this paper explores the use of space time point process models for the short - term forecasting of wildfire hazard in los angeles county , california .the region is especially well suited to such an analysis , since the los angeles county fire department and department of public works have collected and compiled detailed records on the locations burned by large wildfires dating back over a century .the landscape in los angeles county is uniquely vulnerable to high intensity crown - fires , largely because the predominant local vegetation consists of dense , highly flammable contiguous chaparral shrub [ keeley ( ) ] . in addition , the dry summers and early autumns in los angeles county are typically followed by high winds known locally as santa ana winds [ keeley and fotheringham ( ) ] .these offshore winds reach speeds exceeding 100 kph at a relative humidity below 10% , and are annual events lasting several days to several weeks , creating the most severe fire weather in the united states [ schroeder et al .( ) ] .  in order to forecast wildfire hazard, the united states national fire danger rating system ( nfdrs ) , created in 1972 , produces several daily indices that are designed to aid in planning fire control activities on a fire protection unit [ deeming et al .( ) ; bradshaw et al .( ) ; burgan ( ) ] .these include the occurrence index , the burning index ( bi ) , and the fire load index .these indices are derived from three fire behavior components a spread component , an energy release component , and an ignition component , that are in turn computed based on fuel age , environmental parameters ( slope , vegetation type , etc . ) , and meteorological variables such as wind , temperature , and relative humidity .local wildfire management agencies may combine these components in different ways or calibrate the inherent parameters to adapt the system to the local environment for wildfire hazard assessment .fire managers use this information in making decisions about the appropriateness of prescribed burning or alerts for increased preparedness , both in terms of fire suppression staffing and fire prevention activities .since fireline intensity is an important factor in predicting fire containment and the likelihood of fire escape , the burning index is the rating of most interest to many fire managers [ schoenberg et al .( ) ] .this is especially the case for natural crown - fire ecosystems such as southern california shrublands , where bi is commonly employed to assess fire danger [ mees and chase ( ) ] . indeed , in los angeles county , as well as at least of counties nationwide ,the bi is the index primarily used by fire department officials as a measure of overall wildfire hazard , and its use has been justified largely based on its observed empirical correlation with wildfire incidence and burn area in different regions [ haines et al.( ) ; haines , main and simard ( ) ; mees and chase ( ) ; andrews , loftsgaarden and bradshaw ( ) ] .however , several recent investigations have shown that the bi is far from an ideal predictor of wildfire incidence in los angeles county ; schoenberg et al .( ) showed that a simple point process model , which used only the same weather variables as those incorporated by the bi , vastly outperformed the bi in terms of predictive efficacy in los angeles county , using historical data from 19772000 .in fact , the simple model in schoenberg et al .( ) not only offered improvement in terms of likelihood scores such as the akaike information criterion ( aic ) , but the study suggested that substantial improvement in short - term forecasting could be achieved by the simple model using the weather variables directly , compared to a point process model that interpolates bi measurements .  here , we adopt the same basic modeling framework of schoenberg et al .( ) , but extend the models in two important ways .first , we consider not only daily weather variables but also additional covariates with management relevance , such as historical spatial burn patterns and wind direction , using the directional kernel regression method described in schoenberg and xu ( ) .second , unlike the simple models of mees and chase ( ) and schoenberg et al . ( )that average daily weather variables over weather stations within los angeles county , here we explore models that interpolate the records from different weather stations , weighting these data based on their spatial distance from the location where wildfire hazard is to be estimated .thus , the models considered here should have more direct relevance for forecasting wildfire hazard in precise spatial locations within los angeles county , compared to previous work that essentially averaged weather variables and hazard estimates over los angeles county as a whole . as with schoenberg et al .( ) , our results are compared with models using the bi measurements recorded at each of the weather stations , so that the effectiveness of the bi in summarizing the wildfire hazard as a function of the weather variables may be assessed .  while alternative models may be more useful for forecasting long - term wildfire hazard , that is , estimating the number of wildfires occurring within a month , season , or year , the focus here is on forecasting short - term wildfire hazard , that is , the probability of a wildfire occurring within a specific day . to compare the overall performance of the models considered , we employ diagnostics including likelihood - based numerical summaries such as the akaike information criterion ( aic ) , as well as power diagrams summarizing the predictive efficacy of each model for short - term forecasting .residual analysis is also used to highlight specific areas and times where the performance of a model is poor and to suggest areas for improvement .the paper proceeds as follows .section [ sec2 ] describes the wildfire and weather data that are used in the analysis .the models used , as well as methods for their estimation , are outlined in section [ sec3 ] , and methods for goodness - of - fit assessment are discussed in section [ sec4 ] .section [ sec5 ] presents the main results , and a discussion is given in section [ sec6 ] .los angeles county is an ideal test site for models for wildfire hazard , with detailed wildfire data having been collected and compiled by various agencies , including the los angeles county fire department ( lacfd ) and the los angeles county department of public works , the santa monica mountains recreation area , and the california department of forestry and fire protection .regional records of the occurrence of wildfires date back to 1878 , and include information on each fire , including its origin date , the polygonal outline of the resulting area burned , and the centroidal location of this polygon .lacfd officials have noted that the records prior to 1950 are believed to be complete for fires greater than  km ( 100 acres ) , and data since 1950 are believed to be complete for fires burning greater than  km , or 10 acres [ schoenberg et al .( ) ] . as in schoenberg et al .( ) , our analysis in this paper is focused primarily on models for the occurrences of the 592 wildfires burning at least 0.0405 km recorded between january 1976 and december 2000 .the daily burn area is highly right - skewed and closely follows the tapered pareto distribution [ schoenberg , peng and woods ( ) ] . for further details , images of the spatial locations of these wildfires , and information about missing data ,see peng , schoenberg and woods ( ) .since 1976 , daily meteorological observations from the remote automatic weather stations ( raws ) were archived across the united states .the analysis here is based on sixteen raws located within los angeles county , california .the raws record daily measures of many meteorological variables , including air temperature , relative humidity , precipitation , wind speed , and wind direction [ warren and vance ( ) ] . summaries of these records are collected daily at 1300 hr and transmitted by satellite to a central archiving station .these daily raws data are used as inputs by the nfdrs in order to construct fire behavior components that are in turn combined to construct the bi .it should be noted that data were missing on certain days for several of the 16 raws , though the biases resulting from such missing data are likely to be small ; see peng , schoenberg and woods ( ) for details .we follow previous research including schoenberg et al .( ) in modeling the catalog of wildfire centroids in los angeles county as a realization of a point process that may depend on daily meteorological variables .we begin with a basic reference model using merely a spatial background rate and seasonal component , and a model using the burning index in addition to the spatial and seasonal background rates .we then introduce competing models that use daily meteorological variables recorded at the raws , and extend the research of schoenberg et al .( ) by including additional covariates , such as wind direction and fuel age .further , instead of averaging daily weather variables or the burning index over all weather stations within los angeles county , here we explore methods of obtaining an estimated spatial intensity at any location on any particular day by interpolating the meteorological variables from different weather stations , weighting each record based on its distance from the location in question .   a spatial - temporal point process is mathematically defined as a random measure on a spatial - temporal region , taking values in the nonnegative integers or infinity [ daley and vere - jones ( ) ] . in this frameworkthe measure represents the number of points falling in the subset of . since any analytical spatial - temporal point process is characterized uniquely by its associated conditional rate ( or intensity ) , assuming it exists , modeling of such point processes is typically performed by specifying a parametric model for this rate . for the casewhere the spatial region is planar , for any point in time and location in the plane , the conditional rate is defined as a limiting frequency at which events are expected to occur within time range and rectangle , conditional on the prior history , , of the point process up to time . for references on space time point processes and conditional rates , see , for example , daley and vere - jones ( ) or schoenberg , brillinger and guttorp ( ) .given a parametric function for , estimates of the parameters may be obtained by maximizing the log - likelihood function [ see schoenberg , brillinger and guttorp ( ) , page 1576 , or daley and vere - jones ( ) , page 232 , equation 7.24 ] : \,dn(t , x , y ) - \int^{t_1}_{t_0 } \int_x \int_y \lambda(t , x , y ; , \theta ) \,dy \,dx \,dt \\ & = & \sum^n_{i=1 } \log \lambda(t_i , x_i , y_i ; \theta ) - \int^{t_1}_{t_0 } \int_x \int_y \lambda(t , x , y ; \theta ) \,dy \,dx \,dt.