196:
5282:
2729:
3273:: 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.
4988:
1082:
4145:
2086:
3943:
199:
Schematisation of a variogram. The points represent the measured data points (observed) and the curve represents the model function used (empirical). Range stands for the range sought, sill for the plateau value reached at maximum range, nugget for the nugget
5145:
3012:
1582:
2430:
208:, a variogram will give a measure of how much two samples taken from the mining area will vary in gold percentage depending on the distance between those samples. Samples taken far apart will vary more than samples taken close to each other.
2558:
5393:
and hence not directly usable in kriging, without constraints or further processing. This explains why only a limited number of variogram models are used: most commonly, the linear, the spherical, the
Gaussian, and the exponential models.
4619:
need valid semivariograms. In applied geostatistics the empirical variograms are thus often approximated by model function ensuring validity (Chiles&Delfiner 1999). Some important models are (Chiles&Delfiner 1999, Cressie 1993):
4769:
1794:
3175:
828:
4758:
3991:
1357:
5840:
452:
1928:
5689:
1936:
5522:
92:
3306:
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
190:
5602:
3640:
2154:
3778:
1713:
1657:
1477:
3770:
3076:
2542:
1427:
3386:
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
4581:
4503:
4399:
3524:
6086:"Modelling the spatial correlation of earthquake ground motion: Insights from the literature, data from the 2016โ2017 Central Italy earthquake sequence and ground-motion simulations"
2257:
4999:
2840:
3590:
2847:
2303:
2203:
3344:
4318:
4273:
4190:
1839:
are typically not written. The terms are used for all three forms of the function. Moreover, the term "variogram" is sometimes used to denote the semivariogram, and the symbol
1485:
1219:
1190:
1134:
738:
709:
313:
284:
2314:
134:
4425:
3686:
3406:
2724:{\displaystyle 2\gamma (\mathbf {s} _{1},\mathbf {s} _{2})=C(\mathbf {s} _{1},\mathbf {s} _{1})+C(\mathbf {s} _{2},\mathbf {s} _{2})-2C(\mathbf {s} _{1},\mathbf {s} _{2})}
780:
669:
255:
3450:
5240:
2794:
3986:
3706:
1857:
5266:
2768:
2469:
5204:
1161:
1105:
573:
524:
4216:
640:
6262:
5170:
4609:
3966:
3726:
3660:
3544:
3470:
3384:
3364:
3304:
3271:
3246:
3221:
1837:
1817:
820:
800:
758:
617:
593:
544:
495:
475:
333:
6123:
Sokolov, Vladimir; Wenzel, Friedemann (2011-07-25). "Influence of spatial correlation of strong ground motion on uncertainty in earthquake loss estimation".
4983:{\displaystyle \gamma (h)=(s-n)\left(\left({\frac {3h}{2r}}-{\frac {h^{3}}{2r^{3}}}\right)1_{(0,r)}(h)+1_{[r,\infty )}(h)\right)+n1_{(0,\infty )}(h).}
5419:
3184:
As a consequence the semivariogram might be non continuous only at the origin. The height of the jump at the origin is sometimes referred to as
1721:
1077:{\displaystyle 2\gamma (\mathbf {s} _{1},\mathbf {s} _{2})={\text{var}}\left(Z(\mathbf {s} _{1})-Z(\mathbf {s} _{2})\right)=E\left)^{2}\right].}
4140:{\displaystyle N(h\pm \delta )\equiv \{(\mathbf {s} _{i},\mathbf {s} _{j}):|\mathbf {s} _{i},\mathbf {s} _{j}|=h\pm \delta ;i,j=1,\ldots ,N\}}
3096:
4629:
1227:
5713:
2081:{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{1})=\gamma _{i}(0)=E\left((Z(\mathbf {s} _{1})-Z(\mathbf {s} _{1}))^{2}\right)=0}
341:
1874:
6225:
Glass, H.J., 2003, Method for assessing quality of the variogram, The
Journal of The South African Institute of Mining and Metallurgy
5615:
6159:
5329:
5452:
41:
3938:{\displaystyle {\hat {\gamma }}(h\pm \delta ):={\frac {1}{2|N(h\pm \delta )|}}\sum _{(i,j)\in N(h\pm \delta )}|z_{i}-z_{j}|^{2}}
1867:
According to (Cressie 1993, Chiles and
Delfiner 1999, Wackernagel 2003) the theoretical variogram has the following properties:
142:
5535:
3595:
2091:
5307:
1662:
5946:
Bachmaier, Martin; Backes, Matthias (2008-02-24). "Variogram or semivariogram? Understanding the variances in a variogram".
1594:
1432:
671:, all pairs of points at that exact distance would be sampled. In practice it is impossible to sample everywhere, so the
5846:
in which a variogram is of order 2, a madogram is a variogram of order 1, and a rodogram is a variogram of order 0.5.
4611:
and due to variation in the estimation it is not ensured that it is a valid variogram, as defined above. However some
3731:
3020:
2474:
2159:
A function is a semivariogram if and only if it is a conditionally negative definite function, i.e. for all weights
6267:
5303:
1369:
6045:"Characterisation of the spatial variability of material properties of Gilsocarbon and NBG-18 using random fields"
4508:
4430:
4326:
5292:
3478:
5140:{\displaystyle \gamma (h)=(s-n)\left(1-\exp \left(-{\frac {h^{2}}{r^{2}a}}\right)\right)+n1_{(0,\infty )}(h).}
5984:
Nguyen, H.; Osterman, G.; Wunch, D.; O'Dell, C.; Mandrake, L.; Wennberg, P.; Fisher, B.; Castano, R. (2014).
5311:
5296:
4323:
For computational speed, only the unique pairs of points are needed. For example, for 2 observations pairs [
3007:{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{2})=E\left=\gamma (\mathbf {s} _{2},\mathbf {s} _{1})}
2208:
4218:, the square of the difference in the observation (e.g., soil sample content or pixel intensity) is found (
6257:
2799:
1577:{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{2})=\gamma _{s}(\mathbf {s} _{2}-\mathbf {s} _{1}).}
596:
257:
was first defined by
Matheron (1963) as half the average squared difference between the values at points (
6242:
3549:
2262:
2162:
5172:
has different values in different references, due to the ambiguity in the definition of the range. E.g.
3309:
2425:{\displaystyle \sum _{i=1}^{N}\sum _{j=1}^{N}w_{i}\gamma (\mathbf {s} _{i},\mathbf {s} _{j})w_{j}\leq 0}
4278:
4221:
4150:
1195:
1166:
1110:
714:
685:
289:
260:
6097:
6056:
6007:
5429:
5365:
The variogram is the key function in geostatistics as it will be used to fit a model of the temporal/
3452:. For the case where dimensions have different units (e.g., distance and time) then a scaling factor
108:
6181:
Chiles, J. P., P. Delfiner, 1999, Geostatistics, Modelling
Spatial Uncertainty, Wiley-Interscience.
5529:
5366:
3728:
and does not depend on other variables such as center position). Then, the empirical semivariogram
2550:
4404:
3665:
3389:
1107:, this is equivalent to the expectation for the squared increment of the values between locations
763:
3086:
1363:
103:
95:
31:
645:
231:
5425:
Empirical variograms were calculated for the density of a heterogeneous material (Gilsocarbon).
3688:
are used rather than exact distances, and usually isotropic conditions are assumed (i.e., that
3411:
6155:
6025:
5963:
5899:
5696:
5390:
5209:
2773:
3971:
3691:
1842:
6132:
6105:
6064:
6015:
5955:
5891:
5386:
5245:
3177:
corresponds to the variance of the field. The limit of the semivariogram is also called its
2738:
2439:
6224:
6187:
Burrough, P. A. and McDonnell, R. A., 1998, Principles of
Geographical Information Systems.
5175:
1139:
5857:. Should the variable be binary or represent classes of values, one is then talking about
3642:. Generally, plots show the semivariogram values as a function of sample point separation
1090:
549:
500:
5918:
5849:
When a variogram is used to describe the correlation of different variables it is called
4195:
6101:
6060:
6011:
622:
546:
is the separation distance (e.g., in meters or km) of interest. For example, the value
5422:
was used to determine coincidence criteria for satellite and ground-based measurements.
5155:
4594:
3951:
3711:
3645:
3529:
3455:
3369:
3349:
3289:
3256:
3231:
3206:
1822:
1802:
805:
785:
743:
602:
578:
529:
480:
460:
318:
5346:
for describing the spatial or the temporal correlation of observations: these are the
6251:
6044:
5407:
5343:
4612:
4320:. By definition the result is divided by 2 for the semivariogram at this separation.
