natural recharge estimation and uncertainly analysis of an adjudicated groundwater basin using a...

Natural recharge estimation and uncertainty analysis of an adjudicatedgroundwater basin using a regional-scale ow and subsidence model(Antelope Valley, California, USA)

Adam Siade & Tracy Nishikawa & Peter Martin

Abstract Groundwater has provided 5090% of the totalwater supply in Antelope Valley, California (USA). Theassociated groundwater-level declines have led the LosAngeles County Superior Court of California to recentlyrule that the Antelope Valley groundwater basin is inoverdraft, i.e., annual pumpage exceeds annual recharge.Natural recharge consists primarily of mountain-frontrecharge and is an important component of the totalgroundwater budget in Antelope Valley. Therefore, naturalrecharge plays a major role in the Courts decision. Theexact quantity and distribution of natural recharge isuncertain, with total estimates from previous studiesranging from 37 to 200 gigaliters per year (GL/year). Inorder to better understand the uncertainty associated withnatural recharge and to provide a tool for groundwatermanagement, a numerical model of groundwater ow andland subsidence was developed. The transient model wascalibrated using PEST with water-level and subsidencedata; prior information was incorporated through the useof Tikhonov regularization. The calibrated estimate ofnatural recharge was 36 GL/year, which is appreciablyless than the value used by the court (74 GL/year). Theeffect of parameter uncertainty on the estimation of naturalrecharge was addressed using the Null-Space Monte Carlomethod. A Pareto trade-off method was also used to

portray the reasonableness of larger natural recharge rates.The reasonableness of the 74 GL/year value and the effectof uncertain pumpage rates were also evaluated. Theuncertainty analyses indicate that the total natural rechargelikely ranges between 34.5 and 54.3 GL/year.

Keywords Subsidence . Groundwaterow . Groundwaterrecharge/water budget . Inverse modeling . Optimization

Introduction

Prior to 1972, groundwater had provided more than 90%of the overall water supply in Antelope Valley, California,USA (Fig. 1). During this time, groundwater extractionwas primarily used for agricultural purposes and reached apeak of approximately 500 gigaliters per year (GL/year) in1951 (Leighton and Phillips 2003). Since 1951, urbangroundwater use has increased signicantly and agricul-tural use declined. By 1995, groundwater extractionresulted in water-level declines of more than 90m andland subsidence of about 2m in some areas of the basin.Even though the introduction of imported surface water in1972 has alleviated some of the necessity for groundwaterextraction, imported surface water is still relativelyexpensive and difcult to deliver to farms in the basininterior. Total groundwater extraction then declined toapproximately 100 GL/year in 1995 (Leighton andPhillips 2003). Additionally, in recent years, agriculturaldemand has begun to increase once again. Therefore, asboth urban population and agricultural practices increase,and if the quantity of imported surface water is limited, thedemand for groundwater is likely to rise.

Recent increases in the demand for groundwater in theAntelope Valley, combined with continued storage reduc-tions and land subsidence, have caused several entities,both private and governmental, to take legal action toensure their rights to water. As a result, the groundwaterbasin has been adjudicated and the Los Angeles CountySuperior Court of California has subsequently ruled on asafe yield value of 135.7 GL/year (Los Angeles CountySuperior Court of California 2011) based on an estimateof total average annual natural recharge of 74.0 GL/year(Beeby et al. 2010a, b). As dened by the Court, safe

Received: 11 August 2014 /Accepted: 10 June 2015Published online: 24 July 2015

* The Author(s) 2015. This article is published with open access atSpringerlink.com

Published in the theme issue Optimization for GroundwaterCharacterization and Management

A. Siade I T. Nishikawa ()) I P. MartinUS Geological Survey, California Water Science Center,4165 Spruance Rd, Suite 200, San Diego, CA 92101, USAe-mail: [email protected]

A. SiadeNational Centre for Groundwater Research and Training, FlindersUniversity, Adelaide, GPO Box 2100, SA 5001, Australia

A. SiadeSchool of Earth and Environment, University of Western Australia,35 Stirling Highway, Crawley, WA 6009, Australia

Hydrogeology Journal (2015) 23: 12671291DOI 10.1007/s10040-015-1281-y

yield is the amount of annual extractions of water from anaquifer over time equal to the amount of water needed torecharge the groundwater aquifer and maintain it inequilibrium, plus any temporary surplus. In order forthe Court to make appropriate decisions regarding themanagement and distribution of groundwater extraction,the quantity, spatial and temporal distribution of naturalrecharge and return ow must be well understood.

The objectives of this study are to provide an improvedestimate of groundwater recharge and to thoroughly assessits associated uncertainty. The groundwater-ow and land-subsidence model of Leighton and Phillips (2003) was

updated and calibrated for this purpose (Siade et al. 2014).The three-layer model developed by Leighton and Phillips(2003) resulted in an estimate of total natural recharge of37.4 GL/year and provided no quantitative estimate of theposterior uncertainty of this value.

The updated model developed for this study wascalibrated systematically using the Gauss-Marquardt-Levenberg algorithm, which is embedded in PEST(Doherty 2010). The Tikhonov regularization techniquewas used to impose expert knowledge, or prior informa-tion, on the parameter-estimation process (Tikhonov andArsenin 1977). Sources of non-uniqueness, including

Fig. 1 Map showing location of study area including groundwater subbasins, faults, line of geologic section and approximate areal extentof lacustrine deposits in Antelope Valley groundwater basin, California

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Hydrogeology Journal (2015) 23: 12671291 DOI 10.1007/s10040-015-1281-y

parameter correlation and insensitivity, can result in asignicant degree of uncertainty associated with themodels predictions. The primary prediction for this studyis the total average annual natural recharge, which consistsprimarily of the subsurface groundwater ow entering thebasin from the mountain fronts.

The nature of the predictive uncertainty of a model istypically explored using a Monte Carlo type of uncertaintyanalysis with the restriction that each realization reason-ably calibrates the model (i.e., calibration-constrainedMonte Carlo). However, recalibrating the model for a setof randomly generated parameter vectors can be compu-tationally expensive. The null-space Monte Carlo (NSMC)algorithm, contained in the PEST software (Tonkin andDoherty 2009; Doherty 2010; Keating et al. 2010;Herckenrath et al. 2011; Yoon et al. 2013), reduces thiscomputational burden using knowledge of the calibrationnull space, i.e., linear combinations of parameters that arenot estimable given the chosen parameterization and theobservation data provided. It is within this calibration nullspace that the vast majority of the parameter error orparameter uncertainty resides. The NSMC method wasimplemented in this study to evaluate the uncertaintyassociated with natural recharge that stems from parametererror.