\end{aligned}\ ] ] in the case of a poisson process , the intuition behind this formula is that reflects the likelihood associated with the observed events , and represents the probability of no events in any _ other _ portions of the spatial - temporal region , the full likelihood is the product of these two terms , and the logarithm of this product yields above . underrather general conditions , the maximum likelihood estimates ( mles ) are consistent , asymptotically normal , and efficient [ ogata ( ) ] , and estimates of their variance can be derived from the negative of the diagonal elements of the inverse hessian of the likelihood function [ ogata ( ) , rathbun and cressie ( ) ] . in most cases , explicit solutions for mlesare not available and iterative numerical optimization methods are used instead .   in this analysiswe explore several spatial - temporal point process models for predicting wildfire occurrence rates . as an initial baseline model , one may consider an inhomogeneous poisson process , where the conditional intensity at time and at location depends only on the season associated with time , as well as the background rate of wildfires for the location in question .that is , one may consider a baseline model such as where and are parameters to be estimated in modeling fitting .parametric or nonparametric methods can be used to estimate the seasonal pattern and spatial background .while nonparametric methods can be especially flexible for estimating complex patterns such as spatial burn averages , a possible drawback to such methods is their potential for overfitting , particularly when the same data are used for fitting and evaluation of the fit of the model . as in schoenberg et al .( ) , we propose estimating the spatial background for fires between 19762000 by kernel smoothing the centroidal locations of wildfires recorded during the previous 25 years , that is , from january 1950 to december 1975 .that is , where is a kernel function , is a bandwidth to be estimated in modeling fitting , indicates the spatial coordinates of the wildfire between 1950 and 1975 , is the number of observed 19511975 wildfire occurrences , and is the euclidian distance between and .standard kernel functions can be used , and attention is usually limited to functions that are unimodal , symmetric about zero , and that integrate to 1 , such as the gaussian density of the epanechnikov kernel [ hrdle ( ) ] .it is well known that the results are far more sensitive to the choice of bandwidth than the choice of kernel function , and much research has focused on automated methods for choosing bandwidth parameters , including cross - validation , penalty functions , and plug - in methods [ silverman ( ) ; hrdle ( ) ] . here , since the data used in the estimation of is distinct from that used in the rest of the model fitting and in the evaluation , the problem of overfitting is far less severe , and the bandwidth parameter may simply be fitted by maximum likelihood .figure [ fig1 ] shows an estimate of the spatial background rate , with bandwidth estimated by maximum likelihood .one sees the general pattern of fire activity in los angeles county during 19511975 , with most fires occurring in the angeles national forest , as well as parts of the los padres national forest and the santa monica mountains , while many other wildfires were located in or near buckweed , santa clarita , and glendale , california ., with centroid locations of wildfires occurring during 18781976 .( the spatial bandwidth is 0.6 miles ) . ]helmers , magku and zitikis ( ) propose a kernel - based estimate for the consistent estimation of a seasonal time series . here , in order to safeguard against overfitting , we propose estimating the seasonal pattern describing the overall seasonal variation of wildfire activity in a fashion similar to that used for the spatial background rate , that is , by kernel smoothing the times of wildfires during _ previous _ years : in the above equation , represents the date within the year associated with time , that is ,  is the number of days since the beginning of the year for time , is the time of the wildfire occurrence in the data set ( 19501975 ) , and is a bandwidth parameter to be estimated .a wrapped kernel function should be used so that , for instance , january 1 and december 31 are treated as one day apart . the bandwidth may be estimated by maximum likelihood , fitting the kernel smoothing of the 19501975 data to the 19762000 data set .this procedure may be preferable for relatively small data sets such as the one considered here in order to prevent overfitting .   for model ( [ eqn : model1 ] ) , using kernel regression with a wrapped gaussian kernel and bandwidth days , estimated by mle . ]figure [ fig2 ] displays the smoothed function applied to wildfire incidence in los angeles county from january 1950 to december 1975 , with bandwidth estimated by mle by fitting the resulting function to wildfire data from 19762000 .it is evident that the mean number of wildfires is highest between july and october and rapidly decreases during november and december , reaching its minimum in january and february .schoenberg et al . ( ) pointed out that the burning index typically assumes moderate values in december , january , and february , though few wildfires occur during these months .   to evaluate the potential of the burning index ( bi ) in predicting wildfire incidence, one may consider a model such as for some function which interpolates the bi records at time and location , since bi records are only available at fixed raws sites .different methods of interpolation are possible .one possibility is to average the bi records on day , weighing each by the distance between the raws and the location in question .that is , where is the bi value recorded at time from the station , are the coordinates of the station , represents the collection of stations for which bi records are available on day , and is a normalizing constant given by    as an alternative to the model ( [ eqn : model2 ] ) incorporating bi measurements , one may instead consider examining the direct impact on wildfire hazard estimates of meteorological variables used in the computation of the bi , by replacing the function in ( [ eqn : model2 ] ) by functions of the meteorological variables themselves .that is , one may consider models such as where takes into account the contribution of temperature ( ) , relative humidity ( ) , wind speed ( ) , and precipitation ( ) at time  from each raws where the data are available .  since nonlinearities have been detected in the dependence of burn area on climatic variables [ schoenberg et al .( ) ] , one may wish to avoid simple averaging of the meteorological variables in estimating wildfire hazard .instead , one option is to describe the association between each climatic variable and wildfire burn area by an explicit function and weight the information from each raws by the distance to the point to be estimated using kernel smoothing .this suggests a model such as where , , , and are records of temperature , relative humidity , directed wind speed , and precipitation , respectively , on day  at the raws , the parameters , and represent weights associated with these meteorological variables , , and are bandwidths to be estimated , and , and are normalizing constants .  note that the bandwidth parameters are somewhat different here than in ordinary kernel regression models . while ordinarily in kernel regression or kernel density estimation bandwidth parameters may not typically be estimated by maximum likelihood because the likelihood would tend to increase as the bandwidth shrinks to 0 [ silverman ( ) ] , here this is not the case .instead , the bandwidth parameters , and in model ( [ eqn : model3 ] ) merely control the spheres of influence of the relative weather stations in terms of the impact of each on wildfire hazard .that is , if is small , for instance , then each raws station s recorded temperature will affect the wildfire incidence more locally , whereas if is very large , then the wildfire hazard at any particular location will depend more closely on the average temperature throughout los angeles county .  . ]functional forms can be suggested for , and , by individually examining the empirical relationship between daily area burned and each of these variables . in order to smooth these relationships, one possibility would be to use local linear regression or segmented regression , since the relationships between wildfire burn area and temperature , precipitation , and other weather variables appear to have thresholds [ schoenberg et al .