the simultaneous analysis of seismic attributes with wellbore data from multiple wells leads to better predictions of reservoir properties . such seismic guided predictions have higher resolution and predict reservoir properties with higher accuracy , compared to estimates based on well data in which the seismic data were used only for the geometry . for example , when acoustic impedance inverted from seismic data is used to guide the estimate of porosity using the method of the present invention , the error is cut in half as compared to conventional methods . the method identifies statistical correlations between seismic attributes and log properties , estimates linear or nonlinear functional relationships between attributes and properties , uses artificial neural networks for analyzing complex nonlinear relationships , and using these functional relationships and geostatistical technics to predict reservoir properties . based on rock physics , certain seismic attributes are indicative of certain reservoir properties . however , exact functional relationships from theory is unreliable , and a learning phase is therefore valuable in every new reservoir context . examples of known correlations of reservoir properties related to seismic attributes include porosity guided by acoustic impedance , velocity , and volume of clay ; volume of clay and dolomite guided by heterogeneity and instantaneous frequency ; water saturation guided by depth ; thickness guided by amplitude . also , borehole measured dip and azimuth are compared to dip and azimuth inferred from tracking seismic reflections . the 5 principle steps in the method of the present invention are illustrated in fig1 and are as follows . seismic data are measured in time , well data in depth and it is important that the seismic data be related or calibrated in physical space to the wellbore data . relating seismic reflectors to geological markers seen in boreholes is a critical step ; inaccuracy of a few meters may completely change subsequent results . the analysis is based on synthetic seismograms and time to depth conversion based on a velocity model . eventually , the intersections may be refined interactively . fig2 displays a single well view of well logs and acoustic impedance extracted from the seismic data along the well . it is necessary to tie control points in depth ( called geological markers ) to control points in time ( called seismic markers ). as can be seen from fig2 adjustment of the acoustic impedance to match up with well logs is possible . in more detail , fig2 shows a single well display with various logs and a seismic well trace plotted as a function of time and depth . the acoustic impedance log ( first on the left ) is very similar to the porosity log ( second on the left ) and quite similar to the acoustic impedance inverted from surface seismic data with no usage of well data in the inversion . the well trace is displayed in variable area on the right . it contains values from a 3 - d acoustic impedance cube extracted on the trajectory of a deviated well . the acoustic impedance trace resembles the acoustic impedance log , except that it has a lower frequency content . one can identify the high porosity -- low acoustic impedance component on both logs and well trace . this is further indication that acoustic impedance from inversion might be useful to estimate porosity . the volume of clay log ( fourth log from left ) does not resemble the well trace . hence , inverted impedance would not be directly useful for estimation of shaliness . the resolution of well data is higher than that of surface seismic data . to use them together effectively , appropriate averaging is done at the intersections . the well data are averaged between the top and bottom of a geological component ( layer ). the seismic attributes are averaged temporally and spatially to isolate the influence of the geological component . wellbore data from multiple wells are extracted at horizon intersections to give a multidimensional scatter of independent values . in the case of mapping averaged properties within layers , each well contributes a point to this scatter . the purpose of significance estimation is to quantify which attributes are significant to which properties -- i . e ., a match between seismic attributes and wellbore data . for each match , the method calculates the probability of the values not being randomly related . preferably , the kendall tau indicator , τ k , which is a measure of the monotonicity of a scatter , is used . see , m . kendall , and a . stuart , 1977 , the advanced theory of statistics , 4th ed ., griffin & amp ; co ., london . it detects nonlinear relations , as opposed to other indicators which measure how close a relation is to being linear . another advantage is that it is a robust indicator which is not easily affected by a few erroneous measurements , much like the median is more robust than the mean . the significance is the probability of the scatter not being random . it is calculated from the actual τ k and the number of points in the scatter , n , as the probability of having a τ k smaller than the actual one with n random points . each pair of points in a scatter has a slope . if there are n points in the scatter , there are a total of n t = n ( n - 1 )/ 2 slopes . kendall &# 39 ; s τ is defined as , ## equ1 ## where n p , n n , n z , and n . sub .