The uncertainty associated with natural recharge wasalso addressed using Pareto techniques (Moore et al. 2010;Doherty 2010). This analysis portrays the degradation inmodel calibration at larger total volumetric rates of naturalrecharge. It is considered a Pareto curve because itillustrates a trade-off where decreases in model t resultwhen predicted values of natural recharge are increased. Inaddition to the Pareto method, a feasibility analysis wasconducted to specically address the reasonableness of theCourts value for natural recharge (i.e., 74 GL/year).

Another source of uncertainty when making predic-tions of natural recharge resides in the structure of themodel itself, e.g., coarse spatial and temporaldiscretization, improper boundary conditions, inappropri-ate representation of geologic formations, etc. In thisstudy, a potential source of signicant structural errorresides in the a priori estimates of agricultural groundwa-ter extraction. These rates are quite uncertain as they arenot directly measured and are determined indirectly basedon crop type, climatic conditions, etc. The effects of thispotential structural error on predictions of natural rechargeare also addressed in this study by simply considering arange of different scenarios and recalibrating the model foreach scenario.

Description of study area

Antelope Valley is a topographically closed basin, about80 km northeast of Los Angeles, California (Fig. 1). Thevalley is bounded on the south by the San GabrielMountains and on the northwest by the TehachapiMountains. Lower hills, ridges, and buttes form thenorthern and eastern boundaries of the valley. The valley

oor slopes gently toward several playas north and east ofthe center of the basin. Land use in the valley is mainlyurban, agricultural, industrial, and military; Lancaster andPalmdale are the largest cities. All natural channels areephemeral; any surface-water runoff terminates in theplayas. The climate in the valley is semiarid to arid.Estimates of precipitation and evapotranspiration (ET)throughout the valley oor indicate that ET signicantlyexceeds precipitation. Therefore, inltration on the valleyoor is assumed to be negligible and the primary source ofnatural recharge is the underow entering the valley fromthe mountain fronts, along with any inltrated overlandow along ephemeral stream channels (Siade et al. 2014).

HydrogeologyBeneath the Antelope Valley oor exist large sediment-lled structural depressions between the Garlock andthe San Andreas fault zones (Fig. 1; Leighton andPhillips 2003). The bedrock complex in the valleyforms the impervious bottom of the groundwater basinand crops out at higher elevations, surrounding thevalley. The bedrock complex is comprised of pre-Cenozoic igneous and Tertiary sedimentary rocks(Hewett 1954; Dibblee 1963). The basin ll consistsof a series of unconsolidated deposits of Quaternaryage, in some places more than 1,500m thick (Bendaet al. 1960; Mabey 1960; Jachens et al. 2014). Dutcherand Worts (1963) mapped these deposits as eitheralluvial or lacustrine. The alluvium consists of uncon-solidated to moderately indurated, poorly sorted mate-rials with the older, deeper units being more compactedand indurated than the younger shallow units (Dutcherand Worts 1963; Durbin 1978). The ne-grainedlacustrine clay deposits consist of interbedded sands,silts, and clays that have accumulated in a large lakethat periodically covered large parts of the valley center(Dibblee 1967; Orme 2003). These lacustrine depositsconsist primarily of thick layers of a blue-green siltyclay and a brown clay which contains interbedded sandsand silts. These large clay beds are as much as 30mthick and are interbedded with lenses of coarsermaterial. The greater lacustrine deposits are overall asmuch as 90m thick in some areas (Dutcher and Worts1963). These lacustrine deposits are covered by asmuch as 245m of alluvium in the southern part of theLancaster subbasin; they become progressivelyshallower towards the northeast, and are exposed atthe land surface near the southern edge of Rogers (dry)Lake (Fig. 2).

Antelope Valley contains numerous faults (Fig. 1),some of which may act as barriers to groundwater ow(Mabey 1960; Dibblee 1960, 1963; Dutcher and Worts1963; Ward et al. 1993; Nishikawa et al. 2001; Leightonand Phillips 2003). In this study, the aim is to determinewhich faults act as barriers by estimating their conduc-tance via parameter estimation; faults with a low estimatedconductance act as barriers and vice versa.

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Pre-development recharge and dischargeThe primary source of natural recharge to the basin isinltration of precipitation in the surrounding mountains,resulting in subsurface ow into the groundwater basin.Natural recharge may also occur as inltration of runofforiginating from the surrounding mountains in ephemeralstream channels. Since these two types of mountain-frontrecharge comprise the vast majority of natural recharge,they are together referred to, throughout this study, asnatural recharge. Precipitation over the valley oor is low(less that 25 cm/year; Rantz 1969) while ET rates are high;therefore, recharge from direct inltration of precipitationon the valley oor is considered to be negligible (Snyder1955; Durbin 1978). Precipitation in the mountains ishighly variable, but generally greater than 30 cm/year(Rantz 1969).

The quantity of natural recharge in Antelope Valley hasbeen estimated in previous investigations based onrainfall, runoff, channel-geometry data, water-quality data,groundwater age dating, and groundwater-ow modeling.

Bloyd (1967) estimated a natural recharge value of about72 GL/year using the entire valley as the surface-waterdrainage area (1,445 km2), whereas Durbin (1978) esti-mated that natural recharge was about 50 GL/year, usingonly the Antelope Valley groundwater basin as thesurface-water drainage area (997 km2). In a more recentstudy by Leighton and Phillips (2003), the annual naturalrecharge was estimated to be 37.4 GL/year.

The quantity, distribution and source of groundwaterrecharge were estimated for this study using the regional-scale basin characterization model (BCM; Flint and Flint2007). The BCM used a deterministic water-balanceapproach to estimate recharge and runoff from the adjacentmountains. This approach incorporated the distribution ofprecipitation, snow accumulation and melt, potential evapo-transpiration, soil-water storage, and bedrock permeability toestimate a monthly water balance for the groundwatersystem. BCM results are useful for providing boundsassociated with water-balance results of more detailedmodels, evaluating long-term climate conditions, illustrating

Fig. 2 Generalized geologic section showing the relation of lacustrine clays (potential conning units) to aquifers in the Lancaster andNorth Muroc subbasins in Antelope Valley groundwater basin, California

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the mechanisms responsible for recharge in a basin, andcomparing the locations and volumes of recharge and runoffin different basins on a regional scale (Flint et al. 2013). TheBCM-estimated average annual natural recharge was about63 GL/year.

Because the basin is topographically closed, predevel-opment discharge from the Antelope Valley consistedprimarily of ET in the lower parts of the valley where thewater table was within 3m of land surface (Lee 1912).Johnson (1911) mapped the areal extent of artesianconditions by observing numerous shallow owing wellsthroughout the valley. The areal extent of signicant ET isassumed to coincide approximately with this artesianregion. A large area of alkali soils (Durbin 1978) and theexistence of phreatophytes (e.g., mesquite) in the northcentral part of the groundwater basin indicate that thewater table was near land surface at one time and that ETwas signicant (Thompson 1929).