( ) ] .another possibility is to use kernel regression of daily area burned on the average temperature , relative humidity , and precipitation over all raws , respectively .for instance , the impact of temperature may be estimated via where is the area burned on the day during 19762000 , is the average temperature readings over all raws on that day , is the bandwidth of the kernel regression which can be selected by methods such as cross - validation or the plug - in method [ silverman ( ) ] , and is the number of days with records during this period .figure [ fig3 ] displays such kernel regression estimates of , , and .not surprisingly , one sees that daily area burned generally increases as temperature increases , and decreases as relative humidity and precipitation increase , though some local fluctuations are seen in the kernel regressions on temperature and relative humidity .these fluctuations are likely attributable to the high variability of the estimates due to the relatively small sample of large fires contained in the catalog .special care should be taken in estimating , since wind is directional , and this direction may provide important information related to wildfire incidence . one possible way to estimate the relationship between daily area burned and directional wind speed is via directional kernel regression , as outlined in schoenberg and xu ( ) .an example of a two - dimensional directional kernel is the von mises distribution suggested by mardia and jupp ( ) : where denotes the modified bessel function of the first kind and order 0 , is the directional center , and is known as the concentration parameter . following schoenberg and xu ( ) , the corresponding two - dimensional kernel regression function would then be estimated via where and represent the mean wind speed and wind direction , respectively , on day .cross - validation can be used to optimize the estimates of  , , and  .figure [ fig4 ] displays a kernel regression estimate of the relationship between daily burning area and daily mean wind direction , weighted by wind speed , averaged over all raws stations .the sharp increase in mean wildfire burn area , indicated by darker shading in figure [ fig4 ] , is very strongly associated with higher wind speeds .in addition , one sees from figure [ fig4 ] the extent to which winds from the northeast , which are often warm , dry santa ana winds , are associated with higher burn areas . since the impact on average wildfire burn area of wind direction might be different at distinct weather stations , one might wish to estimate distinct kernel regression functions  , one for each raws station .the model ( [ eqn : model3 ] ) described above is additive in each of the weather variables , implying that an extreme value in only one weather variable may lead to a high estimate of wildfire hazard on the corresponding day , which might be questionable .for instance , one would expect few large wildfires occur on days when temperatures are extremely high yet there is some moderate amount of precipitation and relative humidity .an alternative approach is to use a multiplicative component instead , where once again where now the fire weather term has the multiplicative form    in both model ( [ eqn : model3 ] ) and model ( [ eqn : model4 ] ) , , and may be estimated using kernel regression of burn area on average temperature , relative humidity , and precipitation over all weather stations , where each station has the same regression function .however , because of differences in the locations of the weather stations , including differing altitudes of these stations , some stations may have lower average temperatures or higher relative humidities than others throughout the year .hence , a particular temperature and relative humidity at one station might indicate a very different wildfire hazard than the same values observed at a different raws station . in order to deal with this ,one may estimate distinct kernel regression curves for each raws in model ( [ eqn : model3 ] ) and model ( [ eqn : model4 ] ) .that is , one might consider where or where the kernel regression functions such as for each station may be estimated as in models ( [ eqn : model3 ] ) and ( [ eqn : model4 ] ) , that is , by kernel regression of the total daily burn area in los angeles county against the temperature at station  .one may further improve the models by adding fuel age as a covariate .fuel age , or its proxy , the time since the location s last recorded burn , appears to have a nonlinear , threshold - type relationship with burn area [ peng and schoenberg ( ) ] .indeed , burn area appears to increase steadily with fuel age up to ages of approximately 2030 years [ peng and schoenberg ( ) ] .this suggests incorporating the contribution of fuel age into model ( [ eqn : model5 ] ) by a truncated linear function , that is , where is the fuel age at the space time pixel , and where  is an upper truncation time .fuel age may be incorporated similarly into model ( [ eqn : model6 ] ) as well : ( [ eqn : model1])([eq8 ] ) describe eight point process models that may be used to predict wildfire hazard at any time and location within los angeles county . in order to compare the performance of these models ,one commonly used method is the akaike information criterion ( aic ) , which is defined as , where is the log - likelihood and is the number of fitted parameters in the model .smaller values of aic indicate better fit .the aic makes a good trade - off between model complexity and overfitting by rewarding a higher likelihood while penalizing the addition of more parameters [ akaike ( ) ] .  the predictive capacity of competing point process modelsmay also be compared by examining the models performance on the 19762000 wildfire data , as suggested in schoenberg et al .( ) .consider a grid of space time cells , with each cell s center separated by some distance in space and a temporal distance , and let these cells represent locations and times where alarms may potentially be issued .for any such space time point , one may compute the estimated conditional intensity for a particular model . consider issuing an alarm if the value of is above some certain threshold .we say the alarm is successful if a wildfire occurs within the cell ; otherwise , it is a false alarm .the false positive rate of the alarms , defined as the proportion of cells without wildfires where exceeded the alarm threshold , can be compared to the true positive rate , that is , the proportion of wildfires occurring in cells where exceeded the alarm threshold , using traditional receiver operating characteristic ( roc ) curves .each possible alarm threshold represents a single point on the roc curve , and the resulting curve summarizes the potential efficacy of a model in forecasting wildfires .while numerical likelihood scores such as aic and roc curves can be useful in evaluating the overall performance of a point process model , neither method is useful at identifying particular times and locations where a model fits poorly or suggesting ways in which a model might be improved . for these purposes , it is useful to inspect plots of residuals , which may be defined as the difference between the number of events occurring in a certain space time interval and the integral of the estimated conditional intensity over the same interval [ baddeley et al .( ) ] .negative residuals indicate overestimates of wildfire hazard , and very large residuals indicate places and times where the model underestimated wildfire hazard .the maximum likelihood estimates of the parameters for the models ( [ eqn : model1])([eq8 ] ) are listed in tables [ table1 ] and [ table2 ] . in tables[ table1 ] and [ table2 ] , the parameter  was fixed at 22 years for models 7 and 8 , based on peng and schoenberg ( ) ; this parameter was also fit by maximum likelihood , yielding very similar results , so , for simplicity , here we report the fit of the model with  fixed at 22 years .the bandwidths in spatial background range from 0.25 km to 1.20 km and the bandwidth in the seasonal component fall within 8.6 to 34.1 days .the bandwidths related to spatially kernel smoothing the weather variables range from 0.024 km to 0.40 km in models ( [ eqn : model3 ] ) , ( [ eqn : model5 ] ) , and ( [ eq7 ] ) , with the smallest value for wind speed in model ( [ eq7 ] ) and the largest value corresponding to relative humidity in model ( [ eqn : model3 ] ) . as mentioned in section [ sec3 ], these bandwidths can perhaps be interpreted as reflecting the scales of influence of the weather variables in terms of their effect on wildfire incidence.=-1  .3d2.4d2.4d2.4d2.4d2.4d2.4cd2.5@ * model * & & & & & & & & & + ( [ eqn : model1 ] ) & 24 & 2.0 & & & & & & & + & ( 1.3 ) & ( 0.20 ) & & & & & & + ( [ eqn : model2 ] ) & 4.9 & 0.65 & & & & & & & + & ( 0.26 ) & ( 0.039 ) & & & & & & + ( [ eqn : model3 ] ) & 6.4 & 0.66 & & 0.18 & 0.21 & 1.0 & 0.15 & & + & ( 0.64 ) & ( 0.043 ) & & ( 0.017 ) & ( 0.012 ) & ( 0.071 ) & ( 0.010 ) & & + ( [ eqn : model4 ] ) & 6.4 & 2.7 & & & & & & & + & ( 0.31 ) & ( 0.24 ) & & & & & & ( 130 ) & + ( [ eqn : model5 ] ) & 13 & & 0.60 & 0.58 & 0.19 & 17 & 2.1 & & + & ( 0.84 ) & & ( 0.051 ) & ( 0.054 ) & ( 0.011 ) & ( 0.85 ) & ( 0.14 ) & & + ( [ eqn : model6 ] ) & 12 & 1.0 & & & & & & & + & ( 0.15 ) & ( 0.057 ) & & & & & & ( 260 ) & + ( [ eq7 ] ) & 6.9 & 0.60 & & 0.19 & 0.21 & 52 & 0.71 & & 1.0 + & ( 0.44 ) & ( 0.054 ) & & ( 0.018 ) & ( 0.020 ) & ( 2.4 ) & ( 0.068 ) & & ( 0.066 ) + ( [ eq8 ] ) & 4.8 & 0.54 & & & & & & & 0.10 + & ( 0.37 ) & ( 0.049 ) & & & & & & ( ) & ( 0.0094 ) +  .6d2.3d2.6d2.6d2.5d2.4d2.6d2.6@ * model * & & & & & & & & + ( [ eqn : model1 ] ) & 1.20 & 9.86 & & & & & & + & ( 0.004 ) & ( 2.8 ) & & & & & & + ( [ eqn : model2 ] ) & 0.92 & 8.64 & 0.40 & & & & & + & ( 0.00077 ) & ( 3.30 ) & ( 0.0055 ) & & & & & + ( [ eqn : model3 ] ) & 0.36 & 29.6 & & 0.37 & 0.40 & 0.30 & 0.24 & + & ( 0.002 ) & ( 5.6 ) & & ( 0.002 ) & ( 0.001 ) & ( 0.004 ) & ( 0.001 ) & + ( [ eqn : model4 ] ) & 0.92 & 8.6 & & & & & & 0.03 + & ( 0.1 ) & ( 2.3 ) & & & & & & ( 0.003 ) + ( [ eqn : model5 ] ) & 0.34 & 34 & & 0.31 & 0.20 & 0.04 & 0.28 & + & ( 0.002 ) & ( 8.3 ) & & ( 0.001 ) & ( 0.001 ) & ( 0.002 ) & ( 0.0005 ) & + ( [ eqn : model6 ] ) & 0.25 & 20 & & & & & & 0.19 + & ( 0.025 ) & ( 9.7 ) & & & & & & ( 0.003 ) + ( [ eq7 ] ) & 0.46 & 19 & & 0.36 & 0.39 & 0.024 & 0.20 & + & ( 0.00069 ) & ( 5.2 ) & & ( 0.00071 ) & ( 0.0011 ) & & ( 0.00025 ) & + ( [ eq8 ] ) & 0.99 & 13 & & & & & & 0.037 + & ( 0.17 ) & ( 2.6 ) & & & & & & ( 0.00064 ) +  table [ table3 ] presents the relative aic values for models ( [ eqn : model1])([eq8 ] ) . for simplicity and ease of presentation, the aic for the best fitting model ( [ eq8 ] ) has been subtracted from the aic of each model .it is evident that the bi model ( [ eqn : model2 ] ) offers very substantial improvement over the baseline model ( [ eqn : model1 ] ) .however , all the other models that use weather information directly have much better fits than the bi model ( [ eqn : model2 ] ) .the multiplicative model ( [ eq8 ] ) with fuel age appears to offer by far the best fit among these models , without using the bi directly , and only involves one more fitted parameter than the bi model .figure [ fig5 ] shows a comparison of the predictive efficacy of models described in section [ sec3 ] .models ( [ eqn : model6 ] ) and ( [ eq8 ] ) vastly outperform the other models .the performance was evaluated using a regular space time grid , so that each alarm s success or failure was evaluated over a space time window with  km and day ..relative aic values [ cols="<,^,^,^,^,^,^,^,^ " , ]   ) ([eq8 ] ) , with day and km . ]  for any given success rate , the models that directly use the meteorological data offer substantially fewer false alarms than the model ( [ eqn : model2 ] ) that uses the bi .for instance , for a false positive rate fixed at , model ( [ eq8 ] ) correctly signals approximately of the wildfires in the data set , compared to for model ( [ eqn : model2 ] ) .model ( [ eqn : model6 ] ) , which uses only temperature , relative humidity , wind speed , wind direction , and precipitation , but does not use fuel age , signals nearly of the wildfires correctly with a false positive rate of .note that this method of evaluating predictive efficacy over a fine grid of spatial - temporal locations is rather cumbersome for models ( [ eq7])([eq8 ] ) , due to the need to individually estimate the fuel age associated with each wildfire , with respect to each spatial - temporal grid location and time , and each such evaluation requires a rather burdensome computation described in peng and schoenberg ( ).  