∞ are the numbers of positive , negative , zero , and infinite slopes . if all the slopes have the same sign , the scatter is monotonous and the absolute value of τ k is 1 . if there are as many negative as there are positive slopes , τ k will be 0 . if all the slopes are either zero or infinite , τ k will be undetermined . τ k can never exceed 1 . 0 in absolute value . by itself , τ k is not a significance estimator . for example , with two points the absolute value of τ k is ( almost always ) the highest possible 1 , but the significance should be zero . the significance is estimated from a given τ k and the number of points , n , according to the probability of getting the particular τ k from n random points . based on kendall and stuart analysis , ## equ2 ## ( for n & gt ; 4 ). this gives , for example , 44 % for τ k of 0 . 5 with 5 points , but 84 % for τ k of just 0 . 2 with 100 points . the significance values are summarized in a quality matrix ( fig3 ). the columns correspond to reservoir properties and the rows to seismic attributes . previously estimated properties can be loaded as if they were seismic attributes . it is essential to examine the scatters of attribute - property pairs directly , and to disable outlier points that we wish to exclude from the analysis . fig3 depicts the &# 34 ; quality matrix &# 34 ; or significance values useful for determining matches between seismic attributes and reservoir properties . the significance values of each seismic attribute and reservoir property are displayed on buttons , from which a cross plot editor can be popped . for example , acoustic impedance is significant for porosity , depth for water saturation , and instantaneous frequency for volume of clay . an interactive cross plot editor for a workstation is shown in fig4 . fig4 illustrates specifically a technique for calibration editing . the names are well names . volume of clay is expected to affect the reflection heterogeneity seismic attribute . ( a ) before and ( b ) after exclusion of what might be interpreted as out - liars . the significance indicator increases from 13 % to 72 % in the process . to estimate a certain reservoir property , for example porosity , f , guided by velocity v p , and shaliness , vcl , a calibration function f ( v p , vcl ) is sought . a linear calibration function is for example , where the coefficients , c 0 , c 1 and c 2 are found by solving a system of equations . linear calibration is adequate when the relations are linear . nonlinear calibration is needed when the relations are more complex . for example , water saturation increases with depth nonlinearly ; faster at the water cut and slower above and below it . since nonlinear relationships are unknown and varied , instead of prescribing a particular nonlinear model to perform the calibration ( e . g ., a polynomial typically used in regression ), the method uses an , artificial neural network ( ann ) to learn a nonlinear model using example data ( fig5 ). the method perform the nonlinear calibration after the linear calibration using the difference ( residuals ) between the well - measured properties to the estimated properties from linear calibration . the method uses a particular type of ann , called a radial basis function network ( rbfn ). rbfn is a single hidden layer , feed - forward network that uses radially symmetric basis functions in the hidden layer , see , j . moody , and c . darken , 1989 , fast learning in networks of locally - tuned processing units , neural computation 1 , 281 - 294 . the rbfn is &# 34 ; trained &# 34 ; to approximate the unknown multivariate nonlinear target function using a weighted sum of basis functions . the method of the present invention uses the values of seismic attributes and wellbore properties , averaged about the intersections to &# 34 ; train &# 34 ; the network . a hybrid learning procedure is used to train the ann . first , the mean vectors of the rbf nodes are found using the k - means clustering algorithm ( spath , 1980 ). the cluster analysis yields cluster - averaged values and widths . the cluster width is the distance between the mean vector and the mean vector of the nearest cluster . each cluster corresponds to a basis function that occupies a node in the hidden layer of the ann . the width of each basis function is by default the width of the corresponding cluster . however , the nearest neighbor distance is multiplied by a user controlled overlap factor , in order to form a smooth contiguous interpolation over the input space . the approximation learned by the rbfn is given by : ## equ3 ## where r is the residual property to be estimated . a is the attribute vector , n is the number of hidden nodes , w j is the weight associated with the j - th hidden node , aj is the mean - cluster averaged attribute vector , and σ j is the extent of the response of the basis function . n and σ j are found in the cluster analysis but may be reset by the user . the weights w j are found using singular value decomposition to optimally fit the linear combination of basis functions to the residuals . eventually , the basis function widths are determined iteratively by minimizing the approximation error . this local nature of the fit makes the solution generated by the network clearly understood , unlike some other network models which follow a &# 34 ; black - box &# 34 ; approach with little user control . in addition , a fast training procedure is available to determine the network parameters . as can be seen from fig5 seismic