Other types of predevelopment discharge from thebasin included lateral groundwater underow to adja-cent basins and discharge from springs. Bloyd (1967)and Durbin (1978) claimed that groundwater underowoccurred in the northwest corner of the North Murocsubbasin into the Fremont Valley Basin. Estimates ofthis underow were developed by Bloyd (1967) (0.1 to 0.6GL/year), Durbin (1978) (1.2 GL/year) and Leighton andPhillips (2003) (0.5 GL/year). Discharge by springs wasreported to be less than 0.4 GL/year (Johnson 1911;Thompson 1929).

Post-development recharge and dischargeThe history of groundwater extraction throughoutAntelope Valley has caused signicant changes in theamount, distribution, and type of recharge and discharge.In addition to natural recharge, new sources of rechargehave emerged including, irrigation return ow andinltration of treated wastewater. Furthermore, ET hasbeen replaced by groundwater pumping as the primarydischarge from the valley.

Since the development of irrigated agriculture inAntelope Valley, large amounts of irrigation water havebeen applied to crops; a portion of this water may havepercolated below the root zone and contributed recharge tothe groundwater basin. Inltration of treated municipalwastewater may also contribute to groundwater recharge,with the largest producers of treated wastewater being thePalmdale Water Reclamation Plant and the LancasterWater Reclamation Plant (Templin et al. 1995; Fig. 1).Beginning in 1975, treated wastewater has been disposedof in ponds or on spreading grounds such that it eitherevaporates or inltrates below the land surface.

Return ows from urban landscape irrigation and septictanks in urban areas of the Antelope Valley are alsopotential sources of recharge to the groundwater basin.Previous US Geological Survey (USGS) investigationsdid not estimate the quantity of recharge contributed fromthis source; however, as part of the adjudication, returnows from landscape (non-agriculture) irrigation and

septic tanks were estimated based on estimates ofmunicipal and industrial water requirements for the period19192006 (Beeby et al. 2010a, b; Appendix D).

The extraction of groundwater for irrigation in theAntelope Valley began in the 1800s; however, the quantityof groundwater pumpage was initially small. Beginning in1915, the number of wells drilled for agriculture inAntelope Valley increased dramatically, resulting inappreciable increases in annual pumpage. By the 1970s,wells drilled for municipal and industrial use increasedsignicantly (Leighton and Phillips 2003; Templin et al.1995). Historical pumpage, from 1915 through 1995, wasestimated by Leighton and Phillips (2003), who extendedthe work of Snyder (1955), Durbin (1978), and Templinet al. (1995; Fig. 3), and was used directly for this study.Annual pumpage for 1996 through 2005 was estimated bySiade et al. (2014) to extend the Leighton and Phillips(2003) pumping history. The agricultural component oftotal pumpage for 1996 through 2005 for Los AngelesCounty was obtained from the water purveyors themselvesor the California State Water Resources Control Board(2011). Where reported data were not available for LosAngeles County, agricultural pumpage was estimatedindirectly using irrigated crop acreage and cropconsumptive-use data, similar to the techniques used byLeighton and Phillips (2003). Agricultural pumpage for allof Kern County during the 19962005 period wasestimated indirectly since direct pumpage data was notavailable (Siade et al. 2014).

Land subsidence and aquifer-system compactionLand subsidence is the loss of surface elevation as a resultof the removal of subsurface support and is governed bythe principle of effective stress and the theory ofhydrodynamic consolidation (Terzaghi 1925). One of theprincipal causes of land subsidence is the gradualcompaction of compressible aquifer systems resultingfrom declines in hydraulic head caused by groundwaterpumping (Leake 1990; Galloway et al. 1998, 1999). Someof the detrimental effects of land subsidence include theloss of aquifer storage, increased risk of ooding, cracksand ssures, damage to man-made structures, and otherintangible economic costs. The spatial variability in theamount of land subsidence in Antelope Valley is affectedby both the magnitude of declines in hydraulic head, aswell as the distribution of compressible sediments.Between 1930 and 1992, groundwater pumping inAntelope Valley has resulted in as much as 1.8m ofsubsidence and a corresponding loss of groundwaterstorage capacity (Ikehara and Phillips 1994).

Groundwater ow and subsidence model

The three-layer model developed by Leighton and Phillips(2003) was updated for this study; the resulting model wasused to address the uncertainty of natural recharge. TheLeighton and Phillips (2003) model was updated to

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MODFLOW-NWT (Niswonger et al. 2011), which is aNewton formulation of MODFLOW-2005 (Harbaugh 2005)in which an upstream weighted nite-difference method isemployed resulting in greater stability when simulatingcomplex nonlinear systems, especially systems containingmodel cells that transition from dry to wet or vice versa. Thisis particularly important for this study as most of the naturalrecharge occurs along the mountain boundaries where thebasin ll is relatively thin and the model often contains onlyone active layer. In these regions, the model is quitesusceptible to having cells become dry or inactive duringthe parameter estimation process, which can lead toassociated parameter estimation instability.

Model discretizationThe hydrogeologic conceptualization developed byLeighton and Phillips (2003), using stratigraphic, hydro-logic, and water-quality data since the early 1990s, wasused as a starting point for this study, and is dened asfollows. Their model consisted of three primary aquifersystems. The upper aquifer extended from the water tableto an elevation of about 594.0m above sea level (asl) andvaried from unconned to conned depending on thepresence and vertical position of the thick lacustrinedeposits within the aquifer (Fig. 2). The middle aquiferextends from 594.0m asl down to 472.0m asl and iseither unconned or conned by the overlying lacustrinedeposits and the discontinuous interbedded aquitards inthe upper aquifer. The lower aquifer extends from 472.0m

asl down to the bedrock complex. The lower aquifer isconned by the overlying lacustrine deposits and thediscontinuous interbedded aquitards in the middle aquifer.

In order to adequately reproduce measured water levelsthroughout the north-central region of Lancaster subbasin, inthe area of former Lake Thompson (Orme 2003), the three-layer hydrogeologic conceptualization developed by Leightonand Phillips (2003) was modied by dividing the upper aquiferinto two aquifers. Data from electrical-resistivity logs in thearea and geologic logs from a study of the region (CH2MHILL 2005), along with the ndings of Johnson (1911),indicate the presence of a laterally extensive, conning, clayinterbed throughout this region. This interbed was simulated inthis study by dividing model layer 1, as dened by Leightonand Phillips (2003), into two model layers and assigningrelatively low vertical hydraulic conductivity between the newmodel layers 1 and 2 (Figs. 4 and 5). Siade et al. (2014)describe the details of this layer subdivision.