4275:). These squared differences are added together and normalized by the natural number
3079:
6109:
5437:
195:
99:
6069:
678:
The variogram is twice the semivariogram and can be defined, equivalently, as the
3223:: The height of the jump of the semivariogram at the discontinuity at the origin.
5895:
5609:
5347:
5281:
205:
6238:
AI-GEOSTATS: an educational resource about geostatistics and spatial statistics
5410:
as a first estimate of the variogram model needed for spatial interpolation by
5381:
function. Note that the experimental variogram is an empirical estimate of the
6190:
5959:
5433:
5382:
5351:
6243:
Geostatistics: Lecture by Rudolf Dutter at the
Technical University of Vienna
6029:
5967:
5903:
6020:
5985:
5692:
3197:
In summary, the following parameters are often used to describe variograms:
17:
6085:
5418:
Empirical variograms for the spatiotemporal variability of column-averaged
5373:
that is a visualization of a possible spatial/temporal correlation and the
1789:{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{2})=\gamma _{i}(h).}
5369:
of the observed phenomenon. One is thus making a distinction between the
1588:
679:
217:
1591:, then the variogram and semivariogram can be represented by a function
6191:
Isobel Clark, 1979, Practical
Geostatistics, Applied Science Publishers
5862:
5854:
5700:
5411:
5378:
4616:
3170:{\displaystyle \lim _{h\to \infty }\gamma _{s}(h)=var(Z(\mathbf {s} ))}
3090:
5996:
data to ground-based data and its application to ACOS-GOSAT and TCCON"
4753:{\displaystyle \gamma (h)=(s-n)(1-\exp(-h/(ra)))+n1_{(0,\infty )}(h).}
526:
is the value at that point. The triple integral is over 3 dimensions.
6136:
216:"Semivariance" redirects here. For the measure of downside risk, see
2544:
is given by the negative of this double sum and must be nonnegative.
6178:
Cressie, N., 1993, Statistics for spatial data, Wiley
Interscience.
3662:. In the case of empirical semivariogram, separation distance bins
1366:, the variogram and semivariogram can be represented as a function
1352:{\displaystyle 2\gamma (\mathbf {s} _{1},\mathbf {s} _{2})=E\left.}
5835:{\displaystyle 2\gamma (\mathbf {s} _{1},\mathbf {s} _{2})=E\left}
1859:
is sometimes used for the variogram, which brings some confusion.
1479:
between locations only, by the following relation (Cressie 1993):
194:
4591:
The empirical variogram cannot be computed at every lag distance
447:{\displaystyle \gamma (h)={\frac {1}{2V}}\iiint _{V}\left^{2}dV,}
3546:
total different locations. This would provide as set of samples
3472:
can be applied to each to obtain a modified
Euclidean distance.
1923:{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{2})\geq 0}
5684:{\displaystyle |Z(\mathbf {s} _{1})-Z(\mathbf {s} _{2})|^{0.5}}
6237:
5275:
5517:{\displaystyle (Z(\mathbf {s} _{1})-Z(\mathbf {s} _{2}))^{2}}
5440:
and loss assessments of spatially-distributed infrastructure.
2735:
If a stationary random field has no spatial dependence (i.e.
2553:
of a stationary process exists, it is related to variogram by
6184:
Wackernagel, H., 2003, Multivariate Geostatistics, Springer.
3248:: Limit of the variogram tending to infinity lag distances.
575:
could represent the iron content in soil, at some location
87:{\displaystyle 2\gamma (\mathbf {s} _{1},\mathbf {s} _{2})}
185:{\displaystyle \gamma (\mathbf {s} _{1},\mathbf {s} _{2})}
5882:
Matheron, Georges (1963). "Principles of geostatistics".
5597:{\displaystyle |Z(\mathbf {s} _{1})-Z(\mathbf {s} _{2})|}
5428:
Empirical variograms are calculated from observations of
3635:{\displaystyle \mathbf {s} _{1},\ldots ,\mathbf {s} _{k}}
2149:{\displaystyle Z(\mathbf {s} _{1})-Z(\mathbf {s} _{1})=0}
682:
of the difference between field values at two locations (
3286:
is needed for measured data, because sample information
599:
of latitude, longitude, and elevation) over some region
5954:(3). Springer Science and Business Media LLC: 173โ175.
1708:{\displaystyle h=\|\mathbf {s} _{2}-\mathbf {s} _{1}\|}
5716:
5618:
5538:
5455:
5248:
5212:
5206:
is the value used in (Chiles&Delfiner 1999). The
5178:
5158:
5002:
4772:
4632:
4597:
4511:
4433:
4407:
4329:
4281:
4224:
4198:
4153:
3994:
3974:
3954:
3781:
3734:
3714:
3694:
3668:
3648:
3598:
3552:
3532:
3481:
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3414:
3392:
3372:
3352:
3312:
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3259:
3234:
3209:
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2850:
2802:
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2477:
2442:
2317:
2265:
2211:
2165:
2094:
1939:
1877:
1845:
1825:
1805:
1724:
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1488:
1435:
1372:
1230:
1198:
1169:
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1113:
1093:
831:
808:
788:
766:
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581:
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532:
503:
483:
463:
344:
321:
292:
263:
234:
145:
111:
44:
3948:
Or in other words, each pair of points separated by
204:
In the case of a concrete example from the field of
6152:
Geostatistical Glossary and Multilingual Dictionary
6084:Schiappapietra, Erika; Douglas, John (April 2020).
5703:. They can be generalized as a "variogram of order
1652:{\displaystyle \gamma _{i}(h):=\gamma _{s}(he_{1})}
1472:{\displaystyle h=\mathbf {s} _{2}-\mathbf {s} _{1}}
822:) across realizations of the field (Cressie 1993):
5834:
5683:
5596:
5516:
5377:that is further used to define the weights of the
5260:
5234:
5198:
5164:
5139:
4982:
4752:
4603:
4575:
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4419:
4393:
4312:
4267:
4210:
4184:
4139:
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3700:
3680:
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3400:
3378:
3358:
3338:
3298:
3265:
3240:
3215:
3169:
3070:
3006:
2842:everywhere except at the origin, where it is zero.
2834:
2788:
2762:
2723:
2536:
2463:
2424:
2297:
2251:
2197:
2148:
2080:
1922:
1851:
1831:
1811:
1788:
1707:
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1471:
1421:
1351:
1213:
1184:
1155:
1128:
1099:
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814:
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663:
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327:
307:
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249:
184:
128:
86:
6212:Statistical Theory with Engineering Applications
6154:. Oxford University Press. pp. 47, 67, 81.
6125:Earthquake Engineering & Structural Dynamics
5695:based on these lower powers are said to be more
5449:The squared term in the variogram, for instance
3101:
2436:which corresponds to the fact that the variance
3968:(plus or minus some bin width tolerance range
3765:{\displaystyle {\hat {\gamma }}(h\pm \delta )}
3071:{\displaystyle \gamma _{s}(h)=\gamma _{s}(-h)}
2537:{\displaystyle X=\sum _{i=1}^{N}w_{i}Z(x_{i})}
1087:If the spatial random field has constant mean
4583:] do not provide any additional information.
1422:{\displaystyle \gamma _{s}(h)=\gamma (0,0+h)}
8:
6217:Journel, A. G. and Huijbregts, Ch. J., 1978
5979:
5977:
4147:. The number of these points in this bin is
4134:
4016:
1702:
1672:
5524:, can be replaced with different powers: A
5310:. Unsourced material may be challenged and
4576:{\displaystyle (z_{b},z_{a}),(z_{d},z_{c})}
4498:{\displaystyle (z_{a},z_{b}),(z_{c},z_{d})}
4394:{\displaystyle (z_{a},z_{b}),(z_{c},z_{d})}
3988:) are found. These form the set of points
1930:, since it is the expectation of a square.
642:. To obtain the semivariogram for a given
6068:
6019:
5822:
5808:
5803:
5784:
5779:
5747:
5742:
5732:
5727:
5715:
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5330:Learn how and when to remove this message
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3519:{\displaystyle Z(\mathbf {s} _{i})=z_{i}}
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110:
94:, is a function describing the degree of
75:
70:
60:
55:
43:
6043:Arregui Mena, J.D.; et al. (2018).