attributes are the input layer . the hidden layer of a lesser number of nodes represents a set of radial basis functions whose parameters are determined according to the well data . the output layer has a single node for the reservoir property which is predicted . the network parameters are found using the well data for training . after training , the network is used between wells for predicting reservoir properties where only seismic data exists . that is , the reservoir property at a location between existing oil wells is calculated using the trained ann . the seismic attributes at the location ( s ) of interest is input to the trained ann , with the reservoir property output ( c . f . fig5 ). reasonable calibration functions do not fit the intersection data exactly , implying that the estimates do not agree exactly with the borehole measurements . often , an exact agreement will be a requirement , and may be handled by a residual correction . geostatistical methods such as cokriging or gridding the differences between the calibrated attributes and the measured property are used to produce a seismic guided estimate that complies with the well data . see , g . matheron , 1970 , la theorie des variables regionalisees et ses applications , tech . rep . fascicule 5 , les cahiers du centre de morphologie mathematique de fontenebleau , ecole superieure des mines de paris and a . g . journel and c . j . huijbregts , 1978 , mining geostatistics , academic press . these same geostatistical methods are used to estimate the confidence in the prediction of the reservoir property . the following examples were performed on a reservoir modeling workstation , where functionality corresponding to the present method is implemented . rock physics predicts that porosity is related to velocity , acoustic impedance , and petrophysical components such as clay content , see , d . marion , a . nur , h . yin , and d . han , 1992 , compressional velocity and porosity in sand - clay mixtures : geophysics 57 , 554 - 563 . a velocity model is required for imaging seismic data , and can also be used for estimating porosity . acoustic impedance is calculated by inversion programs , and can be used in addition to or instead of velocity . both velocity and acoustic impedance exhibit high significance to porosity ( fig6 ). the porosity values in fig6 are purely from log data . the velocity ( avgv ) and the acoustic impedance ( avg 13 acoustic 13 impedance ) are calculated purely from seismic data . hence , the significance numbers in fig6 are indicative of the quality of the data acquisition , processing , and interpretation , in addition to the rock physics aspect of whether porosity is related to velocity and acoustic impedance . it is reassuring to see that acoustic impedance based on surface seismic data ( called avg 13 acoustic 13 impedance in fig6 ) is indicative for the acoustic impedance based on borehole data ( aimp in fig6 ). the petrophysical composition is an important parameter in the porosity - impedance relations . one difficulty is in estimating the petrophysical composition . for example , volume of clay ( shaliness , or vcl in fig6 and 4 ) and volume of dolomite ( vdol ). sometimes , petrophysical heterogeneity affects the instantaneous frequency and the reflection heterogeneity seismic attributes , because clay lenses or dolomite layers cause a high frequency reflectivity series . in fig2 the volume of clay log ( vcl ) shows what looks like shaley lenses or layers . indeed , the cross plots in fig6 show a trend of increased instantaneous frequency and heterogeneity with increased volume of clay and dolomite . fig6 ( a ) and ( b ) show cross - plots leading to porosity estimation . in fig6 ( a ) porosity and acoustic impedance are inverted from seismic data ; the trend of higher porosity with lower acoustic impedance is predicted by rock physics theory and laboratory measurements . fig6 ( b ) illustrates porosity and average velocity based on moveout velocity analysis . fig6 ( c ) shows a test for the quality of the seismic derived acoustic impedance ; comparing the borehole based values to the surface based values of the same property . fig6 ( d ) is a test for the quality of the velocity estimation ; comparing the average interval velocity from moveout analysis to the average sonic travel time - the higher the velocity , the shorter the travel time . fig6 ( e ) plots volume of clay and instantaneous frequency ; high clay content values come from shale lenses that increase the instantaneous frequency . fig6 ( f ) compares volume of dolomite and heterogeneity ; high dolomite content value indicate more dolomite layers which increase the reflection heterogeneity attribute . for comparison , estimates of porosity with and without seismic guidance are presented in fig7 ( a ) and ( b ). fig7 ( a ) shows porosity estimation guided by acoustic impedance and volume of clay , using linear calibration while fig7 ( b ) shows porosity mapped from well logs without seismic guidance . estimates in fig7 ( a ) and ( b ) agree exactly on the wells , but are quite different away from the wells . as can be seen , the fig7 ( a ) depiction of porosity with seismic guidance computed with the present method is qualitatively superior than fig7 ( b ). the depth map of a particular geological layer can be treated as an attribute . such depth shows a correlation to water saturation , which is driven by gravity mechanisms ( fig8 ). fig8 plots