Model-layer 1, therefore, represents the shallow portion ofthe upper aquifer in the Lancaster subbasin coincident withthe area of former Lake Thompson. This layer represents aconning unit, and consists of unsaturated and saturatedalluvial, lacustrine, and playa deposits. The bottom elevationof model-layer 1 was set to the approximate top of a 315mthick clay, that occurs at an elevation of about 652668m asl,and was spatially distributed (interpolated or extrapolated)based on both the available electrical-resistivity logs and on astudy conducted by CH2M HILL (2005) southwest ofRosamond Lake (Fig. 1). The study conducted by CH2MHILL (2005) indicates that water levels in wells perforated

Fig. 3 Graph comparing estimates of groundwater pumpage in Antelope Valley groundwater basin, California, 19152005. The estimatesobtained by Leighton and Phillips (2003) were used for this study from 1915 to 1995. The pumpage from 1996 to 2005 was estimated usingsimilar methods (Siade et al. 2014). Modied from Leighton and Philips (2003)

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Fig. 4 Cross section of model column 72 and selected hydrographs associated with the 4-layer groundwater ow and subsidence model ofAntelope Valley groundwater basin, California

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above this elevation have higher water levels than wellsperforated below this elevation, suggesting that the clay layersabove 652m asl restrict the vertical ow of groundwater.

The uniform bottom elevations of Leighton andPhillips (2003) for layers 2, 3 and 4 (594.0, 472.0, and305m, respectively) were used, except where recentgravity surveys (Jachens et al. 2014) provided improvedestimates of the lower no-ow boundary. Horizontally, theoriginal model by Leighton and Phillips (2003) used

square 1.6 km model cells, which were rediscretized forthis study into square 1-km cells (Fig. 5).

The groundwater-ow model was used to simulateboth steady-state (i.e., predevelopment) and transientconditions. The simulation period was extended from thatof Leighton and Phillips (2003; i.e., 19151995) to theperiod 19152005. The steady-state results represent early1900 conditions, which were assumed to representpredevelopment conditions in Antelope Valley. Simulated

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Fig. 5 Groundwater-ow and subsidence model geometry, horizontal discretization, active cells, ow barriers, specied-head and general-head boundary conditions, and zonation patterns for hydraulic conductivity and specic yield for the groundwater-ow and land-subsidencemodel, Antelope Valley groundwater basin, California: (a) layer 1, (b) layer 2, (c) layer 3, and (d) layer 4

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hydraulic head results from the steady-state model werethen used as the initial conditions for the transient model.

Model boundariesRecharge and pumping were simulated using specied-ux boundary conditions, and all other groundwaterdischarge was simulated using head-dependent boundaryconditions. Head-dependent ux boundaries were used inthe form of a time-variant specied-head (CHD package),general-head boundary (GHB package), ET (EVT pack-age) and drain (DRN package) boundaries to simulategroundwater owing into or out of the model domain(Harbaugh 2005). A CHD boundary condition was used tosimulate ux exchanges with the Fremont Valley due tothe presence of high quality hydraulic head data at theboundary itself, whereas, due to a lack of high qualitydata, a GHB boundary condition was used for exchangeswith El Mirage (Fig. 5b). EVT boundary conditions areused to simulate discharge due to ET, and DRN boundaryconditions are used to simulate ow through the playasurfaces via evaporation as well as through varioussprings throughout the valley. Reference-head values forthe EVT and DRN packages are based on plant rootingdepths and land-surface elevations, respectively (Fig. 6).

Specied-ux boundaries were used in the form ofmulti-node wells (MNW1 package; Halford and Hanson2002), unsaturated zone ow (UZF1 package; Niswongeret al. 2006), and recharge (RCH package; Harbaugh2005). The MNW1 package was used to simulategroundwater pumping. The MNW1 package internallycalculates the vertical distribution of pumpage for wellsthat are perforated through multiple model layers. MNW1also allows for water ow through boreholes that spanmultiple model layers. Groundwater recharge from agri-cultural irrigation is assumed to be 30% of the waterapplied to crops (i.e., irrigation efciency is assumed to be70% throughout the model domain). This return ow isapplied at land surface using the UZF1 package in order tosimulate delays associated with water traveling throughthe unsaturated zone. Groundwater recharge fromimported surface water is assumed to occur near theirrespective turnouts, and the total annual water delivered toeach turnout was obtained from the Antelope Valley EastKern (AVEK) water agency. Similarly, the irrigationefciency for imported water was assumed to be 70%.Recharge data from treated wastewater spreading pondswas obtained and modeled as an RCH boundary (Beebyet al. 2010a, b).

Urban groundwater recharge volumes and rates (whichare considered to also contribute to return ow) werebased solely on the temporal and spatial distribution andextent of the urban areas. That is, if a model cell isconsidered an urban model cell at a particular stressperiod, it is assumed to contribute groundwater recharge.This return-ow value is constant for all urban modelcells; the assigned value of 182.9mm/year is based on theestimated total basin-wide urban return ow calculatedduring the adjudication (Beeby et al. 2010a, b). Urban

return ow is applied at land surface and the UZF1package simulates delays associated with travel time to thewater table.

Model calibration

The model-independent parameter estimation softwarePEST (Doherty 2010) was used to calibrate thegroundwater-ow and subsidence model. The algorithmemployed in PEST that was chosen for this study isknown as the Gauss-Marquardt-Levenberg method(Levenberg 1944; Marquardt 1963). This algorithm is anonlinear regression algorithm in which parameter valuesare iteratively updated until the sum of the squaredresiduals (i.e., the objective function) is minimized asmuch as possible. Parameter estimation for large-scalenonlinear systems containing many parameters can sufferfrom issues associated with nonuniqueness and insensi-tivity. In this case, the inverse problem is referred to as ill-posed or under-determined (Yeh 1986). To alleviate someof the issues associated with under-determination,Tikhonov regularization is employed (Tikhonov andArsenin 1977). However, it is important to note that thePEST algorithm used in this work is a local-searchalgorithm and will not guarantee a global solution;therefore, the solution may depend largely on the qualityof the initial guess of the parameter values.

Model parameterizationHorizontal and vertical hydraulic conductivity are as-sumed vertically anisotropic and horizontally isotropic andare assigned a value to each model cell. Horizontal andvertical hydraulic conductivity for layers 13 are assignedto model cells using the zonation patterns developed bySiade et al. (2014) depicted in Fig. 5; layer 4 is representedas one homogeneous hydraulic property zone. Specicyield is parameterized using the same zonation pattern ashydraulic conductivity, but specic storage is assumed tobe homogeneous for each entire model layer.