1221:are points in space and possibly time):
672:
5874:
4505:] need to be considered, as the pairs [
4401:] taken from locations with separation
5358:. The last is also more simply called
2252:{\displaystyle \sum _{i=1}^{N}w_{i}=0}
6205:Geostatistical Ore Reserve Estimation
2796:), the semivariogram is the constant
7:
6263:Statistical deviation and dispersion
5308:adding citations to reliable sources
2835:{\displaystyle var(Z(\mathbf {s} ))}
5986:"A method for colocating satellite
5861:. Indicator variograms are used in
3585:{\displaystyle z_{1},\ldots ,z_{k}}
2298:{\displaystyle s_{1},\ldots ,s_{N}}
2198:{\displaystyle w_{1},\ldots ,w_{N}}
6214:, John Wiley & Sons, New York.
6000:Atmospheric Measurement Techniques
5117:
4960:
4918:
4730:
3339:{\displaystyle \mathbf {s} =(x,y)}
3111:
2088:at distance 0 is always 0, since
477:is a point in the geometric field
25:
4313:{\displaystyle |N(h\pm \delta )|}
4268:{\displaystyle |z_{i}-z_{j}|^{2}}
4185:{\displaystyle |N(h\pm \delta )|}
1871:The semivariogram is nonnegative
5804:
5780:
5743:
5728:
5656:
5632:
5576:
5552:
5491:
5467:
5280:
4624:The exponential variogram model
4077:
4062:
4039:
4024:
3772:can be calculated for each bin:
3622:
3601:
3490:
3475:Sample observations are denoted
3408:could be a 4 dimensional vector
3394:
3314:
3157:
2991:
2976:
2935:
2911:
2874:
2859:
2822:
2708:
2693:
2666:
2651:
2627:
2612:
2588:
2573:
2393:
2378:
2127:
2103:
2044:
2020:
1963:
1948:
1901:
1886:
1748:
1733:
1692:
1677:
1558:
1543:
1512:
1497:
1459:
1444:
1318:
1294:
1257:
1242:
1214:{\displaystyle \mathbf {s} _{2}}
1201:
1185:{\displaystyle \mathbf {s} _{1}}
1172:
1129:{\displaystyle \mathbf {s} _{1}}
1116:
1040:
1016:
983:
959:
916:
892:
858:
843:
768:
733:{\displaystyle \mathbf {s} _{2}}
720:
704:{\displaystyle \mathbf {s} _{1}}
691:
308:{\displaystyle \mathbf {s} _{2}}
295:
279:{\displaystyle \mathbf {s} _{1}}
266:
169:
154:
119:
71:
56:
6110:10.1016/j.earscirev.2020.103139
5853:. Cross-variograms are used in
4192:. Then for each pair of points
740:, note change of notation from
129:{\displaystyle Z(\mathbf {s} )}
5814:
5799:
5790:
5775:
5753:
5723:
5671:
5666:
5651:
5642:
5627:
5620:
5590:
5586:
5571:
5562:
5547:
5540:
5505:
5501:
5486:
5477:
5462:
5456:
5229:
5223:
5131:
5125:
5120:
5108:
5030:
5018:
5012:
5006:
4974:
4968:
4963:
4951:
4932:
4926:
4921:
4909:
4898:
4892:
4887:
4875:
4800:
4788:
4782:
4776:
4764:The spherical variogram model
4744:
4738:
4733:
4721:
4707:
4704:
4701:
4692:
4678:
4663:
4660:
4648:
4642:
4636:
4570:
4544:
4538:
4512:
4492:
4466:
4460:
4434:
4388:
4362:
4356:
4330:
4306:
4302:
4290:
4283:
4255:
4226:
4178:
4174:
4162:
4155:
4088:
4056:
4049:
4019:
4010:
3998:
3925:
3896:
3890:
3878:
3869:
3857:
3845:
3841:
3829:
3822:
3806:
3794:
3788:
3759:
3747:
3741:
3500:
3485:
3439:
3415:
3333:
3321:
3164:
3161:
3153:
3147:
3132:
3126:
3108:
3065:
3056:
3040:
3034:
3001:
2971:
2950:
2945:
2930:
2921:
2906:
2899:
2884:
2854:
2829:
2826:
2818:
2812:
2751:
2745:
2718:
2688:
2676:
2646:
2637:
2607:
2598:
2568:
2531:
2518:
2458:
2452:
2403:
2373:
2137:
2122:
2113:
2098:
2058:
2054:
2039:
2030:
2015:
2009:
1995:
1989:
1973:
1943:
1911:
1881:
1780:
1774:
1758:
1728:
1646:
1630:
1614:
1608:
1587:If the process is furthermore
1568:
1538:
1522:
1492:
1416:
1398:
1389:
1383:
1328:
1313:
1304:
1289:
1267:
1237:
1057:
1053:
1050:
1035:
1026:
1011:
1005:
996:
993:
978:
969:
954:
948:
945:
926:
911:
902:
887:
868:
838:
658:
652:
562:
556:
513:
507:
421:
415:
406:
394:
354:
348:
244:
238:
179:
149:
123:
115:
81:
51:
1:
6200:, Applied Science Publishers.
6070:10.1016/j.jnucmat.2018.09.008
5926:faculty.washington.edu/edford
4994:The Gaussian variogram model
6049:Journal of Nuclear Materials
5612:of the absolute difference,
5436:. These models are used for
5342:Three functions are used in
4420:{\displaystyle h\pm \delta }
3681:{\displaystyle h\pm \delta }
3401:{\displaystyle \mathbf {s} }
3346:for a 2D sample space where
775:{\displaystyle \mathbf {s} }
218:Variance ยง Semivariance
5896:10.2113/gsecongeo.58.8.1246
27:Spatial statistics function
6284:
3526:. Samples may be taken at
1163:(Wackernagel 2003) (where
664:{\displaystyle \gamma (h)}
250:{\displaystyle \gamma (h)}
215:
6150:Olea, Ricardo A. (1991).
5960:10.1007/s11119-008-9056-2
5919:"The Empirical Variogram"
5389:. As such, it may not be
3445:{\displaystyle (x,y,z,t)}
5235:{\displaystyle 1_{A}(h)}
3014:is a symmetric function.
2789:{\displaystyle h\not =0}
315:) separated at distance
192:is half the variogram.
6198:Practical Geostatistics
6021:10.5194/amt-7-2631-2014
3981:{\displaystyle \delta }
3701:{\displaystyle \gamma }
3085:If the random field is
1852:{\displaystyle \gamma }
619:with element of volume
6207:, Elsevier Publishing.
