometric index 1; SDGI2: spatial dependence geometric index 2.