Flow through the unsaturated zone is assumed to bevertical and is simulated with a kinematic wave approxima-tion of Richards equation (Niswonger et al. 2006). The rateat which water fronts or waves move through, andaccumulate in, the unsaturated zone is dependent on thesaturated vertical hydraulic conductivity of the unsaturatedzone (Kuz), the Brooks-Corey coefcient, and the saturatedwater content of the unsaturated zone (Niswonger et al.2006). The Brooks-Corey coefcient was set to 3.5everywhere, which is consistent with the sedimentarydeposits found in Antelope Valley (Tindall et al. 1999).The saturated water content was assumed to be a constant25%, a value just larger than the specic yield of typicalAntelope Valley sediments. Kuz was parameterized usingzones that are consistent with the zonation patterns ofhydraulic conductivity for layers 1 and 2 (Fig. 7).

The SUB Package was used to simulate landsubsidence, which, in addition to instantaneous

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compaction, also allows for the simulation of delayeddewatering of the thicker, ne-grained interbeds(Hoffmann et al. 2003). The simulation of subsidencein the model assumes that compaction occurs from thedeformation of the conning clay layers and the ne-grained deposits of the aquifers. The compressible

deposits in model-layer 1 consist primarily of the visibleplayas on land surface; these deposits are assumed tocompact instantaneously. The compressible deposits inmodel-layer 2 consist of the relatively young, thin,shallow interbeds that span most of the Lancastersubbasin and the older, thicker, deeper lacustrine

Fig. 6 The distribution of estimated average annual natural mountain-front recharge, evapotranspiration and spring discharge for thegroundwater-ow and subsidence model of Antelope Valley, California

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deposits. The former is assumed to compact instanta-neously and the latter is assumed delayed. Model-layer 3consists of the relatively old, thicker lacustrine depositsand is assumed to have delayed compaction. Model-layer 4 was assumed non-susceptible to compactionbecause the continental deposits of this aquifer rangefrom moderately to very well consolidated.

Storage and other subsidence-related properties mustbe dened in order to simulate subsidence. These includedelastic and inelastic skeletal storage coefcients (Ske andSkv respectively, unitless), preconsolidation head, and thevertical hydraulic conductivity associated with the

compressible deposits. Ske is the product of elastic skeletalspecic storage and the saturated thickness of thecompressible deposits; and, Skv is the product of theinelastic skeletal specic storage and the saturatedthickness of the compressible deposits. The spatiallydistributed thickness of the lacustrine deposits wasobtained from an interpolation and extrapolation of well-log data (Fig. 8). Assuming that these deposit thicknessesare known and entered directly as model input, only thespecic storage values needed to be parameterized andwere assumed homogeneous for each model layer.Because it is unknown, initial preconsolidation head is

Fig. 7 The zonation pattern used to assign the saturated vertical hydraulic conductivity values of the unsaturated zone to numerical modelcells in the groundwater-ow and subsidence model of Antelope Valley, California

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parameterized using pilot-point interpolation (Dohertyet al. 2010) to produce a smoothly varying eld; however,this interpolation is zoned based on model faults (Fig. 9).

Pilot-point interpolation (Doherty et al. 2010) was usedto distribute average annual natural recharge across therecharge model cells. The values assigned to the pilotpoints were treated as parameters. Each catchmentdepicted in Fig. 6 has a pilot-point interpolation zoneassociated with it. Most of the zones used in this processcontain only a single pilot-point and, therefore, nointerpolation is conducted (which is equivalent to usinga zone for the parameterization). However, some catch-ments contain distinct channels in which some overlandow is observed during signicant rainfall events. In thesezones, the natural recharge is interpolated along thechannel such that the magnitude of recharge diminishesas the channel extends into the model domain (Fig. 6).

Observation dataObservation data consisted of measured water levels inwells (Fig. 10) and measured changes in land-surfaceelevation, i.e., land-surface deformations (Fig. 11).Observed water-level data were used in two ways: as

direct observations of hydraulic head and as observationsof drawdown. Drawdown data were used in this study tohighlight information about water-level dynamics byremoving the impact of overall hydraulic head magni-tudes. Land-surface deformations were measured at selectedbenchmarks (Fig. 11) by sequential leveling surveys, exten-someter measurements, and interferometric synthetic apertureradar (InSAR) data. An extensometer located in Edwards AirForce Base (EAFB), known as the Holly site, directlymeasures compaction at this location and depth (Fig. 11).

Synthetic observation data are also added to controlunreasonably high water levels that could potentiallyresult in discharge to the land surface. Currently,MODFLOW-NWT is designed such that when the watertable rises above the land surface elevation, surface leakagewill occur (Niswonger et al. 2006). This phenomenon hasnot occurred in Antelope Valley since pre-developmentbecause of the appreciable depth to water in the modeldomain, with the exception of springs, which are beingsimulated using the DRN package. The potential for waterto be lost to surface leakage presents a problem whenconducting parameter estimation due to the fact that anunreasonably large amount of natural recharge can bespecied in the model while achieving a reasonable level of

Layer 1Instantaneous compaction

Layer 2

0

0

10

10

20

20 KILOMETERS

MILES

a

N


Model fault

Modelgrid Model boundary

EXPLANATION

Compressibleclay thickness,

in meters0 to 12.1

12.2 to 24.324.4 to 36.536.6 to 48.7

48.8 to 6061 to 73.1

73.2 to 85.385.4 to 97.5

97.6 to 109.7109.8 to 122

Layer 2Delayed compaction

Layer 3


Model fault

Modelgrid Model boundary

0

0

10

10

20

20 KILOMETERS

MILES

EXPLANATION

b

Compressibleclay thickness,

in meters0 to 12.1

12.2 to 24.324.4 to 36.536.6 to 48.7

48.8 to 6061 to 73.1

73.2 to 85.385.4 to 97.5

97.6 to 109.7109.8 to 122

N

Fig. 8 Simulated thicknesses of compressible geologic units within each layer by type of compaction for the groundwater-ow andsubsidence model of Antelope Valley groundwater basin, California: (a) instantaneous compaction thickness (layer 1 and layer 2), and (b)delayed compaction thickness (layer 2 and layer 3)

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calibration, because most of the water is leaving the modelas surface leakage. In this study, this phenomenon iscontrolled by introducing a penalty into the objectivefunction. This penalty is implemented in PEST as a seriesof above-land-control observations. Each model cell,where water should never rise above land surface, containsone of these observations. After each steady-statesimulation within the parameter-estimation process, thedistance between the steady-state water table and landsurface is calculated in each model cell outside of the ETand DRN cells. If the steady-state water table is near landsurface, a non-zero residual is assigned to the above-land-

control observation. This residual then increases as thehydraulic head increases.