5836:
5685:
5598:
5518:
5371:experimental variogram
5262:
5261:{\displaystyle h\in A}
5236:
5200:
5166:
5141:
4984:
4754:
4605:
4577:
4499:
4421:
4395:
4314:
4269:
4212:
4186:
4141:
3982:
3962:
3939:
3766:
3722:
3708:is only a function of
3702:
3682:
3656:
3636:
3586:
3540:
3520:
3466:
3446:
3402:
3380:
3360:
3340:
3300:
3267:
3242:
3217:
3171:
3072:
3008:
2836:
2790:
2764:
2763:{\displaystyle C(h)=0}
2732:
2725:
2538:
2504:
2465:
2464:{\displaystyle var(X)}
2426:
2359:
2338:
2299:
2253:
2232:
2199:
2150:
2082:
1924:
1853:
1833:
1813:
1790:
1709:
1653:
1578:
1473:
1423:
1353:
1215:
1186:
1157:
1130:
1101:
1078:
816:
796:
776:
754:
734:
705:
665:
636:
613:
597:geographic coordinates
589:
569:
540:
520:
491:
471:
448:
329:
309:
280:
251:
201:
186:
130:
88:
6090:Earth-Science Reviews
5948:Precision Agriculture
5837:
5686:
5599:
5519:
5263:
5237:
5201:
5199:{\displaystyle a=1/3}
5167:
5142:
4985:
4755:
4606:
4578:
4500:
4422:
4396:
4315:
4270:
4213:
4187:
4142:
3983:
3963:
3940:
3767:
3723:
3703:
3683:
3657:
3637:
3587:
3541:
3521:
3467:
3447:
3403:
3381:
3361:
3341:
3301:
3268:
3243:
3218:
3172:
3073:
3009:
2837:
2791:
2765:
2726:
2554:
2539:
2484:
2466:
2427:
2339:
2318:
2300:
2254:
2212:
2200:
2151:
2083:
1925:
1854:
1834:
1814:
1791:
1715:only (Cressie 1993):
1710:
1654:
1579:
1474:
1424:
1354:
1216:
1187:
1158:
1156:{\displaystyle s_{2}}
1131:
1102:
1079:
817:
797:
777:
755:
735:
706:
666:
637:
614:
590:
570:
541:
521:
492:
472:
449:
330:
310:
281:
252:
198:
187:
131:
89:
6219:Mining Geostatistics
5859:indicator variograms
5714:
5616:
5608:is defined with the
5536:
5528:is defined with the
5453:
5430:strong ground motion
5304:improve this section
5246:
5210:
5176:
5156:
5000:
4770:
4630:
4595:
4509:
4431:
4405:
4327:
4279:
4222:
4196:
4151:
3992:
3972:
3952:
3779:
3732:
3712:
3692:
3666:
3646:
3596:
3550:
3530:
3479:
3456:
3412:
3390:
3370:
3350:
3310:
3290:
3257:
3232:
3207:
3097:
3021:
2848:
2800:
2774:
2739:
2559:
2475:
2440:
2315:
2263:
2209:
2163:
2092:
1937:
1875:
1843:
1823:
1803:
1722:
1663:
1595:
1486:
1433:
1370:
1228:
1196:
1167:
1140:
1111:
1100:{\displaystyle \mu }
1091:
829:
806:
786:
764:
744:
715:
686:
646:
623:
603:
579:
568:{\displaystyle f(M)}
550:
530:
519:{\displaystyle f(M)}
501:
481:
461:
342:
319:
290:
261:
232:
143:
109:
42:
6102:2020ESRv..20303139S
6061:2018JNuM..511...91A
6012:2014AMT.....7.2631N
5530:absolute difference
5404:empirical variogram
5367:spatial correlation
4211:{\displaystyle i,j}
3284:empirical variogram
3278:Empirical variogram
2551:covariance function
1933:The semivariogram
673:empirical variogram
5832:
5681:
5594:
5514:
5258:
5232:
5196:
5162:
5137:
4980:
4750:
4601:
4573:
4495:
4417:
4391:
4310:
4265:
4208:
4182:
4137:
3978:
3958:
3935:
3894:
3762:
3718:
3698:
3678:
3652:
3632:
3582:
3536:
3516:
3462:
3442:
3398:
3376:
3356:
3336:
3296:
3263:
3238:
3213:
3167:
3115:
3068:
3004:
2832:
2786:
2760:
2721:
2534:
2461:
2422:
2295:
2249:
2195:
2146:
2078:
1920:
1849:
1829:
1809:
1786:
1705:
1649:
1574:
1469:
1429:of the difference
1419:
1364:stationary process
1349:
1211:
1182:
1153:
1126:
1097:
1074:
812:
792:
772:
750:
730:
701:
661:
635:{\displaystyle dV}
632:
609:
585:
565:
536:
516:
487:
467:
444:
325:
305:
276:
247:
202:
182:
126:
104:stochastic process
96:spatial dependence
84:
32:spatial statistics
6268:Spatial processes
6221:, Academic Press.
6203:David, M., 1978,
6196:Clark, I., 1979,
5863:indicator kriging
5391:positive definite
5340:
5339:
5332:
5268:and 0 otherwise.
5242:function is 1 if
5165:{\displaystyle a}
5085:
4863:
4831:
4604:{\displaystyle h}
3961:{\displaystyle h}
3852:
3850:
3791:
3744:
3721:{\displaystyle h}
3655:{\displaystyle h}
3539:{\displaystyle k}
3465:{\displaystyle B}
3379:{\displaystyle y}
3359:{\displaystyle x}
3299:{\displaystyle Z}
3266:{\displaystyle r}
3241:{\displaystyle s}
3216:{\displaystyle n}
3188:or nugget effect.
3100:
1832:{\displaystyle s}
1812:{\displaystyle i}
1659:of the distance
1362:In the case of a
877:
815:{\displaystyle Z}
795:{\displaystyle f}
753:{\displaystyle M}
675:is used instead.
612:{\displaystyle V}
588:{\displaystyle M}
539:{\displaystyle h}
490:{\displaystyle V}
470:{\displaystyle M}
373:
328:{\displaystyle h}
16:(Redirected from
6275:
6210:Hald, A., 1952,
6166:
6165:
6147:
6141:
6140:
6137:10.1002/eqe.1074
6120:
6114:
6113:
6081:
6075:
6074:
6072:
6040:
6034:
6033:
6023:
6006:(8): 2631โ2644.
5981:
5972:
5971:
5943:
5937:
5936:
5934:
5932:
5923:
5914:
5908:
5907:
5890:(8): 1246โ1266.
5884:Economic Geology
5879:
5841:
5839:
5838:
5833:
5831:
5827:
5826:
5821:
5817:
5813:
5812:
5807:
5789:
5788:
5783:
5752:
5751:
5746:
5737:
5736:
5731:
5690:
5688:
5687:
5682:
5680:
5679:
5674:
5665:
5664:
5659:
5641:
5640:
5635:
5623:
5603:
5601:
5600:
5595:
5593:
5585:
5584:
5579:
5561:
5560:
5555:
5543:
5523:
5521:
5520:
5515:
5513:
5512:
5500:
5499:
5494:
5476:
5475:
5470:
5445:Related concepts
5387:Gaussian process
5335:
5328:
5324:
5321:
5315:
5284:
5276:
5267:
5265:
5264:
5259:
5241:
5239:
5238:
5233:
5222:
5221:
5205:
5203:
5202:
5197:
5192:
5171:
5169:
5168:
5163:
5146:
5144:
5143:
5138:
5124:
5123:
5096:
5092:
5091:
5087:
5086:
5084:
5080:
5079:
5069:
5068:
5059:
4989:
4987:
4986:
4981:
4967:
4966:
4939:
4935:
4925:
4924:
4891:
4890:
4869:
4865:
4864:
4862:
4861:
4860:
4847:
4846:
4837:
4832:
4830:
4822:
4814:
4759:
4757:
4756:
4751:
4737:
4736:
4691:
4615:methods such as
4610:
4608:
4607:
4602:
4587:Variogram models
4582:
4580:
4579:
4574:
4569:
4568:
4556:
4555:
4537:
4536:
4524:
4523:
4504:
4502:
4501:
4496:
4491:
4490:
4478:
4477:
4459:
4458:
4446:
4445:
4426:
4424:
4423:
4418:
4400:
4398:
4397:
4392:
4387:
4386:
4374:
4373:
4355:
4354:
4342:
4341:
4319:
4317:
4316:
4311:
4309:
4286:
4274:
4272:
4271:
4266:
4264:
4263:
4258:
4252:
4251:
4239:
4238:
4229:
4217:
4215:
4214:
4209:
4191:
4189:
4188:
4183:
4181:
4158:
4146:
4144:
4143:
4138:
4091:
4086:
4085:
4080:
4071:
4070:
4065:
4059:
4048:
4047:
4042:
4033:
4032:
4027:
3987:
3985:
3984:
3979:
3967:
3965:
3964:
3959:
3944:
3942:
3941:
3936:
3934:
3933:
3928:
3922:
3921:
3909:
3908:
3899:
3893:
3851:
3849:
3848:
3825:
3813:
3793:
3792:
3784:
3771:
3769:
3768:
3763:
3746:
3745:
3737:
3727:
3725:
3724:
3719:
3707:
3705:
3704:
3699:
3687:
3685:
3684:
3679:
3661:
3659:
3658:
3653:
3641:
3639:
3638:
3633:
3631:
3630:
3625:
3610:
3609:
3604:
3591:
3589:
3588:
3583:
3581:
3580:
3562:
3561:
3545:
3543:
3542:
3537:
3525:
3523:
3522:
3517:
3515:
3514:
3499:
3498:
3493:
3471:
3469:
3468:
3463:
3451:
3449:
3448:
3443:
3407:
3405:
3404:
3399:
3397:
3385:
3383:
3382:
3377:
3365:
3363:
3362:
3357:
3345:
3343:
3342:
3337:
3317:
3305:
3303:
3302:
3297:
3272:
3270:
3269:
3264:
3247:
3245:
3244:
3239:
3222:
3220:
3219:
3214:
3176:
3174:
3173:
3168:
3160:
3125:
3124:
3114:
3077:
3075:
3074:
3069:
3055:
3054:
3033:
3032:
3013:
3011:
3010:
3005:
3000:
2999:
2994:
2985:
2984:
2979:
2964:
2960:
2959:
2958:
2953:
2944:
2943:
2938:
2920:
2919:
2914:
2902:
2883:
2882:
2877:
2868:
2867:
2862:
2841:
2839:
2838:
2833:
2825:
2795:
2793:
2792:
2787:
2769:
2767:
2766:
2761:
2730:
2728:
2727:
2722:
2717:
2716:
2711:
2702:
2701:
2696:
2675:
2674:
2669:
2660:
2659:
2654:
2636:
2635:
2630:
2621:
2620:
2615:
2597:
2596:
2591:
2582:
2581:
2576:
2543:
2541:
2540:
2535:
2530:
2529:
2514:
2513:
2503:
2498:
2470:
2468:
2467:
2462:
2431:
2429:
2428:
2423:
2415:
2414:
2402:
2401:
2396:
2387:
2386:
2381:
2369:
2368:
2358:
2353:
2337:
2332:
2304:
2302:
2301:
2296:
2294:
2293:
2275:
2274:
2258:
2256:
2255:
2250:
2242:
2241:
2231:
2226:
2204:
2202:
2201:
2196:
2194:
2193:
2175:
2174:
2155:
2153:
2152:
2147:
2136:
2135:
2130:
2112:
2111:
2106:
2087:
2085:
2084:
2079:
2071:
2067:
2066:
2065:
2053:
2052:
2047:
2029:
2028:
2023:
1988:
1987:
1972:
1971:
1966:
1957:
1956:
1951:
1929:
1927:
1926:
1921:
1910:
1909:
1904:
1895:
1894:
1889:
1858:
1856:
1855:
1850:
1838:
1836:
1835:
1830:
1818:
1816:
1815:
1810:
1795:
1793:
1792:
1787:
1773:
1772:
1757:
1756:
1751:
1742:
1741:
1736:
1714:
1712:
1711:
1706:
1701:
1700:
1695:
1686:
1685:
1680:
1658:
1656:
1655:
1650:
1645:
1644:
1629:
1628:
1607:
1606:
1583:
1581:
1580:
1575:
1567:
1566:
1561:
1552:
1551:
1546:
1537:
1536:
1521:
1520:
1515:
1506:
1505:
1500:
1478:
1476:
1475:
1470:
1468:
1467:
1462:
1453:
1452:
1447:
1428:
1426:
1425:
1420:
1382:
1381:
1358:
1356:
1355:
1350:
1345:
1341:
1340:
1335:
1331:
1327:
1326:
1321:
1303:
1302:
1297:
1266:
1265:
1260:
1251:
1250:
1245:
1220:
1218:
1217:
1212:
1210:
1209:
1204:
1191:
1189:
1188:
1183:
1181:
1180:
1175:
1162:
1160:
1159:
1154:
1152:
1151:
1135:
1133:
1132:
1127:
1125:
1124:
1119:
1106:
1104:
1103:
1098:
1083:
1081:
1080:
1075:
1070:
1066:
1065:
1064:
1049:
1048:
1043:
1025:
1024:
1019:
992:
991:
986:
968:
967:
962:
933:
929:
925:
924:
919:
901:
900:
895:
878:
875:
867:
866:
861:
852:
851:
846:
821:
819:
818:
813:
801:
799:
798:
793:
781:
779:
778:
773:
771:
759:
757:
756:
751:
739:
737:
736:
731:
729:
728:
723:
710:
708:
707:
702:
700:
699:
694:
670:
668:
667:
662:
641:
639:
638:
633:
618:
616:
615:
610:
594:
592:
591:
586:
574:
572:
571:
566:
545:
543:
542:
537:
525:
523:
522:
517:
496:
494:
493:
488:
476:
474:
473:
468:
453:
451:
450:
445:
434:
433:
428:
424:
384:
383:
374:
372:
361:
334:
332:
331:
326:
314:
312:
311:
306:
304:
303:
298:
285:
283:
282:
277:
275:
274:
269:
256:
254:
253:
248:
191:
189:
188:
183:
178:
177:
172:
163:
162:
157:
135:
133:
132:
127:
122:
93:
91:
90:
85:
80:
79:
74:
65:
64:
59:
34:the theoretical
21:
6283:
6282:
6278:
6277:
6276:
6274:
6273:
6272:
6248:
6247:
6234:
6175:
6173:Further reading
6170:
6169:
6162:
6149:
6148:
6144:
6131:(9): 993โ1009.
6122:
6121:
6117:
6083:
6082:
6078:
6042:
6041:
6037:
5995:
5994:
5983:
5982:
5975:
5945:
5944:
5940:
5930:
5928:
5921:
5916:
5915:
5911:
5881:
5880:
5876:
5871:
5851:cross-variogram
5802:
5778:
5771:
5767:
5766:
5762:
5741:
5726:
5712:
5711:
5669:
5654:
5630:
5614:
5613:
5574:
5550:
5534:
5533:
5504:
5489:
5465:
5451:
5450:
5447:
5400:
5375:variogram model
5336:
5325:
5319:
5316:
5301:
5285:
5274:
5244:
5243:
5213:
5208:
5207:
5174:
5173:
5154:
5153:
5103:
5071:
5070:
5060:
5054:
5050:
5037:
5033:
4998:
4997:
4946:
4904:
4870:
4852:
4848:
4838:
4823:
4815:
4812:
4808:
4807:
4803:
4768:
4767:
4716:
4628:
4627:
4593:
4592:
4589:
4560:
4547:
4528:
4515:
4507:
4506:
4482:
4469:
4450:
4437:
4429:
4428:
4403:
4402:
4378:
4365:
4346:
4333:
4325:
4324:
4277:
4276:
4253:
4243:
4230:
4220:
4219:
4194:
4193:
4149:
4148:
4075:
4060:
4037:
4022:
3990:
3989:
3970:
3969:
3950:
3949:
3923:
3913:
3900:
3817:
3777:
3776:
3730:
3729:
3710:
3709:
3690:
3689:
3664:
3663:
3644:
3643:
3620:
3599:
3594:
3593:
3572:
3553:
3548:
3547:
3528:
3527:
3506:
3488:
3477:
3476:
3454:
3453:
3410:
3409:
3388:
3387:
3368:
3367:
3348:
3347:
3308:
3307:
3288:
3287:
3280:
3255:
3254:
3230:
3229:
3205:
3204:
3195:
3116:
3095:
3094:
3046:
3024:
3019:
3018:
2989:
2974:
2948:
2933:
2909:
2897:
2893:
2872:
2857:
2846:
2845:
2798:
2797:
2772:
2771:
2737:
2736:
2706:
2691:
2664:
2649:
2625:
2610:
2586:
2571:
2557:
2556:
2521:
2505:
2473:
2472:
2438:
2437:
2406:
2391:
2376:
2360:
2313:
2312:
2285:
2266:
2261:
2260:
2233:
2207:
2206:
2185:
2166:
2161:
2160:
2125:
2101:
2090:
2089:
2057:
2042:
2018:
2008:
2004:
1979:
1961:
1946:
1935:
1934:
1899:
1884:
1873:
1872:
1865:
1841:
1840:
1821:
1820:
1801:
1800:
1764:
1746:
1731:
1720:
1719:
1690:
1675:
1661:
1660:
1636:
1620:
1598:
1593:
1592:
1556:
1541:
1528:
1510:
1495:
1484:
1483:
1457:
1442:
1431:
1430:
1373:
1368:
1367:
1316:
1292:
1285:
1281:
1280:
1276:
1255:
1240:
1226:
1225:
1199:
1194:
1193:
1170:
1165:
1164:
1143:
1138:
1137:
1114:
1109:
1108:
1089:
1088:
1056:
1038:
1014:
981:
957:
944:
940:
914:
890:
883:
879:
856:
841:
827:
826:
804:
803:
784:
783:
762:
761:
742:
741:
718:
713:
712:
689:
684:
683:
644:
643:
621:
620:
601:
600:
577:
576:
548:
547:
528:
527:
499:
498:
479:
478:
459:
458:
390:
386:
385:
375:
365:
340:
339:
317:
316:
293:
288:
287:
264:
259:
258:
230:
229:
221:
214:
167:
152:
141:
140:
107:
106:
69:
54:
40:
39:
28:
23:
22:
15:
12:
11:
5:
6281:
6279:
6271:
6270:
6265:
6260:
6250:
6249:
6246:
6245:
6240:
6233:
6232:External links
6230:
6229:
6228:
6222:
6215:
6208:
6201:
6194:
6188:
6185:
6182:
6179:
6174:
6171:
6168:
6167:
6160:
6142:
6115:
6076:
6035:
5992:
5990:
5973:
5938:
5909:
5873:
5872:
5870:
5867:
5844:
5843:
5830:
5825:
5820:
5816:
5811:
5806:
5801:
5798:
5795:
5792:
5787:
5782:
5777:
5774:
5770:
5765:
5761:
5758:
5755:
5750:
5745:
5740:
5735:
5730:
5725:
5722:
5719:
5678:
5673:
5668:
5663:
5658:
5653:
5650:
5647:
5644:
5639:
5634:
5629:
5626:
5622:
5592:
5588:
5583:
5578:
5573:
5570:
5567:
5564:
5559:
5554:
5549:
5546:
5542:
5511:
5507:
5503:
5498:
5493:
5488:
5485:
5482:
5479:
5474:
5469:
5464:
5461:
5458:
5446:
5443:
5442:
5441:
5426:
5423:
5420:carbon dioxide
5399:
5396:
5338:
5337:
5288:
5286:
5279:
5273:
5270:
5257:
5254:
5251:
5231:
5228:
5225:
5220:
5216:
5195:
5191:
5187:
5184:
5181:
5161:
5152:The parameter
5150:
5149:
5148:
5147:
5136:
5133:
5130:
5127:
5122:
5119:
5116:
5113:
5110:
5106:
5102:
5099:
5095:
5090:
5083:
5078:
5074:
5067:
5063:
5057:
5053:
5049:
5046:
5043:
5040:
5036:
5032:
5029:
5026:
5023:
5020:
5017:
5014:
5011:
5008:
5005:
4992:
4991:
4990:
4979:
4976:
4973:
4970:
4965:
4962:
4959:
4956:
4953:
4949:
4945:
4942:
4938:
4934:
4931:
4928:
4923:
4920:
4917:
4914:
4911:
4907:
4903:
4900:
4897:
4894:
4889:
4886:
4883:
4880:
4877:
4873:
4868:
4859:
4855:
4851:
4845:
4841:
4835:
4829:
4826:
4821:
4818:
4811:
4806:
4802:
4799:
4796:
4793:
4790:
4787:
4784:
4781:
4778:
4775:
4762:
4761:
4760:
4749:
4746:
4743:
4740:
4735:
4732:
4729:
4726:
4723:
4719:
4715:
4712:
4709:
4706:
4703:
4700:
4697:
4694:
4690:
4686:
4683:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4638:
4635:
4613:geostatistical
4600:
4588:
4585:
4572:
4567:
4563:
4559:
4554:
4550:
4546:
4543:
4540:
4535:
4531:
4527:
4522:
4518:
4514:
4494:
4489:
4485:
4481:
4476:
4472:
4468:
4465:
4462:
4457:
4453:
4449:
4444:
4440:
4436:
4416:
4413:
4410:
4390:
4385:
4381:
4377:
4372:
4368:
4364:
4361:
4358:
4353:
4349:
4345:
4340:
4336:
4332:
4308:
4304:
4301:
4298:
4295:
4292:
4289:
4285:
4262:
4257:
4250:
4246:
4242:
4237:
4233:
4228:
4207:
4204:
4201:
4180:
4176:
4173:
4170:
4167:
4164:
4161:
4157:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4090:
4084:
4079:
4074:
4069:
4064:
4058:
4054:
4051:
4046:
4041:
4036:
4031:
4026:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3977:
3957:
3946:
3945:
3932:
3927:
3920:
3916:
3912:
3907:
3903:
3898:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3862:
3859:
3855:
3847:
3843:
3840:
3837:
3834:
3831:
3828:
3824:
3820:
3816:
3811:
3808:
3805:
3802:
3799:
3796:
3790:
3787:
3761:
3758:
3755:
3752:
3749:
3743:
3740:
3717:
3697:
3677:
3674:
3671:
3651:
3629:
3624:
3619:
3616:
3613:
3608:
3603:
3579:
3575:
3571:
3568:
3565:
3560:
3556:
3535:
3513:
3509:
3505:
3502:
3497:
3492:
3487:
3484:
3461:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3396:
3375:
3355:
3335:
3332:
3329:
3326:
3323:
3320:
3316:
3295:
3282:Generally, an
3279:
3276:
3275:
3274:
3262:
3249:
3237:
3224:
3212:
3194:
3191:
3190:
3189:
3182:
3166:
3163:
3159:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3123:
3119:
3113:
3110:
3107:
3103:
3083:
3067:
3064:
3061:
3058:
3053:
3049:
3045:
3042:
3039:
3036:
3031:
3027:
3017:Consequently,
3015:
3003:
2998:
2993:
2988:
2983:
2978:
2973:
2970:
2967:
2963:
2957:
2952:
2947:
2942:
2937:
2932:
2929:
2926:
2923:
2918:
2913:
2908:
2905:
2901:
2896:
2892:
2889:
2886:
2881:
2876:
2871:
2866:
2861:
2856:
2853:
2843:
2831:
2828:
2824:
2820:
2817:
2814:
2811:
2808:
2805:
2785:
2782:
2779:
2759:
2756:
2753:
2750:
2747:
2744:
2733:
2720:
2715:
2710:
2705:
2700:
2695:
2690:
2687:
2684:
2681:
2678:
2673:
2668:
2663:
2658:
2653:
2648:
2645:
2642:
2639:
2634:
2629:
2624:
2619:
2614:
2609:
2606:
2603:
2600:
2595:
2590:
2585:
2580:
2575:
2570:
2567:
2564:
2546:
2545:
2533:
2528:
2524:
2520:
2517:
2512:
2508:
2502:
2497:
2494:
2491:
2487:
2483:
2480:
2460:
2457:
2454:
2451:
2448:
2445:
2434:
2433:
2432:
2421:
2418:
2413:
2409:
2405:
2400:
2395:
2390:
2385:
2380:
2375:
2372:
2367:
2363:
2357:
2352:
2349:
2346:
2342:
2336:
2331:
2328:
2325:
2321:
2307:
2306:
2292:
2288:
2284:
2281:
2278:
2273:
2269:
2259:and locations
2248:
2245:
2240:
2236:
2230:
2225:
2222:
2219:
2215:
2192:
2188:
2184:
2181:
2178:
2173:
2169:
2157:
2145:
2142:
2139:
2134:
2129:
2124:
2121:
2118:
2115:
2110:
2105:
2100:
2097:
2077:
2074:
2070:
2064:
2060:
2056:
2051:
2046:
2041:
2038:
2035:
2032:
2027:
2022:
2017:
2014:
2011:
2007:
2003:
2000:
1997:
1994:
1991:
1986:
1982:
1978:
1975:
1970:
1965:
1960:
1955:
1950:
1945:
1942:
1931:
1919:
1916:
1913:
1908:
1903:
1898:
1893:
1888:
1883:
1880:
1864:
1861:
1848:
1828:
1808:
1797:
1796:
1785:
1782:
1779:
1776:
1771:
1767:
1763:
1760:
1755:
1750:
1745:
1740:
1735:
1730:
1727:
1704:
1699:
1694:
1689:
1684:
1679:
1674:
1671:
1668:
1648:
1643:
1639:
1635:
1632:
1627:
1623:
1619:
1616:
1613:
1610:
1605:
1601:
1585:
1584:
1573:
1570:
1565:
1560:
1555:
1550:
1545:
1540:
1535:
1531:
1527:
1524:
1519:
1514:
1509:
1504:
1499:
1494:
1491:
1466:
1461:
1456:
1451:
1446:
1441:
1438:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1380:
1376:
1360:
1359:
1348:
1344:
1339:
1334:
1330:
1325:
1320:
1315:
1312:
1309:
1306:
1301:
1296:
1291:
1288:
1284:
1279:
1275:
1272:
1269:
1264:
1259:
1254:
1249:
1244:
1239:
1236:
1233:
1208:
1203:
1179:
1174:
1150:
1146:
1123:
1118:
1096:
1085:
1084:
1073:
1069:
1063:
1059:
1055:
1052:
1047:
1042:
1037:
1034:
1031:
1028:
1023:
1018:
1013:
1010:
1007:
1004:
1001:
998:
995:
990:
985:
980:
977:
974:
971:
966:
961:
956:
953:
950:
947:
943:
939:
936:
932:
928:
923:
918:
913:
910:
907:
904:
899:
894:
889:
886:
882:
873:
870:
865:
860:
855:
850:
845:
840:
837:
834:
811:
791:
770:
749:
727:
722:
698:
693:
660:
657:
654:
651:
631:
628:
608:
584:
564:
561:
558:
555:
535:
515:
512:
509:
506:
486:
466:
455:
454:
443:
440:
437:
432:
427:
423:
420:
417:
414:
411:
408:
405:
402:
399:
396:
393:
389:
382:
378:
371:
368:
364:
359:
356:
353:
350:
347:
324:
302:
297:
273:
268:
246:
243:
240:
237:
213:
210:
181:
176:
171:
166:
161:
156:
151:
148:
125:
121:
117:
114:
83:
78:
73:
68:
63:
58:
53:
50:
47:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6280:
6269:
6266:
6264:
6261:
6259:
6258:Geostatistics
6256:
6255:
6253:
6244:
6241:
6239:
6236:
6235:
6231:
6226:
6223:
6220:
6216:
6213:
6209:
6206:
6202:
6199:
6195:
6192:
6189:
6186:
6183:
6180:
6177:
6176:
6172:
6163:
6161:9780195066890
6157:
6153:
6146:
6143:
6138:
6134:
6130:
6126:
6119:
6116:
6111:
6107:
6103:
6099:
6095:
6091:
6087:
6080:
6077:
6071:
6066:
6062:
6058:
6054:
6050:
6046:
6039:
6036:
6031:
6027:
6022:
6017:
6013:
6009:
6005:
6001:
5997:
5989:
5980:
5978:
5974:
5969:
5965:
5961:
5957:
5953:
5949:
5942:
5939:
5927:
5920:
5917:Ford, David.