water saturation ( sw ) and depth . within a hydro - carbon bearing reservoir component , the depth is significant for water saturation , as more water are in the deep and hydrocarbons on top . arguably , one can see a water cut at about 3100 meters . one rarely sees a water cut on a single well water saturation log . however , the multi - well cross plot displays water saturation in an interval that spans nearly 200 meters , while a single well crosses a reservoir component in 5 - 40 meters as can be seen in fig1 . water saturation mapping can be guided by depth as shown in fig9 . fig9 illustrates water saturation estimation where fig9 ( a ) is guided by depth using an artificial neural network for nonlinear calibration and 9 ( b ) is without seismic guidance . estimations shown in fig9 ( a ) and ( b ) agree exactly on the wells but are quite different away from the wells . avo indicators may be effective to estimate gas saturation in a manner similar to fig9 . in thick reservoirs , the thickness can be obtained from tracking reflections . in thin reservoirs , reflection amplitude is affected by thickness tuning . see , a . r . brown , r . m . wright , k . d . burkarl , and w . l . abriel , 1984 , interactive seismic mapping of net producible gas sand in the gulf of mexico , geophysics 49 , 686 - 714 . to use thickness tuning , the reflection amplitude is used as a seismic attribute , and the well measured thickness is used as the reservoir property to be mapped . a cross plot of thickness versus amplitude is shown in fig1 . in fig1 , the amplitude is affected by the thickness because of interfering reflections from the top and the bottom of a component . fluid and gas transmissibility are important reservoir properties , but seismic data are not directly affected by permeability and viscosity . permeability is often proportional to porosity and this is , perhaps , why it seems to be related to seismic velocity . seismic attenuation will be a useful attribute for the estimate of permeability distributions , see , n . akbar , j . dvorkin , and a . nur , 1993 , relating p - wave attenuation to permeability , geophysics 58 , 20 - 29 . when well log and core sample estimates of permeability are available , an effective match with seismic alternation is believed possible . attribute calibration offers an opportunity to compare estimations of dip and azimuth based on wellbore data to those based on surface seismic data . dip and azimuth estimated from pure seismic data ( by tracking a reflection ) versus those estimated purely from wellbore data ( by tracking resistivity with a formation microscanner ) are shown in fig1 . in fig1 , the amplitude is affected by the thickness because of interfering reflections from the top and the bottom of a component . unfortunately , the borehole dips and azimuths were acquired in only 4 wells which do not offer a sufficient sample , except to show that the results have the right order of magnitude for the azimuth , and that the dips from surface seismic data are underestimated , which might be due to underestimation of seismic velocity at the wells agate and sapphire . the improvement in the reservoir property estimation by using seismic data is illustrated in fig1 . fig1 illustrates the results of a controlled study where the porosity estimations with and without seismic guidance are illustrated . the control wells ( not used in the estimations ) are in yellow . some wells are assigned as a validation group , then estimated porosity and water saturation without the validation wells , with and without seismic guidance , and compared seismic guided and unguided estimates to the measured values in the validation wells . five examples were made that differ in the wells assigned as control wells . the improvement in the accuracy of the estimates varied from - 3 % ( the estimates without seismic guided were slightly more accurate ) to a factor of 2 . 5 reduction in average error . the results in fig1 indicate a significant improvement in the quality of the property estimates , when seismic data are used in accordance with the present invention . in this case the average absolute value of the error in estimating porosity was twice smaller when acoustic impedance was used to guide the estimate . the average error in estimating water saturation was 2 . 5 times smaller when the depth was used . better estimates of reservoir properties using seismic data are obtained using the method of the present invention . statistical correlations between seismic attributes and log properties are identified , estimates of linear or nonlinear functional relationships between attributes and properties are made , an ann is used for the nonlinear relationships , and these functional relationships are used to predict reservoir properties . seismic guided estimates have higher resolution and predict reservoir properties more accurately than estimates based on interpolation of well measurements on structures estimated from seismic data . this has been verified in a controlled study , in which some borehole data were withheld in the process and were later compared to the estimations with and without seismic guidance . for example , when acoustic impedance inverted from seismic data was used to guide estimation of porosity , the error was cut in half . all patent references cited herein are incorporated by reference . all nonpatent references cited herein are incorporated by reference for background .