days . ]  days . ]the fit of the models can be evaluated by examining their spatial - temporal residuals over a relatively coarse grid .for instance , figure [ fig6 ] shows the medians of the absolute values of the residuals in each month , for models ( [ eqn : model1])([eq8 ] ) , where each residual is computed over a space time grid of sqkm  days .it is evident that models ( [ eqn : model3 ] ) , ( [ eqn : model5 ] ) , ( [ eqn : model6 ] ) , ( [ eq7 ] ) , and ( [ eq8 ] ) outperform the other three models , especially in the late summer and fall months when most wildfires in los angeles county occur .the months of october and november are especially critical , since santa ana winds prevail and can cause catastrophic wildfires .figure [ fig7 ] shows a spatial plot of the medians of the absolute values of the residuals , over all months . from figure [ fig7 ]one sees that the eight models perform surprisingly comparably in terms of the median absolute residual , though models ( [ eqn : model3])([eq8 ] ) , which use meteorological data directly , have better performance than model ( [ eqn : model2 ] ) in the sense that the corresponding residuals are generally closer to zero in most areas .these residual plots also indicate that several of the models may require improvement in the northwest portion of the angeles national forest , as well as near the border with san bernardino county on the eastern side of los angeles county . from the residuals in figure [ fig7 ], one can observe that the residuals for models ( [ eqn : model5 ] ) , ( [ eq7 ] ) , and ( [ eq8 ] ) are highly concentrated around zero except a few large values that occur in the northwest part and east part of los angeles county .the models explored here use the identical information recorded at the raws stations and used as inputs into the computation of the bi .hence , it seems relevant to compare the fit of such models with comparable spatial interpolations of bi measurements , and the fact that the models using the weather variables directly appear to offer superior fit suggests that the bi may not be effective as a short - term forecasting measure of wildfire hazard in los angeles county .it should be noted that the empirical relationship between a fire danger rating index such as the bi and wildfire incidence is only one way to evaluate the effectiveness of such an index ; alternatives may include assessing the cost - effectiveness of staffing or other decisions made based on the index .furthermore , the use of fire danger ratings by fire department officials for wildfire suppression and prevention activities may confound the empirical relationship between fire danger ratings and observed wildfire activity .nevertheless , most evaluation studies of fire danger rating systems relate such indices to ultimate fire responses , including fire incidence and fire size .indeed , andrews and bradshaw ( ) , whose work was instrumental in the current implementation of the bi , suggested that the value of a fire danger index be evaluated according to its relationship with fire activity , which may be defined as the incidence of large wildfires .such empirical relationships have been used as support for the use of such rating systems for predictive purposes [ haines et al .( ) ; haines , main and simard ( ) ; mees and chase ( ) ; mandallaz and ye ( , ) ; viegas et al .( ) ; andrews , loftsgaarden and bradshaw ( ) ] .the results here suggest that , for the purpose of forecasting wildfire hazard , point process models using raws records and previous wildfire activity as covariates may represent a promising alternative to existing indices that use essentially the same information .however , we must emphasize that the point process models proposed here remain rather simplistic and could potentially be improved by incorporating a host of other important variables , such as detailed vegetation type , vegetation cover , soil characteristics , other weather variables such as cloud cover and lightning , as well as human factors such as land use and public policy .the exclusion of such variables from this analysis is solely motivated by our aim to optimize forecasts given the same remote , automatically - recorded information used in the computation of the bi .the models considered here could also perhaps be improved in various ways .for instance , one might allow long - term temporal trends and/or allow the seasonal component to vary from year to year .in addition , one may consider estimating the kernel function in models ( [ eqn : model5 ] ) and ( [ eqn : model6 ] ) for each station using only local wildfires close to the corresponding station , or perhaps by some more sophisticated weighting scheme where nearby fires are given higher weight in the estimation of this function .because daily burn areas are right - skewed [ schoenberg , peng and woods ( ) ] , perhaps kernel regressions where the response variable is some transformation of the daily burn area might yield superior results .an additional important direction for future work is the exploration of similar point process models for wildfire occurrences in other locations and for other vegetation types or alternative wildfire regimes , as well as the use of such models for actual _ prospective _ predictions of wildfire activity , rather than merely the empirical assessment of goodness of fit to historical data .thanks to larry bradshaw at the usda forest service for generously providing us with raws data and helping us to process it . thanks also to james woods , roger peng , and members of the lacfd and ladpw ( especially mike takeshita , frank vidales , and herb spitzer ) for sharing their data and expertise .andrews , p. l. and bradshaw , l. s. ( 1997 ) .fires : fire information retrieval and evaluation system a program for fire danger rating analysis .technical report int - gtr-367 .ogden , ut ; u.s . dept .agriculture , forest service , intermountain research station .64 p.    bradshaw , l. s. , deeming , j. e. , burgan , r. e. and cohen , j. d. ( 1983 ) . _ the 1978 national fire - danger rating system : technical documentation ._ united states dept .agriculture forest service general technical report int-169 .intermountain forest and range experiment station , ogden , utah .46 p.       haines , d. a. , main , w. a. and simard , a. j. ( 1986 ) .fire - danger rating observed wildfire behavior in the northeastern united states .pap . nc-274 .paul , mn : u.s .agriculture , forest service , 23 p.     keeley , j. e. and fotheringham , c. j. ( 2003 ) .impact of past , present , and future fire regimes on north american mediterranean shrublands . in : _ fire and climatic change in temperate ecosystems of the western americas _ ( t. t. veblen , w. l. baker , g. montenegro and t. w. swetnam , eds . ) 218262 .springer , new york .schoenberg , f. p. , brillinger , d. r. and guttorp , p. m. ( 2002 ) .point processes , spatial - temporal . in _ encyclopedia of environmetrics _ ( a. el - shaarawi and w. piegorsch , eds . ) * 3 * 15731577 .wiley , new york .schroeder , m. j. , glovinsky , m. , hendricks , v. , hood , f. , hull , m. , jacobson , h. , kirkpatrick , r. , krueger , d. , mallory , l. , oertel , a. , reese , r. , sergius , l. and syverson , c. ( 1964 ) .synoptic weather types associated with critical fire weather .institute for applied technology , national bureau of standards , u.s .commerce , washington , dc .warren , j. r. and vance , d. l. ( 1981 ) .remote automatic weather station for resource and fire management agencies .united states dept .agriculture forest service technical report .intermountain forest and range experiment station .

Abstract: the burning index ( bi ) produced daily by the united states government s national fire danger rating system is commonly used in forecasting the hazard of wildfire activity in the united states . however , recent evaluations have shown the bi to be less effective at predicting wildfires in los angeles county , compared to simple point process models incorporating similar meteorological information . here , we explore the forecasting power of a suite of more complex point process models that use seasonal wildfire trends , daily and lagged weather variables , and historical spatial burn patterns as covariates , and that interpolate the records from different weather stations . results are compared with models using only the bi . the performance of each model is compared by akaike information criterion ( aic ) , as well as by the power in predicting wildfires in the historical data set and residual analysis . we find that multiplicative models that directly use weather variables offer substantial improvement in fit compared to models using only the bi , and , in particular , models where a distinct spatial bandwidth parameter is estimated for each weather station appear to offer substantially improved fit .  and  .