5913:
5910:
5905:
5901:
5897:
5893:
5889:
5885:
5878:
5875:
5868:
5866:
5864:
5860:
5856:
5852:
5847:
5828:
5823:
5818:
5809:
5796:
5793:
5785:
5772:
5768:
5763:
5759:
5756:
5748:
5738:
5733:
5720:
5717:
5710:
5709:
5708:
5706:
5702:
5698:
5694:
5676:
5661:
5648:
5645:
5637:
5624:
5611:
5607:
5581:
5568:
5565:
5557:
5544:
5531:
5527:
5509:
5496:
5483:
5480:
5472:
5459:
5444:
5439:
5435:
5431:
5427:
5424:
5421:
5417:
5416:
5415:
5413:
5409:
5408:geostatistics
5405:
5397:
5395:
5392:
5388:
5384:
5380:
5376:
5372:
5368:
5363:
5361:
5357:
5356:semivariogram
5353:
5349:
5345:
5344:geostatistics
5334:
5331:
5323:
5313:
5309:
5305:
5299:
5298:
5294:
5289:This section
5287:
5283:
5278:
5277:
5271:
5269:
5255:
5252:
5249:
5226:
5218:
5214:
5193:
5189:
5185:
5182:
5179:
5159:
5134:
5128:
5114:
5111:
5104:
5100:
5097:
5093:
5088:
5081:
5076:
5072:
5065:
5061:
5055:
5051:
5047:
5044:
5041:
5038:
5034:
5027:
5024:
5021:
5015:
5009:
5003:
4996:
4995:
4993:
4977:
4971:
4957:
4954:
4947:
4943:
4940:
4936:
4929:
4915:
4912:
4905:
4901:
4895:
4884:
4881:
4878:
4871:
4866:
4857:
4853:
4849:
4843:
4839:
4833:
4827:
4824:
4819:
4816:
4809:
4804:
4797:
4794:
4791:
4785:
4779:
4773:
4766:
4765:
4763:
4747:
4741:
4727:
4724:
4717:
4713:
4710:
4698:
4695:
4688:
4684:
4681:
4675:
4672:
4669:
4666:
4657:
4654:
4651:
4645:
4639:
4633:
4626:
4625:
4623:
4622:
4621:
4618:
4614:
4598:
4586:
4584:
4565:
4561:
4557:
4552:
4548:
4541:
4533:
4529:
4525:
4520:
4516:
4487:
4483:
4479:
4474:
4470:
4463:
4455:
4451:
4447:
4442:
4438:
4414:
4411:
4408:
4383:
4379:
4375:
4370:
4366:
4359:
4351:
4347:
4343:
4338:
4334:
4321:
4299:
4296:
4293:
4287:
4260:
4248:
4244:
4240:
4235:
4231:
4205:
4202:
4199:
4171:
4168:
4165:
4159:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4092:
4082:
4072:
4067:
4052:
4044:
4034:
4029:
4013:
4007:
4004:
4001:
3995:
3975:
3955:
3930:
3918:
3914:
3910:
3905:
3901:
3887:
3884:
3881:
3875:
3872:
3866:
3863:
3860:
3853:
3838:
3835:
3832:
3826:
3818:
3814:
3809:
3803:
3800:
3797:
3785:
3775:
3774:
3773:
3756:
3753:
3750:
3738:
3715:
3695:
3675:
3672:
3669:
3649:
3627:
3617:
3614:
3611:
3606:
3592:at locations
3577:
3573:
3569:
3566:
3563:
3558:
3554:
3533:
3511:
3507:
3503:
3495:
3482:
3473:
3459:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3373:
3353:
3330:
3327:
3324:
3318:
3293:
3285:
3277:
3260:
3253:
3250:
3235:
3228:
3225:
3210:
3203:
3200:
3199:
3198:
3192:
3187:
3183:
3180:
3150:
3144:
3141:
3138:
3135:
3129:
3121:
3117:
3105:
3092:
3088:
3084:
3081:
3080:even function
3062:
3059:
3051:
3047:
3043:
3037:
3029:
3025:
3016:
2996:
2986:
2981:
2968:
2965:
2961:
2955:
2940:
2927:
2924:
2916:
2903:
2894:
2890:
2887:
2879:
2869:
2864:
2851:
2844:
2815:
2809:
2806:
2803:
2783:
2780:
2777:
2757:
2754:
2748:
2742:
2734:
2731:
2713:
2703:
2698:
2685:
2682:
2679:
2671:
2661:
2656:
2643:
2640:
2632:
2622:
2617:
2604:
2601:
2593:
2583:
2578:
2565:
2562:
2552:
2548:
2547:
2526:
2522:
2515:
2510:
2506:
2500:
2495:
2492:
2489:
2485:
2481:
2478:
2455:
2449:
2446:
2443:
2435:
2419:
2416:
2411:
2407:
2398:
2388:
2383:
2370:
2365:
2361:
2355:
2350:
2347:
2344:
2340:
2334:
2329:
2326:
2323:
2319:
2311:
2310:
2309:
2308:
2290:
2286:
2282:
2279:
2276:
2271:
2267:
2246:
2243:
2238:
2234:
2228:
2223:
2220:
2217:
2213:
2190:
2186:
2182:
2179:
2176:
2171:
2167:
2158:
2143:
2140:
2132:
2119:
2116:
2108:
2095:
2075:
2072:
2068:
2062:
2049:
2036:
2033:
2025:
2012:
2005:
2001:
1998:
1992:
1984:
1980:
1976:
1968:
1958:
1953:
1940:
1932:
1917:
1914:
1906:
1896:
1891:
1878:
1870:
1869:
1868:
1862:
1860:
1846:
1826:
1806:
1783:
1777:
1769:
1765:
1761:
1753:
1743:
1738:
1725:
1718:
1717:
1716:
1697:
1687:
1682:
1669:
1666:
1641:
1637:
1633:
1625:
1621:
1617:
1611:
1603:
1599:
1590:
1571:
1563:
1553:
1548:
1533:
1529:
1525:
1517:
1507:
1502:
1489:
1482:
1481:
1480:
1464:
1454:
1449:
1439:
1436:
1413:
1410:
1407:
1404:
1401:
1395:
1392:
1386:
1378:
1374:
1365:
1346:
1342:
1337:
1332:
1323:
1310:
1307:
1299:
1286:
1282:
1277:
1273:
1270:
1262:
1252:
1247:
1234:
1231:
1224:
1223:
1222:
1206:
1177:
1148:
1144:
1121:
1094:
1071:
1067:
1061:
1045:
1032:
1029:
1021:
1008:
1002:
999:
988:
975:
972:
964:
951:
941:
937:
934:
930:
921:
908:
905:
897:
884:
880:
871:
863:
853:
848:
835:
832:
825:
824:
823:
809:
789:
747:
725:
696:
681:
676:
674:
655:
649:
629:
626:
606:
598:
582:
559:
553:
533:
510:
504:
484:
464:
441:
438:
435:
430:
425:
418:
412:
409:
403:
400:
397:
391:
387:
380:
376:
369:
366:
362:
357:
351:
345:
338:
337:
336:
322:
300:
271:
241:
235:
228:
227:semivariogram
223:
219:
211:
209:
207:
197:
193:
174:
164:
159:
146:
139:
138:semivariogram
112:
105:
101:
98:of a spatial
97:
76:
66:
61:
48:
45:
37:
33:
19:
6218:
6211:
6204:
6197:
6151:
6145:
6128:
6124:
6118:
6093:
6089:
6079:
6052:
6048:
6038:
6003:
5999:
5987:
5951:
5947:
5941:
5929:. Retrieved
5925:
5912:
5887:
5883:
5877:
5858:
5850:
5848:
5845:
5704:
5605:
5525:
5448:
5438:seismic risk
5403:
5401:
5398:Applications
5374:
5370:
5364:
5359:
5355:
5341:
5326:
5317:
5302:Please help
5290:
5151:
4590:
4322:
3947:
3474:
3283:
3281:
3251:
3226:
3201:
3196:
3185:
3178:
2555:
1866:
1799:The indexes
1798:
1586:
1361:
1086:
677:
456:
226:
224:
222:
203:
137:
100:random field
35:
29:
18:Semivariance
5610:square root
5434:earthquakes
5406:is used in
5348:correlogram
2205:subject to
335:. Formally
206:gold mining
38:, denoted
6252:Categories
6096:: 103139.