Tikhonov regularizationTikhonov regularization (Tikhonov and Arsenin 1977;Doherty 2003) is a form of Bayesian estimation in whicha composite objective function is minimized (Yeh 1986).This composite objective function consists of the sum ofsquared residuals discussed previously, which is oftenreferred to as the least-squares objective function and aBayesian term that penalizes the composite objective

a

b

Fig. 9 Estimated distribution of preconsolidation head and associated pilot point locations for the groundwater-ow and subsidence modelof Antelope Valley groundwater basin, California: (a) layers 1 and 2, and (b) layer 3

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function when parameters deviate from their expectedvalue:

r m 1

where, is the composite or overall objective function, r isthe penalty function for parameter deviations from expectedvalues, m is the least-squares objective function, and is thetrade-off or regularization weight factor. Tikhonov regulariza-tion determines the optimal regularizationweight factor given amodeler-specied level of calibrationthat is, a desired value

for m, denoted as m1. Therefore, the inverse problem is

considered to have converged when mm1; however, inpractice it is often mm1. Throughout the process ofachieving this, the Tikhonov regularization algorithmcontained in PEST will determine the optimal such thatr is minimized as much as possible (Doherty 2010).

Prior estimates, or expected values, of the modelparameters throughout the Antelope Valley groundwater-ow and subsidence model are assigned, for the most part,using the values reported in Leighton and Phillips (2003).Additional parameters, resulting from the modications of the

Fig. 10 Location of observation wells where water levels were used to calibrate the 19152005 transient groundwater-ow andsubsidence model of Antelope Valley groundwater basin, California

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Leighton and Phillips (2003) model made for this study, wereassigned values based on professional judgment and geologicknowledge of the area. Prior estimates of natural rechargewere obtained from the results of both the BCM model andthe results of Leighton and Phillips (2003). The BCM resultswere important for providing prior information about therelative distribution of natural recharge, which was also usedto develop the upper and lower bounds of each naturalrecharge pilot point. Any remaining parameters without priorinformation were assigned an expected value similar to nearbyparameters of the same type. This association tends toward theuse of a simpler model parameterization by interjecting a

precondition for local homogeneity. There were a total of 203independent parameters estimated for this study, which consistof 24 recharge, 92 hydraulic conductivity, 25 storage, 27 faultconductance, and 35 subsidence-related parameters.

Calibrated model simulation results

The comparisons between observations and their corre-sponding model-simulated equivalents for transient waterlevels and subsidence are displayed in Figs. 4, 12 and 13,respectively; additionally, detailed calibration results can

Fig. 11 Locations of benchmarks used to calibrate the transient-state groundwater-ow and subsidence model of Antelope Valleygroundwater basin, California

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be found in Siade et al. (2014). Overall, the modelreproduces historical observations with a reasonable levelof accuracy. Simulated hydraulic head contours for model-layer 2 (the principal aquifer for groundwater extraction)show good agreement with both water-level measurementsand total simulated drawdown at the end of the simulationperiod (Fig. 14). Dividing model-layer 1 of the original 3-layer model by Leighton and Phillips (2003) into twolayers allowed for improved simulated water levels in theLancaster subbasin (Fig. 4). For example, water levels attwo neighboring wells, 7N/11W-9P2 (9P2) and 7N/11W-21E1 (21E1; Figs. 4 and 10), could not have been simulatedaccurately if the model layer they are perforated in isvertically homogeneous. This is due to the fact that theimpedance and conning effects of the shallow clay lenses inthis area cannot be simulated with a vertically homogeneousmodel layer. Simulated total land subsidence also shows

good agreement with observed land-surface deformationthroughout the simulation period (Fig. 13); contoured resultsare displayed for conditions in 1951, when pumpage was atits maximum, and 2005 (Fig. 15).

The simulated hydraulic heads deviate from observedwater levels in the northwestern region of the FingerButtes subbasin and along the mountain-front boundariesin the Pearland and Buttes subbasins (Fig. 1). The modelunderpredicts the transient water levels in the northwest-ern region of the Finger Buttes subbasin by approximately33m at well 10N/15W-33D1 (33D1, Fig. 10). The largedifference between simulated hydraulic heads and mea-sured water levels could be due to the fact that the bedrockslope in this region is relatively steep, or to the presence ofa previously unmapped fault downgradient of well 33D1.Additionally, the model signicantly overpredicts thesteady-state water level at well 08N/16W-10E1 (10E1,

Fig. 12 Relation between measured and simulated hydraulic head values for 4 years spanning the simulation period, for the groundwater-ow and subsidence model of Antelope Valley groundwater basin, California

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Fig. 10); this is also likely due to the steep slope of thebedrock or a misrepresentation of the fault structure in thisregion.

Discrepancies in the Pearland and Buttes subbasins arelikely due to the fact that this entire region is simulated asa single model layer. Some of the observation wells in thePearland and Buttes subbasins are also located near streamchannels where natural recharge occurs (Fig. 10). Themeasured water levels in observation wells along thestream channels vary in response to wet and dry years; forexample well 5N/9W-20K1 (20K1) in Pearland subbasinand wells 6 N/10W-17 N1 (17 N1), 6 N/10W-20P1(20P1), and 6N/9W-30 F1 (30 F1) in Buttes subbasin(Fig. 10). Because natural recharge in the basin issimulated as an average, temporally constant distributionthroughout the simulation (which is a valid assumption forthis study), it is impossible to reproduce any subtleobserved naturally occurring temporal variability inrecharge rates.

The estimated total average annual natural rechargewas about 36 GL/year. This is consistent with the value of37.4 GL/year estimated by Leighton and Phillips (2003).However, this value was used as prior information, i.e., asa starting regularization target; therefore, there may beother values or predictions of natural recharge that alsoreasonably calibrate the model. This issue is addressed inthe following section using systematic predictive uncer-tainty procedures.

Major components of the time-varying groundwaterbudget are shown in Fig. 16. Prior to signicantgroundwater development in the valley, the averageannual natural discharge due to ET, groundwaterunderow, springs, and evaporation through the playasurface was 28.5, 3.1, 1.9, and 0.1 GL/year, respectively.Additionally, at predevelopment (i.e., steady state), asmall amount of water (2.3 GL/year) discharges as

surface leakage in areas not mapped for spring dis-charge. This discharge likely did not exist and is perhapsthe product of model-structure error; however, this rateof discharge is relatively small compared with thesimulated total natural recharge (36 GL/year) and haslittle effect on the overall model-simulated results andestimated natural recharge.