solute dispersion in porous media is of importance in many fields of science , such as chemistry , groundwater hydrology , oil recovery , etc .tracers , dissolved in a fluid flowing through a porous material , will experience dispersion in both the longitudinal ( downstream ) and transverse directions , due to thermal or fickian diffusion and the variability in both flow velocity and direction in the pores .tracers injected at one point ( e.g. , a leaking tank polluting groundwater ) will thus spread out into a plume .  over the past decades, many models have been proposed to model tracer transport .most models relate the dispersion coefficients to the flow velocity ( ) , a characteristic length scale ( the grain size or pore length ) , and the molecular diffusion coefficient ( ) via the dimensionless pclet number pe . due to the use of the microscale , the pclet number is a microscopic property .the most basic approach to computing the spatiotemporal evolution of the concentration of tracers is provided by the advection - diffusion equation ( ade ) , using separate values of and for the dispersion coefficients in the longitudinal and transverse direction , respectively .to explain the apparently anomalous behavior of the dispersion coefficients observed in experiments , more sophisticated models have been developed , e.g. , flow through random capillaries  .the analysis of  provided an initial approach to evaluate the macroscopic effects of microscopic pore geometry by means of an averaging procedure .percolation theory and continuous time random walks allowed handling long range correlations  .a range of numerical methods have also been applied to the problem , such as network models in which pore connectivity plays an essential role ( e.g. , ) .finally , full microscopic modeling of incompressible flow through porous materials has become feasible thanks to modern computing , and considerable successes have been obtained with this method ( e.g. , ) .  in this paperwe propose a new approach to understand the first - order relationship between dispersion and flow velocity ( and hence pclet number ) .it is based on a straightforward physical argument involving the competition of diffusion and advection in the pore channels , and leads to an accurate prediction of the observed behavior over the full relevant range of pclet numbers .the analysis also reveals the existence of a new material property related to the cited competition and the mesoscopic structure of pore geometry .consider the laminar flow of fluid through a regular or random homogeneous porous material .the flow is produced by a pressure difference ( head ) applied at opposite ends of the sample of porous material .this pressure difference induces a complex flow pattern inside the pores that can be computed exactly using the appropriate equations for an incompressible fluid , with appropriate boundary conditions , cf . .  due to the complexity of the pore structure ,the flow lines go apart and come together again as they traverse the material , dictated by the pressure head and the pore structure , so that the dispersion of the longitudinal distance among different flow lines tends to grow linearly as a function of the mean longitudinal distance , taken in the direction of the negative pressure gradient .regardless of the complexity of the flow pattern inside the porous material , the incompressible flow at every point is linear in the applied pressure head , so the geometry of this flow pattern does not vary as the head is varied .this leads to what is known as ` mechanical ' dispersion , with a diffusion coefficient given by where the mean longitudinal flow velocity has been taken as a measure for the pressure head , and is a geometric proportionality constant .thus , the longitudinal dispersion is linear in the mean longitudinal flow velocity . a similar argumentcan also be applied to the transverse dispersion , that is likewise linear in the flow velocity , but with a different ( smaller ) proportionality constant due to the significant geometrical angle between the direction of the mean pressure gradient and the direction of the transport : .  into this flow ,tracers are injected .these tracers are not merely advected with the flow , but in addition are subject to random ( thermal ) brownian motion . at zero or very small flow ,pure thermal diffusion will result , with diffusion coefficient . at large flow velocity , but still in the laminar flow regime , the thermal motion of the tracers can be neglected with respect to the advection , and nearly pure ` mechanical ' dispersion will result , or .we expect there to be an intermediate or transitional regime , in which global tracer dispersion should lie somewhere between the thermal and mechanical diffusion limits , but _ a priori _ it is not clear how exactly this dispersion scales with flow velocity .  following a tracer path through the pore channels of the porous material , there will be channels oriented at a small angle to the driving force , in which the mechanical dispersion will tend to dominate over thermal diffusion .other channels will be oriented nearly at right angles to the driving force , and there , thermal diffusion will tend to dominate over mechanical diffusion .thus , we propose modeling the mixed behavior in this intermediate regime by assuming that transport in each of the successive pore channels traversed by a tracer is dominated by one of the two mechanisms .in other words , we assume that a tracer alternately experiences thermal and mechanical diffusion .thus , the tracer experiences thermal diffusion for a fraction of time , with thermal diffusion coefficient . alternating with these phases of thermal diffusion, the tracer is advected by the flow for a fraction of time , during which it experiences a mechanical dispersion ( proportional to ) . therefore , the total dispersion of the longitudinal distance is the weighted sum of these two : so the net or total longitudinal diffusion is now , as the net longitudinal flow velocity increases , we make the essential assumption that the time fraction becomes progressively longer with respect to as increases .this is motivated as follows . as the drive is increased , transport in ever more pore channelsbecomes dominated by mechanical dispersion .so the time fraction ratio of mechanical to thermal dispersion a tracer experiences will be an increasing function of the velocity : where is a monotonically increasing function of such that and , and is a critical velocity ( the velocity where mechanical diffusion starts to dominate globally over thermal diffusion ) . in the limit , one has , and . in the limit , one has , and .both these limits agree with expectation . in the following, we will set for simplicity , but leave open the possibility that future studies may reveal more complex functional dependencies .to better understand the intermediate regime , we insert , and find which is recognized as a pad approximation .( [ longitudinal ] ) has _asymptotic behavior . for ,one has . for ,one has .but there is also an intermediate regime where .this regime occurs for and . summarizing : the intermediate asymptote only appears in full when the two corresponding limits are sufficiently far apart , i.e. , when .physically , the left and right asymptotes correspond to the limits in which one of the transport mechanisms ( thermal diffusion and mechanical dispersion , respectively ) dominates .the intermediate regime arises when the strength of the mechanical dispersion increases simultaneously with the fraction of time that the tracer is experiencing mechanical dispersion as compared to thermal diffusion .experimental data are commonly expressed in terms of versus pe , where is the molecular diffusion coefficient and pe is the dimensionless pclet number .the above expressions can be recast in this dimensionless form by substituting and \{pe , pe}. thus , the dimensionless form of eq .( [ longitudinal ] ) is :   to test the validity of eq .( [ longitudinal_dl ] ) , we have fitted our expression to the numerical simulation data of and ( based on micro computer tomography ( ct ) scans of porous sandstone ) . due to the very low noise level of the numerical simulation ,this test is probably more stringent than testing the model against data from laboratory experiments .the fits , shown in fig .[ ovaysi ] , are a very good match to the simulated data .note , in particular , that the intermediate region ( with logarithmic slope ) is reproduced in full detail .transverse dispersion is different from longitudinal dispersion in that the drive is not very efficient in the transverse direction .therefore , the transverse mechanical dispersion is not able to compete with thermal diffusion , and thermal diffusion can not be neglected along any part of the tracer trajectory .the drive , however , is only effective in pore channels directed at angles less than 90 with respect to the mean flow vector , during a fraction of time .thus , we get which expresses that thermal diffusion is always operative in the transverse direction , while mechanical dispersion ( with characteristic diffusion coefficient ) is operative only in specific channels , i.e. , for a fraction of the total time . due to the relative inefficiency of the drive, this approach is expected to yield less precise results than eq .( [ longitudinal ] ) , and minor corrections to this expression may be needed  .this is left to future work .following the same reasoning as before , we immediately find   we have assumed that ( or ) has the same value here as with longitudinal dispersion , based on the assumption that the geometrical flow pattern is one and the same for longitudinal and transverse dispersion , although this is only strictly true in the limit of small thermal diffusion .the transverse dispersion has two extreme asymptotes that are equivalent to those of longitudinal dispersion , namely : for , one has , and for , one has .however , here there is no ` intermediate asymptote ' , but only a very gradual transition from one asymptote to the other , specified via the factor containing in eq .( [ transverse ] ) .the transition between asymptotes occurs around .   in dimensionless form , eq .( [ transverse ] ) becomes :   a fit of the analytic expressions to an ample collection of measurement data is shown in fig .[ delgado ] .the figure contains data from 18 different experiments on longitudinal dispersion and 15 on transverse dispersion , spanning an ample range of the schmidt number ,  .the parameters , pe and were determined from a least - squares fit to the longitudinal dataset . for the transverse dataset ,the parameters and pe were held fixed at these values , while only was varied . by making a joint fit to all data ,the obtained parameters represent mean values over the various individual datasets .this procedure illustrates the generic dispersional behavior , covering a range of different materials .but much better results , with less scatter , are to be expected for fits to an individual dataset .  to illustrate the preceding remark , fig .[ carvalho ] shows an example of a fit to an individual longitudinal dispersion dataset .the fit is rather satisfactory .in previous work , the available experimental data were analyzed by subdividing the range of pe numbers into individual regimes and describing the dispersional behavior in each of these sections heuristically ( using , e.g. , power - law expressions  ) .  in fig .[ exponent ] we show ( top ) the typical shape of the analytical curves , eqs .( [ longitudinal_dl ] ) and ( [ transverse_dl ] ) , with parameters similar to those used in the examples above ( , pe , , ) .the bottom panel in this figure shows the local power - law exponent , computed as the evolution of the local power - law exponent with pe number displays roughly the same behavior as the successive regimes described in .the intermediate regime does not develop fully as the condition is only marginally fulfilled . thus , the intermediate power - law exponent for longitudinal dispersion only reaches a value of about 1.2 , instead of the maximum of 2 predicted by the model .this value ( 1.2 ) is consistent with the value cited in literature for the corresponding range of pe numbers  .in this work , we present a unified model for longitudinal and transverse tracer dispersion in laminar flow in porous media . based on a straightforward physical argument , an analytical expression is obtained that describes the observed behavior over the full available range of laminar flow velocities or pclet numbers .  in literature , it has been customary to use power - law scalings to describe the behavior of , each of which being valid only in a limited range of pe values  .the generic appearance of power - law scalings seemed to indicate that tracer dispersion in porous media was an anomalous transport process , characterized by fractional exponents  .however , in view of the present work , it seems that no anomalous transport mechanisms need be invoked to explain the observed behavior .this is a very satisfactory situation , at least for sandbox or glass bead experiments , in which the structure of the medium is not patently fractal .the present work has revealed the existence of two numbers that determine the dispersion curves : the critical velocity or pclet number ( or pe ) and .pe is a material property that depends on the pore geometry , so that different materials will have different pe values .the exploration of the dependence of pe on other material properties may provide further insight into its significance .it is of interest to note that we generally find that pe , rather than equal to one , as one might expect for the transition from dominant diffusive to dominant advective transport  .this indicates that the scale at which this transition occurs is larger ( by a factor pe ) than the grain or pore scale .also note that the family of three - parameter curves given by and may explain a significant part of the observed data spread mentioned in e.g.  .finally , the intermediate region and the maximum value of the local longitudinal power - law exponent depend exclusively on .the analysis we present is consistent with the idea , proposed also by other authors  , that the main effects of porosity on dispersion can be modelled using a single characteristic mesoscopic length scale ( pe ) , and that the details of the microscopic pore geometry and inertial effects only lead to corrections to this generic behavior .the present modeling approach allows determining this mesoscale from the dispersion data . the present work does not pretend to provide the same modeling precision as the empirical scalings provided in the ample literature , but merely to provide improved insight into the generic behavior of tracer dispersion in driven laminar flow through porous mediathis study was carried out within the framework of dgmk ( german society for petroleum and coal science and technology ) research project 718 `` mineral vein dynamics modelling '' , which is funded by the companies exxonmobil production deutschland gmbh , gdf suez e&p deutschland gmbh , rwe dea ag and wintershall holding gmbh , within the basic research program of the weg wirtschaftsverband erdl- und erdgasgewinnung e.v .we thank the companies for their financial support and their permission to publish these results .this research was partially supported by grant ene2009 - 07247 , ministerio de ciencia e innovacin ( spain ) .