6055:: 91โ108.
5931:31 October
5869:References
5855:co-kriging
5693:Estimators
5383:covariance
5354:, and the
5352:covariance
5272:Discussion
3193:Parameters
3087:stationary
1863:Properties
212:Definition
6030:1867-8548
5968:1385-2256
5904:1554-0774
5824:α
5794:−
5721:γ
5697:resistant
5646:−
5566:−
5481:−
5360:variogram
5291:does not
5253:∈
5118:∞
5056:−
5048:
5042:−
5025:−
5004:γ
4961:∞
4919:∞
4834:−
4795:−
4774:γ
4731:∞
4682:−
4676:
4670:−
4655:−
4634:γ
4415:δ
4412:±
4300:δ
4297:±
4241:−
4172:δ
4169:±
4126:…
4102:δ
4099:±
4014:≡
4008:δ
4005:±
3976:δ
3911:−
3888:δ
3885:±
3873:∈
3854:∑
3839:δ
3836:±
3804:δ
3801:±
3789:^
3786:γ
3757:δ
3754:±
3742:^
3739:γ
3696:γ
3676:δ
3673:±
3615:…
3567:…
3118:γ
3112:∞
3109:→
3060:−
3048:γ
3026:γ
2969:γ
2925:−
2852:γ
2680:−
2566:γ
2486:∑
2417:≤
2371:γ
2341:∑
2320:∑
2305:it holds:
2280:…
2214:∑
2180:…
2117:−
2034:−
1981:γ
1941:γ
1915:≥
1879:γ
1847:γ
1766:γ
1726:γ
1703:‖
1688:−
1673:‖
1622:γ
1600:γ
1589:isotropic
1554:−
1530:γ
1490:γ
1455:−
1396:γ
1375:γ
1308:−
1235:γ
1095:μ
1030:−
1000:−
973:−
906:−
836:γ
650:γ
410:−
377:∭
346:γ
236:γ
147:γ
49:γ
36:variogram
5701:outliers
5606:rodogram
5604:, and a
5526:madogram
5320:May 2015
2781:≠
680:variance
6098:Bibcode
6057:Bibcode
6008:Bibcode
5412:kriging
5379:kriging
5312:removed
5297:sources
4617:kriging
3091:ergodic
2549:If the
200:effect.
6158:
6028:
5966:
5902:
5350:, the
4427:only [
3202:nugget
3186:nugget
3093:, the
3078:is an
595:(with
497:, and
457:where
136:. The
5922:(PDF)
5432:from
5385:of a
3252:range
6156:ISBN
6026:ISSN
5964:ISSN
5933:2017
5900:ISSN
5402:The
5295:any
5293:cite
3366:and
3227:sill
3179:sill
3089:and
1192:and
1136:and
782:and
711:and
286:and
225:The
6133:doi
6106:doi
6094:203
6065:doi
6053:511
6016:doi
5956:doi
5892:doi
5707:",
5699:to
5677:0.5
5414:.
5306:by
5045:exp
4673:exp
3102:lim
2770:if
2471:of
1819:or
876:var
802:to
760:to
102:or
30:In
6254::
6129:40
6127:.
6104:.
6092:.
6088:.
6063:.
6051:.
6047:.
6024:.
6014:.
6002:.
5998:.
5991:CO
5976:^
5962:.
5950:.
5924:.
5898:.
5888:58
5886:.
5865:.
5691:.
5532:,
5362:.
3810::=
1618::=
6227:.
6193:.
6164:.
6139:.
6135::
6112:.
6108::
6100::
6073:.
6067::
6059::
6032:.
6018::
6010::
6004:7
5993:2
5988:X
5970:.
5958::
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5935:.
5906:.
5894::
5842:,
5829:]
5819:|
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5810:2
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5800:(
5797:Z
5791:)
5786:1
5781:s
5776:(
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5769:|
5764:[
5760:E
5757:=
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5468:s
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5318:(
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5300:.
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5250:h
5230:)
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5224:(
5219:A
5215:1
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5190:/
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5180:a
5160:a
5135:.
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5129:h
5126:(
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5115:,
5112:0
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5101:n
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5073:r
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5052:(
5039:1
5035:(
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5028:n
5022:s
5019:(
5016:=
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5010:h
5007:(
4978:.
4975:)
4972:h
4969:(
4964:)
4958:,
4955:0
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4156:|
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4120:1
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3807:)
3798:h
3795:(
3760:)
3751:h
3748:(
3716:h
3670:h
3650:h
3628:k
3623:s
3618:,
3612:,
3607:1
3602:s
3578:k
3574:z
3570:,
3564:,
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3555:z
3534:k
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3508:z
3504:=
3501:)
3496:i
3491:s
3486:(
3483:Z
3460:B
3440:)
3437:t
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3431:z
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3425:y
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3395:s
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3319:=
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3151:Z
3148:(
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3136:=
3133:)
3130:h
3127:(
3122:s
3106:h
3082:.
3066:)
3063:h
3057:(
3052:s
3044:=
3041:)
3038:h
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3030:s
3002:)
2997:1
2992:s
2987:,
2982:2
2977:s
2972:(
2966:=
2962:]
2956:2
2951:|
2946:)
2941:2
2936:s
2931:(
2928:Z
2922:)
2917:1
2912:s
2907:(
2904:Z
2900:|
2895:[
2891:E
2888:=
2885:)
2880:2
2875:s
2870:,
2865:1
2860:s
2855:(
2830:)
2827:)
2823:s
2819:(
2816:Z
2813:(
2810:r
2807:a
2804:v
2784:0
2778:h
2758:0
2755:=
2752:)
2749:h
2746:(
2743:C
2719:)
2714:2
2709:s
2704:,
2699:1
2694:s
2689:(
2686:C
2683:2
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2644:C
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2618:1
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2605:C
2602:=
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2563:2
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2523:x
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2496:1
2493:=
2490:i
2482:=
2479:X
2459:)
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2453:(
2450:r
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2420:0
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2408:w
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2399:j
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2389:,
2384:i
2379:s
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2366:i
2362:w
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2351:1
2348:=
2345:j
2335:N
2330:1
2327:=
2324:i
2291:N
2287:s
2283:,
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2272:1
2268:s
2247:0
2244:=
2239:i
2235:w
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2224:1
2221:=
2218:i
2191:N
2187:w
2183:,
2177:,
2172:1
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2096:Z
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2073:=
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2059:)
2055:)
2050:1
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2040:(
2037:Z
2031:)
2026:1
2021:s
2016:(
2013:Z
2010:(
2006:(
2002:E
1999:=
1996:)
1993:0
1990:(
1985:i
1977:=
1974:)
1969:1
1964:s
1959:,
1954:1
1949:s
1944:(
1918:0
1912:)
1907:2
1902:s
1897:,
1892:1
1887:s
1882:(
1827:s
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1784:.
1781:)
1778:h
1775:(
1770:i
1762:=
1759:)
1754:2
1749:s
1744:,
1739:1
1734:s
1729:(
1698:1
1693:s
1683:2
1678:s
1670:=
1667:h
1647:)
1642:1
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1634:h
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1626:s
1615:)
1612:h
1609:(
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1572:.
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1564:1
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1549:2
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1534:s
1526:=
1523:)
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1465:1
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1393:=
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1324:2
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1278:[
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1033:Z
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20:)
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