About 18,500 GL of cumulative groundwater pumpagewas specied during the transient simulation period of19152005. The estimated cumulative depletion ingroundwater storage is 10,700 GL. The decline inhydraulic head in the groundwater basin (Fig. 14) is theresult of this depletion in groundwater storage. In turn, thedecline in hydraulic head in the groundwater basin hasresulted in a decrease in natural discharge from the basinand caused compaction of aquitards, resulting in landsubsidence (Fig. 15).

Uncertainty of natural-recharge estimates

The model developed in this study can be used to helpevaluate water-management scenarios throughoutAntelope Valley. However, in order to more effectivelyuse this model, the uncertainty associated with itspredictions should be estimated. In particular, the predic-tion of the distribution and quantity of average annualnatural mountain-front recharge is important to evaluatebecause it is the principal source of natural recharge.

Null-space Monte Carlo analysisPredictive uncertainty stemming from potential parametererror associated with non-uniqueness and insensitivity canbe signicant, particularly when the parameterization iscomplex and the observation data is insufcient to

Fig. 13 Relation between measured and simulated subsidence values for the entire simulation for the groundwater-ow and subsidencemodel of Antelope Valley groundwater basin, California

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uniquely estimate all parameters. To quantitatively assessthe resulting predictive uncertainty, one can apply a basiccalibration-constrained Monte Carlo analysis; however,this approach can be computationally expensive, especial-ly for highly parameterized models such as the modelpresented in this study.

The NSMC method reduces the computational burdenof conducting calibration-constrained Monte Carlo analy-sis using subspace techniques (Tonkin and Doherty 2009).This method employs a nonlinear extension of principalcomponents regression (PCR), in which the originalparameter vector is projected onto a vector subspace witha reduced dimensionality (Jolliffe 2002). Using PCR inlinear regression, the inverse problem is reparameterizedbased on the eigenvectors that span the row space of theregression matrix. For nonlinear regression and NSMC,the regression matrix (conditioned on a given set ofparameter values) becomes the Jacobian matrix containing

the sensitivities of each model parameter to the model-simulated equivalent of each observation (Tonkin andDoherty 2009). This Jacobian matrix can be decomposedusing singular-value decomposition into its respectivesingular values and eigenvectors. The mutually orthogonalunit eigenvectors (that span the row space of the Jacobianmatrix), whose corresponding singular values are signif-icantly non-zero, are assumed to span the calibrationsolution space. Therefore, these eigenvectors dene thetransformed parameters, also known as superparameters,which reside in the solution space. The remaining eigenvec-tors whose corresponding singular values are zero or nearzero, are assumed to span the calibration null space and,therefore, represent superparameters that are relatively ines-timable. As a result, parameter perturbations that have beenprojected onto the calibration null space may have little or noeffect on the least-squares objective function thus, maintain-ing the calibrated state of the model. However, since the

a

b

Fig. 14 Contours of simulated 2005 (a) hydraulic heads and (b) drawdown, for layer 2 from the groundwater-ow and subsidence modelof Antelope Valley groundwater basin, California

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ab

Fig. 15 Contours of simulated (a) 1951 and (b) 2005 land subsidence from the groundwater-ow and subsidence model of AntelopeValley groundwater basin, California

Fig. 16 Simulated groundwater budget components for the Antelope Valley groundwater basin, California, 19152005

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inverse problem is nonlinear, the denition of thesuperparameters is not always valid and parameter perturba-tions projected onto the null space cannot always beguaranteed to calibrate themodel. Therefore, further iterationsof the parameter-estimation algorithm may need to beundertaken to achieve an acceptable level of calibration;however, only the superparameters in the solution space areestimated, which are fewer than the original base parameters.

The methodology that was used to conduct the NSMCanalysis for the Antelope Valley groundwater-ow andsubsidence model using PEST and its suite of utilities isdocumented in Doherty et al. (2010); however, a briefoverview is as follows. First, thousands of randomrealizations of parameter vectors are obtained based onprior probability distributions. For each realization, thecorresponding parameter perturbations from the calibratedparameters are calculated and projected onto the calibra-tion null space. A single iteration of recalibration isconducted if necessary. If the model is sufcientlycalibrated, the prediction (i.e., total annual natural re-charge) is recorded; if the model is not calibrated, therealization is discarded. The predictions associated withthe acceptable realizations are collected and analyzedstatistically. In this study, a realization is assumedsufciently calibrated if the weighted least-squares objec-tive function is below a certain value, which was chosenbased on the calibration of each data type (e.g., transientheads, land-surface deformations, etc.); if the weightedleast-squares objective for a particular data type wasgreater than twice that of the calibrated value, therealization was discarded. Additionally, the prior proba-bility distributions for the parameters were assumed log-uniform between upper and lower bounds.

Prior to conducting the analysis described in thepreceding, the singular-value truncation level (i.e., the

dimension of the solution space and null space) must bedetermined. Depending on how underdetermined theinverse problem is, there may be many near-zero singularvalues. Determining where to divide the solution spacefrom the null space is based on the number of near-zerosingular values; however, dening near-zero is notstraightforward. If the truncation level is set too high,the resulting randomly generated, projected parameterperturbations could signicantly affect the objectivefunction, requiring several iterations of the PEST algo-rithm for recalibration, which is often computationallyinfeasible. However, if the truncation level is set too low,these parameter perturbations may result in an overlynarrow exploration of the predictive uncertainty (Tonkinand Doherty 2009). However, based on the assumption ofa linear model (about the calibrated parameter vector), thepredictive error variance as a function of solution spacedimensionality can be approximated using PEST and itsutilities (Doherty 2010; Fig. 17). Based on this result, thepredictive error variance associated with total naturalrecharge is minimized when the solution space dimen-sionality is 186, which is the truncation level chosen forthis study.

Due to limited computational resources, only oneiteration is conducted in the recalibration step of theNSMC method. This single-iteration approach may notbe adequate to achieve a model with an acceptablelevel of calibration for every Monte Carlo realization.Subsequently, the resulting histogram may not perfectlyrepresent the true histogram (see Tonkin and Doherty(2009)) for an example comparison between one- andtwo-iteration NSMC results). Of the 4,251 realizationstested in the NSMC process, 1,022 were deemedacceptable. The simulated mean natural recharge forthe acceptable realizations was about 40.0 GL/year,

b

a

Fig. 17 ab Contributions of superparameters to the predictive error variance associated with average annual natural mountain-frontrecharge in Antelope Valley groundwater basin, California, using the tools contained in PEST

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with a standard deviation of about 2.5 GL/year. Ahistogram of the simulated predictions of naturalrecharge shows an overall range of about 3549 GL/year (Fig. 18).