Abstract: in this work , we present a novel analytical model for tracer dispersion in laminar flow through porous media . based on a straightforward physical argument , it describes the generic behavior of dispersion over a wide range of pclet numbers ( exceeding 8 orders of magnitude ) . in particular , the model accurately captures the intermediate scaling behavior of longitudinal dispersion , obviating the need to subdivide the dispersional behavior into a number of disjunct regimes or using empirical power law expressions .   the analysis also reveals the existence of a new material property , the critical pclet number , which reflects the mesoscale geometric properties of the microscopic pore structure .


in this work we study the mixed formulation of poisson s equation with a multiscale diffusion coefficient , i.e. where the diffusion coefficient is highly varying on a continuum of different scales . for such coefficients ,the solution is typically also highly varying and standard galerkin methods fail to converge to the correct solution , unless the features on the finest scale are resolved by the underlying computational mesh .a classical application is the flow in a porous medium , modeled by darcy s law . in this case , the multiscale coefficient describes a permeability field , which is heterogeneous , rapidly varying and has high contrast .classical discretizations that involve the full fine scale often lead to a vast number of degrees of freedom , which limits the performance and feasibility of corresponding computations . in this paper, we address this kind of problems in the context of mixed finite elements .we will interpret the mixed formulation of poisson s equation in a darcy flow setting , referring to the vector component as flux , and the scalar component as pressure . in darcy flow applicationsthe flux solution is of particular interest since it tells us how a fluid is transported through the medium .it is desirable and common to use flux conservative discretization schemes .the proposed method is based on the raviart thomas finite element which is locally flux conservative . concerning the mixed formulation of poisson s equation , corresponding multiscale methods were for instance proposed in .these methods are based on the raviart thomas finite element and fit into the framework of the multiscale finite element method ( msfem , cf .another family of multiscale methods is derived from the framework of the variational multiscale method ( vms ) .multiscale methods for mixed finite elements based on vms are proposed and studied in .inspired by the results presented in , a new multiscale framework arose .we refer to this framework as localized orthogonal decomposition ( lod ) .it is based on the idea that a finite element space is decomposed into a low dimensional space that incorporates multiscale features and a high dimensional remainder space which is given as the kernel of an interpolation or quasi - interpolation operator .the multiscale space can be used for galerkin - approximations and allows for cheap computations .various realizations have been proposed so far . for corresponding formulations and rigorous convergence results for elliptic multiscale problems ,we refer to for galerkin finite element methods , to for discontinuous galerkin methods and to for galerkin partition of unity methods . among the various applications we refer to the realizations for eigenvalue problems , for semilinear equations , for the wave equation and for the helmholtz equation .  in this paperwe introduce a two level discretization of the mixed problem , that is we work with two meshes : a fine mesh ( mesh size ) which resolves all the fine scale features in the solution and a coarse mesh ( mesh size ) which is of computationally feasible size .this gives us a fine and a coarse raviart thomas function space for the flux .we denote them respectively by ( high dimensional ) and ( low dimensional ) .the kernel of the ( standard ) nodal raviart thomas interpolation operator onto is the detail space .this space can be interpreted as all fine scale features that can not be captured in the coarse space .a low dimensional ideal multiscale space is constructed as the orthogonal complement to the divergence free fluxes in .we prove that this space has good approximation properties in the sense that the energy norm of the error converges with without pre - asymptotic effects due to the multiscale features .however , the basis functions of the ideal multiscale space have global support and are expensive to compute .we show exponential decay of these basis functions allowing them to be truncated to localized patches with a preserved order of accuracy for the convergence .the resulting space is called the localized multiscale space .the problems that are associated with the localized basis functions have a small number of degrees of freedom and can be solved in parallel with reduced computational cost and memory requirement .once computed , the low dimensional localized multiscale space can be reused in a nonlinear or time iterative scheme .we prove inf - sup stability and a priori error estimates ( of linear order in ) for both the ideal and the localized method .the local -instability of the nodal raviart thomas interpolation operator leads to instabilities as decreases for the localized method .we show that these instabilities can be compensated by increasing the patch size or using clment - type interpolators instead . in the numerical examples we verify that the localized method has the theoretically derived order of accuracy .we confirm our theoretical findings by performing experiments on the unit square and an l - shaped domain , as well as using a diffusion coefficient with high contrast noise and channel structures .the proposed method is also compared numerically with results from an msfem - based approach using a permeability field from the spe10 benchmark problem .we consider a bounded lipschitz domain ( dimension or ) with a piecewise polygonal boundary and let denote the outgoing normal vector of . for any subdomain , we shall use standard notation for lebesgue and sobolev spaces , i.e. for  ] .we equip this space with the usual norm , where .additionally , for , we introduce the subspace of functions with zero flux on the boundary , where should be interpreted in the sense of traces .we denote by the quotient space of by .the continuous dual space of a banach space is denoted by .   with these definitions we are ready to state the continuous problem , which is poisson s equation in mixed form with neumann boundary conditions on the full boundary .find , such that for all , .we pose the following assumptions on the coefficient and data . [ ass : data ]  1 .^{d \times d} ] .we introduce the -conforming lowest ( zeroth ) order raviart thomas finite element . for each fine element and coarse element ,the spaces of raviart thomas shape functions are given by ^d + x { \mathbb{p}^{0}(t)}\ } \text { and } \\  { \mathcal{rt}}_{h}(t ) & = \{{\mathbf{v}}|_t = [ { \mathbb{p}^{0}(t)}]^d + x { \mathbb{p}^{0}(t)}\ } ,  \end{aligned}\ ] ] respectively , where is the space coordinate vector .the raviart thomas finite element spaces on and are then defined as the degrees of freedom ( in the coarse and fine raviart thomas spaces ) are given by the averages of the normal fluxes over the edges ( respectively faces for ) .we denote the degrees of freedom by for the fine and coarse discretization , respectively .the direction of the normal ( respectively ) can be fixed arbitrarily for each edge ( respectively face ) . here , and are bounded linear functionals on the space , for some .note , that the additional regularity ( i.e.  for ) is necessary for the edge integrals to be well - defined ( cf .we introduce the ( standard ) nodal raviart thomas interpolation operators and by fixing the degrees of freedom in the natural way , i.e.  and are defined such that additionally , we let be the space of all piecewise constant functions on and with zero mean . we denote by and the -projections onto and , respectively . using the fine spaces , we define the fine scale discretization of , which will be referred to as the reference problem .find and , such that for all and .a similar problem can be stated with the coarse spaces and with flux solution .the remainder of this section treats only the fine discretization .however , all results hold also for the coarse discretization .we denote the space of divergence free functions on the fine grid by   [ rem : kernel ] a natural definition of for our purposes is . however , since we have for all ( due to the definition of the raviart thomas element ) , we can characterize equivalently as done in .  to establish existence and uniqueness of a solution to the reference problem, we use that is divergence compatible , i.e. we have the commuting property for , and that is bounded on ( but not on ! ) , i.e. there exists a generic -independent constant such that for . using this ,the inf - sup stability of with respect to and follows : for , where is chosen such that and .this is possible by letting for a solution to in with homogeneous neumann boundary conditions .now , applying lemma [ lem : existence ] with , , , we can derive the constants , and and establish existence and uniqueness of a solution to the reference problem .note that the inf - sup stability constant is independent of and hence also holds for the pair of spaces and .  in the following , we are mainly interested in approximating the flux component of the solution .we treat as a reliable reference to the exact solution .note that the -norm of the divergence error is controlled by the data for the energy norm of the flux error , we have the following error estimate in the energy norm for the lowest order raviart thomas element : where is independent of . for a problem with a coefficient that has fast variations at a scale of size , we have in general that .hence , we require before we can observe the linear convergence in numerically .we call the regime with a pre - asymptotic regime . the goal of this work is the construction of a discrete space which does not suffer from such pre - asymptotic effects triggered by . in the following ,we assume that the fine mesh is fine enough ( in the sense that ) so that is sufficiently small and hence a sufficiently accurate reference solution . with the same argument, the accuracy of the coarse solution will not be satisfying as long as .note that reference problem never needs to be solved .it just serves as a reference .  in the next section, we will construct the ideal multiscale space of the same ( low ) dimension as , but which yields approximations that are of similar accuracy as the reference solution ( in particular in the regime ) . throughout the paper, we do not consider errors that arise from numerical quadrature .for simplicity , we assume that all integrals can be computed exactly .in this section , we construct a low dimensional space that can capture the fine scale features of the true multiscale solution . we focus on constructing a good multiscale representation of the flux solution only .we call it ideal since the reference flux solution is in this space for all .this should be contrasted to a localized multiscale space to be introduced in section [ sec : localized ] .in addition to the spaces and defined above we introduce the following detail space as the intersection of the fine space and the kernel of the coarse raviart thomas interpolation operator ,  is the kernel of a projection , it induces the splitting , where is low dimensional and is high dimensional .we refer to as the detail space . in this sectionwe aim at constructing a modified splitting , where is replaced by a multiscale space which incorporates fine scale features .we will construct the ideal multiscale space by applying fine scale correctors to all coarse functions in , i.e. so that is the desired multiscale space for a linear corrector operator .the corrector operator is constructed using information from the coefficient , and has divergence free range in order to keep the flux conservation property of the coarse space .the definition of the corrector requires us to construct the splitting with next , we introduce an ideal corrector operator .we distinguish between local ( element - wise ) correctors and a global corrector .