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Water In Ecosystems: A Non Renewable Resource
E{I(x;z)}
= I . Prob {Z(x) ~ z } + O. Prob { Z(x) > z }
= Prob {Z(x) ~ z } = F(z)
The local estimation of the cumulative distribution function is very important in
many applications. The indicator map would separate at high thresholds the
distribution of the maximum values in the studied area, while at low thresholds, it
would show the distribution of the minimum values.
4 STOCHASTIC SIMULATION USING DSI
4.1
Goals and Constraints
Let D = {U n K} be a domain on which the RV Z(x) has been sampled at location
{Xi E K}. The purpose is to estimate the value of Z(x) at an unknown location x in
U. Figure 1 shows a real case study in which the values are known on a regular
grid 50x50. The RV is sampled by 140 data reported in Figure 2. Classical
interpolation techniques such as kriging are applied to reconstruct the initial image
(Fig. 3). It is obvious that the estimator used in Figure 3 is inefficient to
reconstruct the initial image because the result is too smooth, especially for the
maximum and the minimum values. Moreover, kriging provides an estimated
value plus a standard deviation for the estimate but not a probability distribution
of the errors (or of the risk).
The proposed technique consists in estimating the local cdr of Z(x) at each
unknown location using the indicator function approach. However, this involves
some theoretical difficulty because general estimator techniques (kriging,
cokriging) cannot be applied directly to estimate I(x,z) because they do not
assume the following properties:
Order inequality constraints: for a set of thresholds {Zi; i= l,n} the
different indicators must verify the cdf properties: I(x,z) E [0, I], I(x,z) ~
I(x,z') for z < z';
• Global cd! constraint: The mean of the estimated indicators I(x,z) must
be equal to the cdf of Z: F(z), according to relation (I);
These different constraints have been implemented in the DSI algorithm (Royer
and Shtuka, 1994; Shtuka, 1994).
4.2 Indicator based simulation using OSI
The DSI (Discrete Smooth Interpolation) method was developed by Mallet (1989,
1992) to model complex 3D geological surfaces (reverse faults, salt domes, ... ) and
to interpolate physical properties (velocity, porosity, grades, ... ). This method is a
new interpolation method based on the optimization of an objective function (such
as the roughness of the surface) taking into account precise and imprecise data
(hard or soft data) using several local linear constraints. Examples on modeling
the geometry of 3D complex objects both in Earth Sciences and Medecine can
also be found in Royer et aI., 1995 (CODATA Vol. I, Chambery, 1994). To take
into account the constraints defined in § 4.1, two types of constraints were added
to the classical DSI approach.
Water In Ecosystems: A Non Renewable Resource
E{I(x;z)}
= I . Prob {Z(x) ~ z } + O. Prob { Z(x) > z }
= Prob {Z(x) ~ z } = F(z)
The local estimation of the cumulative distribution function is very important in
many applications. The indicator map would separate at high thresholds the
distribution of the maximum values in the studied area, while at low thresholds, it
would show the distribution of the minimum values.
4 STOCHASTIC SIMULATION USING DSI
4.1
Goals and Constraints
Let D = {U n K} be a domain on which the RV Z(x) has been sampled at location
{Xi E K}. The purpose is to estimate the value of Z(x) at an unknown location x in
U. Figure 1 shows a real case study in which the values are known on a regular
grid 50x50. The RV is sampled by 140 data reported in Figure 2. Classical
interpolation techniques such as kriging are applied to reconstruct the initial image
(Fig. 3). It is obvious that the estimator used in Figure 3 is inefficient to
reconstruct the initial image because the result is too smooth, especially for the
maximum and the minimum values. Moreover, kriging provides an estimated
value plus a standard deviation for the estimate but not a probability distribution
of the errors (or of the risk).
The proposed technique consists in estimating the local cdr of Z(x) at each
unknown location using the indicator function approach. However, this involves
some theoretical difficulty because general estimator techniques (kriging,
cokriging) cannot be applied directly to estimate I(x,z) because they do not
assume the following properties:
Order inequality constraints: for a set of thresholds {Zi; i= l,n} the
different indicators must verify the cdf properties: I(x,z) E [0, I], I(x,z) ~
I(x,z') for z < z';
• Global cd! constraint: The mean of the estimated indicators I(x,z) must
be equal to the cdf of Z: F(z), according to relation (I);
These different constraints have been implemented in the DSI algorithm (Royer
and Shtuka, 1994; Shtuka, 1994).
4.2 Indicator based simulation using OSI
The DSI (Discrete Smooth Interpolation) method was developed by Mallet (1989,
1992) to model complex 3D geological surfaces (reverse faults, salt domes, ... ) and
to interpolate physical properties (velocity, porosity, grades, ... ). This method is a
new interpolation method based on the optimization of an objective function (such
as the roughness of the surface) taking into account precise and imprecise data
(hard or soft data) using several local linear constraints. Examples on modeling
the geometry of 3D complex objects both in Earth Sciences and Medecine can
also be found in Royer et aI., 1995 (CODATA Vol. I, Chambery, 1994). To take
into account the constraints defined in § 4.1, two types of constraints were added
to the classical DSI approach.