[ def : idealcorrector ] let for .for each such , we define an _ ideal element corrector operator _ by the equation for all .furthermore , we define the _ ideal global corrector operator _ by summing the local contributions , i.e.  .the ideal corrector operators are well - defined since equation is guaranteed a unique solution by the lax milgram theorem due to the coercivity and boundedness of on . using the ideal global corrector operator, we can define the discrete multiscale function space by where is the identity operator .this space has the same dimension as .furthermore , it allows for the splitting .note that the ideal multiscale space is the orthogonal complement of with respect to , i.e. for all and .in this section , we use the previously defined ideal multiscale space to define a ( preliminary ) multiscale approximation .the ideal multiscale problem reads as follows .find and , such that for all and .[ lem : uniqueideal ] under assumptions ( a1)(a3 ) and ( b1)(b3 ) , the ideal multiscale problem has a unique solution .in particular , we have i.e. inf - sup stability independent of and .we let .the coercivity of on follows immediately from its coercivity on since .the operator is stable in with constant , since and for all . combining these results with the inf - sup stability of on and , we get i.e.  is inf - sup stable with constant independent of and .we note that , since ( see remark [ rem : kernel ] ) .finally , we apply lemma [ lem : existence ] with , , and constants , and .   in this section ,we show that the flux solution of the ideal multiscale problem above converges in the energy norm with linear order in to the reference solution .this convergence is independent of the variations of , i.e. we do not have any pre - asymptotic effects from the multiscale features .[ lem : idealerrorestimate ] under assumptions ( a1)(a3 ) and ( b1)(b3 ) , let solve and solve , then where and are independent of and .  parametrizing the solutions and by the data , we use the triangle inequality to obtain the two last terms will be shown to equal zero .  for the first term , we proceed in several steps .let us define , which is the flux solution for the data .the corresponding pressure solution shall be denoted by .first , we observe in order to bound the term , we let be the weak solution to . then we have defining we get and .next , we use a pair of projection operators and that commute with respect to the divergence operator , allows for to be non quasi - uniform , and where is -stable , i.e.  and , with independent of .the existence of such operators is proved in . exploiting this stability and the fact that ( since is a projection on and ) , we obtain combining this estimate with yields   for the second term , since the correctors are divergence free , we have .this implies , hence i.e.  .now , from first the equations in and in combination with the -orthogonality between and , we get since , we obtain for all .choosing , we see that , thus the second term equals zero .  to show that the third term is zero , it is sufficient to show that .the data is -orthogonal to the test space and it enters the equation only in an scalar product with test functions . hence .the ideal corrector problems are at least as expensive to solve as the original reference problem .hence , we require to localize these problems to very small patches , without sacrificing the good approximation properties .if we can achieve this , the corrector problems can be solved with low computational costs and fully in parallel . in this section ,we show that this is indeed possible .we prove that we can truncate the computational domain in the local corrector problems to a small environment of a coarse element .this is possible , since the solutions of decay with exponential rate outside the coarse element .we obtain a new localized corrector operator which can be used analogously to the ideal corrector operator to construct a localized multiscale space .this localization reduces the computational effort for assembling the multiscale space significantly .in addition to the assumptions ( a1)(a3 ) and ( b1)(b3 ) , we require additional assumptions on the computational domain and the mesh . more precisely we assume the following for the analysis .  1.  we consider and a simply - connected domain .the fine grid is quasi - uniform , i.e. the ratio between the maximum and the minimum diameter of a grid element is bounded by a generic constant .we note that assumption ( a4 ) is crucial for our proof .assumption ( b4 ) on the other hand could be dropped with a more careful analysis . in this casethe estimates ( and in particular the decay ) will depend on the inverse of the minimum mesh size of the fine grid in a patch .for simplicity of the presentation , we do not elaborate this case and restrict ourselves to quasi - uniform meshes , i.e. to ( b4 ) .note that even though we fix , we keep the general notation to illustrate how the results are influenced by the dimension .the localized method can be formulated analogously for .  in order to localize the detail space , we use admissible patches .we call this restriction to patches _ localization_. for each we pick a connected patch consisting of coarse grid elements and containing .more precisely , for positive we define -coarse - layer patches iteratively in the following way . for all ( which are assumed to be closed sets ), we define the element patch in the coarse mesh by   .4 -coarse - layer patches .dark gray subdomain is .light gray subdomain is .,title="fig : " ]  .4 -coarse - layer patches .dark gray subdomain is .light gray subdomain is .,title="fig : " ]  see figure [ fig : patch ] for an illustration of patches . for a given patch , we define the restriction of to by accordingly , we also define using this localized space , we define the localized corrector operators .localized quantities are indexed by the patch layer size .[ def : loccorrector ] for each and layers , we define a _ localized element corrector operator _ : for all .further , we define the _ localized global corrector operator _ .the localized corrector operators are again well - defined by the lax - milgram theorem , exploiting that is a weighted -scalar product .note that the definition of implies neumann boundary conditions on the localized corrector problems .we define a localized multiscale function space by and state the localized multiscale problem as follows .[ def : locproblem ] the localized multiscale problem reads : find and , such that for all and .definitions [ def : loccorrector ] and [ def : locproblem ] constitute the proposed multiscale method .next , we show that the above stated problem is well - posed .[ lem : uniquelocalized ] under assumptions ( a1)(a3 ) and ( b1)(b3 ) , the localized multiscale problem has a unique solution for all , and .we use similar arguments as in lemma [ lem : uniqueideal ] .the basic difference is that we need to show stability for the localized corrector operator .we start with the stability of the localized element corrector operators . herewe have for arbitrary  now , we can prove -stability of the localized global operator .we get where is a constant only depending on the shape regularity constant of the coarse mesh .similar to we derive inf - sup stability with observe that the inf - sup stability constant depends on this time .the inf - sup stability constant depends on due to overlapping patches .we come back to another estimate of the inf - sup stability constant in section [ sec : infsup ] after proving the decay of the correctors .it is important to note that in the localized case we do not have orthogonality between and as in the ideal case ( cf .equation ) .this orthogonality was crucial in the error estimate for the ideal method presented in lemma [ lem : idealerrorestimate ] . in the localized case ,we rely on the exponential decay of the localized element correctors , which justifies localization to patches .   in this section we state the main result of this paper , which is an a priori error estimate in the energy norm between the reference solution and the localized multiscale approximation .we first present a logarithmic stability result for the nodal raviart thomas interpolation operator for fine scale functions and then state a lemma on the exponential decay of the correctors. then the main theorem follows .the proof of the exponential decay is contained in section [ sec : decay ] .the notation stands for with some constant that might depend on , , , and coarse and fine mesh regularity constants , but not on the mesh sizes and .in particular it does not depend on the possibly rapid oscillations in .we recall a well known stability result for the nodal raviart thomas interpolation operator .[ lemma - wohlmuth ] assume ( b1)(b4 ) . for any given element exists a constant that only depends on the regularity of and the quasi - uniformity of , such that with for all with .  a proof for this can be found in ( * ? ? ?* lemma 4.1 ) .this result holds for both and .[ rem : clement ] there exist unconditionally -stable clment - type interpolation operators for which we could define for all and instead , see . in particular , the operators introduced in are projections and were used as a technical tool in the proof of lemma [ lem : idealerrorestimate ] above . however , these operators are hard to implement in practice and hence are not used in the proposed numerical method .[ trunc - applied ] under assumptions ( a1)(a4 ) and ( b1)(b4 ) , there exists a generic constant depending on the contrast , but not on or such that for all positive : for all .the lemma is a direct consequence of lemma [ trunc - lemma-3 ] in section [ sec : decay ] .now , combining the error estimate for the ideal multiscale method in lemma [ lem : idealerrorestimate ] and lemma [ trunc - applied ] we get the following a priori error estimate of the localized multiscale method .[ thm : errorestimate ] under assumptions ( a1)(a4 ) and ( b1)(b4 ) , for a positive , let solve and solve , then for some depending on the contrast , but not on , and .before stating the proof , we discuss the role and choice of .the second term in the error estimate is an effect of the localization .this term can be made small by choosing large values of , i.e. large patch sizes .a natural question is how to choose to make the second term of order to some power .we write for convenience .let , where is a constant independent of and .we are interested in the asymptotic behavior , so we consider .setting the second term in equal to yields that is , where is the lambert -function . in terms of the number of layers , we get .this equation has two solutions for sufficiently small .since we require , we pick the branch .another , more practical option is to choose for some constant .then the expression will be asymptotically ( as ) dominated by the power .choosing sufficiently large yields arbitrary order of accuracy of the term .the fine mesh size is often fixed and we can choose for some bases and of the two logarithms .if clment - type interpolation operators are used , we have independent of .choosing makes the second term in proportional to . for an appropriate we can make the first term in dominate the error estimatelet , then is divergence free .hence , by galerkin orthogonality we have and obtain the first term can be bounded by by lemma [ lem : idealerrorestimate ] . regarding the second term , using ( * ? ? ?* lemma 4.1 ) and stability of the ideal multiscale solution , we get and can combine this with lemma [ trunc - applied ] to get    this section consists of four lemmas , lemma [ existence - of - skew - symmetric - matrix][trunc - lemma-3 ] , of which the last one is the main result .the two first lemmas are auxiliary and are motivated by steps in the proofs of the latter two . before starting , we need to set some notation and introduce some tools .we use the notation . note that we will use the letter to denote arbitrary triangles of the coarse mesh .the first lemma says that every divergence free function in is the divergence of a skew - symmetric matrix .[ existence - of - skew - symmetric - matrix ] let be a simply connected domain with lipschitz boundary and let with in .then there exists a skew - symmetric matrix ^{d \times d} ] and such that here , the divergence of is defined along the rows .note that the above lemma is the only instance , where we require the restriction . even though the existence of the skew - symmetric matrix is also available for , we could not prove localized estimates of the type .the result is a combination of well - known results .first , we extend the divergence - free vector field to a divergence - free vector field . in particular we have ^d ] with ^d ] with denote the corresponding skew - symmetric matrix as in lemma [ existence - of - skew - symmetric - matrix ] .let furthermore and let ^{d \times d} ] and the source function is given by ^ 2,\\   -1 & \text{if } x \in [ 3/4,1]^2,\\   0 & \text{otherwise}.  \end{cases}\ ] ] we consider three different diffusion coefficients :  1 .constant : in the whole domain .2 .  noise : is piecewise constant on a uniform rectangular grid . in each grid cell , the value of is equal to a realization of , where is a cell - specific standard uniformly distributed variable .channels : is as is shown in figure [ fig : structured_a ] .it is piecewise constant on a uniform rectangular grid .the coefficient for in black cells and for in white cells .+  defined on a grid of .,scaledwidth=30.0% ]  .4   .4   figure [ fig : mesh ] shows the mesh used in the experiment .both fine and coarse meshes are constructed as shown in the figure . a reference solution computed with the standard raviart thomas spaces and with .solutions to the localized multiscale problem were computed using .the patch size was chosen as rounded to the nearest integer with and .the relative error ( using the reference solution in place of the exact solution ) in energy norm , i.e.  was computed .see figure [ fig : convergence ] for the resulting convergence of this error with respect to for the two values of .note that since for all examples , the first term in vanishes .the error is hence bounded by , which allows for a careful investigation of the influence of , and . a reference line proportional to plotted for guidance .we can see that we achieve convergence for both choices of .however , since is rounded to an integer , the convergence plots have a staggered appearance .this example shows that the error due to localization can be kept small and decreases with .the plots also show the relative error in energy norm for the standard raviart thomas discretization on the coarse mesh .it is evident that the localized multiscale space has good approximation properties since it permits convergence while the standard space of the same dimension does not .   in this experimentwe show how singularity - like features can appear in the solution , probably as a result of high contrast in combination with the -instability of the nodal raviart thomas interpolator .again , we consider the unit square ^ 2 ] , while the multiscale solution has a spike reaching magnitude at .interesting to note is that the singularities vanish for the ideal multiscale method , i.e.without localization , see lemma [ lem : idealerrorestimate ] .   decreases . ].47  .47   next , we consider an l - shaped domain with noisy diffusion coefficient ( case 2 . in section [ sec : firstexperiment ] ) and with . in this experiment, we show that the localization error investigated in the previous section can be dominated by errors from projecting .we use the domain ^ 2 \setminus [ 1/2,1]\times[0,1/2] ] and the permeability coefficient is given in a uniform rectangular grid of size by the 85th permeability layer in model 2 of spe10 .the method proposed in is based on a fine and a coarse mesh with quadrilateral elements .the fine mesh is uniform , i.e. aligned with the permeability data , and the coarse mesh is , so that each coarse element is subdivided into fine elements .the implementation of the method proposed in this work uses triangular meshes , which is why we divide each of the rectangular elements into two triangular elements by a diagonal line drawn from the upper left corner to the lower right corner .as coarse mesh , we use a similar triangular mesh that is constructed from a rectangular mesh such that the fine mesh is a conforming refinement of the coarse mesh .  the ( quasi - singular ) source data is equal to in the lower left and in the upper right fine quadrilateral element .note that such is a discretization of point sources that model production wells .in particular , the source terms on the continuous level are mathematically described by dirac delta functions .hence , for , we only have for , opposed to as is required for our analysis . to account for this difference , we follow and compute the localized source corrections on -coarse - layer patches for , for all and , where is the restriction of to , analogous to the definition of .( the pressure solution is not needed for correcting the flux and is discarded after its use as lagrange multiplier ) . since is non - zero only for the two triangles and in the lower left and upper right corners , only two such corrector problems need to be solved. the total localized source correction is .the localized corrector problems are unaffected by the source correction .the right hand side of the localized multiscale problem is appended with the localized source corrections and instead reads : find such that using a value of will be referred to as an ad - hoc source correction , since we do not expect to have any decay of the correction already within the source triangle itself .the source corrected solution is .we emphasize that the need for source correctors for singular source terms is not an exclusive drawback for our approach , but it is a common necessity shared by all comparable multiscale methods in this setting . in particularthey are also used for the msfem - based approach in that we use for our comparative study .the proposed localized multiscale method was used to solve for the flux in the described problem for three corrector patch sizes : and .three variants of source correction were used : i ) without source correction , i.e.  , ii ) with ad - hoc source correction , i.e.  for ( without interpolation constraint ) , and iii ) with source correction , i.e.  for . a reference solution computed on the fine mesh .table [ tbl : speresults ] shows the relative energy norm and -norm of the difference between the localized multiscale solution and the reference solution for the different values of and . the corresponding -norm of the error for the msfem method with oversampling he0-os proposed in is also presented in the table . note that he0-os is based on a discretization with roughly 33% less degrees of freedom than the proposed method , since it uses quadrilaterals instead of triangles ( however , since this holds for both the fine and the coarse mesh , the relative change in the amount of degrees of freedom with respect to the reference solution is the same ) .the flux solutions are plotted in figure [ fig : speresults ] .the results show that the proposed method even without error correction compares favorably with the homogenization based approach .ad - hoc error correction gives small errors for this problem in both norms .for source correction with patch size , instabilities similar to that studied in section [ sec : instability ] cause the error to increase .however , letting is enough to get errors that compare favorably with .[ tbl : speresults ]  .relative error in energy norm and -norm for the spe10 - 85 problem . [ cols="<,^,^,^,^",options="header " , ]   .30  and , and different source corrections ( means no source correction and means ad - hoc error correction ) .the color maps to the magnitude of the flux at the midpoint of the triangular elements .the colors map from ( white ) to ( black ) and is saturated at white and black for lower and higher values , respectively.,title="fig : " ]  .30  and , and different source corrections ( means no source correction and means ad - hoc error correction ) .the color maps to the magnitude of the flux at the midpoint of the triangular elements .the colors map from ( white ) to ( black ) and is saturated at white and black for lower and higher values , respectively.,title="fig : " ]  .30  and , and different source corrections ( means no source correction and means ad - hoc error correction ) .the color maps to the magnitude of the flux at the midpoint of the triangular elements .the colors map from ( white ) to ( black ) and is saturated at white and black for lower and higher values , respectively.,title="fig : " ]   +  .30  and , and different source corrections ( means no source correction and means ad - hoc error correction ) .the color maps to the magnitude of the flux at the midpoint of the triangular elements .the colors map from ( white ) to ( black ) and is saturated at white and black for lower and higher values , respectively.,title="fig : " ]  .30  and , and different source corrections ( means no source correction and means ad - hoc error correction ) .the color maps to the magnitude of the flux at the midpoint of the triangular elements .the colors map from ( white ) to ( black ) and is saturated at white and black for lower and higher values , respectively.,title="fig : " ]  .30  and , and different source corrections ( means no source correction and means ad - hoc error correction ) .the color maps to the magnitude of the flux at the midpoint of the triangular elements .the colors map from ( white ) to ( black ) and is saturated at white and black for lower and higher values , respectively.,title="fig : " ]we gratefully acknowledge the anonymous reviewers for their careful reading and insightful suggestions that improved the manuscript .p. henning , p. morgenstern , and d. peterseim .multiscale partition of unity . in m.griebel and m. a. schweitzer , editors , _ meshfree methods for partial differential equations vii _ , volume 100 of _ lecture notes in computational science and engineering _ , pages 185204 .springer international publishing , 2015 .p. a. raviart and j. m. thomas .a mixed finite element method for 2-nd order elliptic problems . in i.galligani and e. magenes , editors , _ mathematical aspects of finite element methods _ , volume 606 of _ lecture notes in mathematics _ , pages 292315 .springer berlin heidelberg , 1977 .

Abstract: in this work , we propose a mixed finite element method for solving elliptic multiscale problems based on a localized orthogonal decomposition ( lod ) of raviart thomas finite element spaces . it requires to solve local problems in small patches around the elements of a coarse grid . these computations can be perfectly parallelized and are cheap to perform . using the results of these patch problems , we construct a low dimensional multiscale mixed finite element space with very high approximation properties . this space can be used for solving the original saddle point problem in an efficient way . we prove convergence of our approach , independent of structural assumptions or scale separation . finally , we demonstrate the applicability of our method by presenting a variety of numerical experiments , including a comparison with an msfem approach .   multiscale mixed finite elements +   fredrik hellman , patrick henning , axel mlqvist  [ [ keywords ] ] keywords + + + + + + + +  mixed finite elements , multiscale , numerical homogenization , raviart thomas spaces , upscaling  [ [ ams - subject - classifications ] ] ams subject classifications + + + + + + + + + + + + + + + + + + + + + + + + + + +  35j15 , 35m10 , 65n12 , 65n30 , 76s05