The low standard deviation and relatively narrow rangeof predicted quantities of natural recharge indicate that thepredictive uncertainty associated with parameter error islikely to be relatively small. However, it is important tonote that many realizations of the NSMC analysis wereomitted due to failure to recalibrate the model in a singleiteration. Conducting more iterations may result in fewerrejected realizations and a slightly different estimate of theposterior probability distribution for annual naturalrecharge.

Pareto trade-off uncertainty analysisIn addition to the NSMC method, predictive uncertainty ofnatural recharge was also analyzed using a feasibilityanalysis known as a Pareto trade-off analysis (Moore et al.2010; Doherty 2010). This analysis portrays the degrada-tion in model calibration at larger total volumetric rates ofnatural recharge, and is conducted by imposing a penalty,within the objective function, that increases as the model-prediction of interest (i.e., total natural recharge) deviatesfrom a modeler-specied value. For example, if the weighton this penalty is high, the parameter-estimation procedurewill degrade the model t in an attempt to match thedesired, greater recharge rate as closely as possible.Conversely, if this weight is low, the penalty will havevery little effect on the model calibration and the model

predicted recharge rate will likely remain at the initial,calibrated value of 36.0 GL/year.

The Pareto analysis contained in the PEST softwarecan begin with a relatively small weight on this penaltyand incrementally increase this weight until the specied,greater natural recharge rate is nearly met. For each weightvalue considered by PEST, the model is recalibrated suchthat the objective function (which now includes theprediction of interest along with the observations andprior information) becomes as small as possible; for eachweight/calibration, PEST records the prediction of interest.

The observations used in this analysis consisted of thecalibrated-model outcomes, corresponding to each mea-sured value, rather than the measured values themselves(as suggested by Doherty 2010). The Pareto procedurebegins with the calibrated model; therefore, the initialobjective function value is, by denition, zero. Themaximum prediction for natural recharge was set verylarge at about 200 GL/year, and for each solution (that is,each point on the curves in Fig. 19), four iterations of therecalibration procedure were conducted. The choice offour iterations was sufcient due to the fact that the modelcalibration at each point on the Pareto curve stoppedimproving after only a few iterations.

The Pareto curves indicate that the calibration potentialfor very large volumes of annual natural recharge is notlikely, and is driven primarily by degradation in themodels ability to t observations of transient water levels(Fig. 19). A clear inection point in the overall Paretocurve is observed when the natural recharge reaches about54 GL/year. This value is consistent with the largest valueobserved during the NSMC analysis of about 49 GL/year,

Fig. 18 Histogram showing the results of the NSMC method where the prediction of interest is the total annual average mountain-frontrecharge for Antelope Valley groundwater basin, California

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which suggests that the one-iteration approach for therecalibration step of the NSMC procedure may have beensufcient for adequately characterizing the predictiveuncertainty associated with natural recharge for this study.As the natural recharge increases from this point, modelcalibration begins to deteriorate rapidly. The noisybehavior in the Pareto curves associated with extremevalues of natural recharge is likely a result of theinstability and/or nonlinearity associated with subsidencesimulations (Fig. 19).

Natural recharge feasibility testsThe feasibility of the volume of natural recharge used bythe Court was also tested directly by attempting tocalibrate the model with this value using Tikhonovregularization. The Los Angeles County Superior Courtof California ruled a safe yield value of 135.7 GL/yearbased on an estimate of total average annual naturalrecharge of 74.0 GL/year (Beeby et al. 2010a, b). Usingthe distribution of natural recharge resulting from theBCM simulation, the corresponding natural recharge pilotpoint values were calculated such that the total rechargewas 74.0 GL/year. These values were then used as bothinitial values and regularization targets for the parameter-estimation process (i.e., the prior information used in theprevious analyses for the natural recharge pilot points hasnow been altered to support the estimate of natural rechargeused by the Court). The initial values for the remainingparameters were set equal to their respective calibratedvalues. The regularization targets for these remainingparameters also were set equal to their calibrated values(i.e., the prior information associated with the remainingparameters was altered to reect the calibrated model). Inother words, the model was recalibrated in an attempt toproduce a total average natural recharge of 74.0 GL/year,while reasonably matching observed historical water levels

and land-surface deformations. Additionally, the upperbounds associated with horizontal hydraulic conductivitywere also increased to about 305m/day to allow for greaterexibility in obtaining a calibrated model; however, this alsoprovides the potential for values that are not realistic. Thisfeasibility analysis, or recalibration of the model, differsfrom the Pareto analysis discussed previously because theprior information (or regularization targets) and some of theupper bounds of the parameters have been changed.

There were two simulations conducted for this exercise.The rst simulation consisted of the same formulation of theparameter-estimation process used to calibrate the modeloriginally (with the alterations mentioned previously). Thissimulation converged to an objective function value similarto the original calibrated model. The resulting estimatednatural recharge was about 56 GL/year. This value is lessthan that associatedwith the regularization targets, indicatingthat natural recharge values above 56 GL/year will not likelyproduce a reasonably calibrated model. Furthermore, thisvalue is consistent with both the inection point in the Paretocurves and the maximum value observed in the NSMCanalysis. However, the horizontal hydraulic conductivityparameters did not change signicantly from their startingvalues and regularization targets. This is likely due to the factthat PESTs Tikhonov procedure adjusts the inter-regularization group weights independently, which couldlead to relatively high weights placed on matching priorvalues of hydraulic conductivity.

To further address the feasibility of 74 GL/year of naturalrecharge, a second calibration simulation was conducted inwhich the greatest emphasis for matching prior informationwas placed on the natural recharge pilot points. An option inPEST, known as IREGADJ, allows for the relative inter-regularization group weighting to be controlled somewhat bythe user (Doherty 2010). For this second simulation, IREGADJwas set to three, which requires PEST, at each iteration, tohonor the relative inter-regularization group weights set by the

Fig. 19 Pareto curve or trade-off function that results from different rates of total average annual mountain-front recharge for observedearly water levels, transient water levels, drawdown, and total subsidence in the Antelope Valley groundwater basin, California

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user at the outset of the parameter estimation process. In orderto use this option, the inter-regularization weight ratios mustnow be chosen carefully. Since the objective is to match the74.0 GL/year total natural recharge rate as closely as possible,the second simulation was conducted such that the weightsassociated with natural recharge targets were ten times largerthan those associated with the rest of the parameters.

This second simulation converged to an objectivefunction value slightly larger than that of the originalcalibrated model; the largest discrepancy or mist was thatassociated with the transient water-level observations. Theresulting estimated annual natural recharge was about 66GL/year. This value is still less than that associated withthe regularization targets, indicating that a natural rechargerate of 74.0 GL/year may be overestimated. Furthermore,at 66 GL/year, many of the resulting estimated horizontalhydraulic conductivity values were unreasonably large forthe type of geologic deposits in Antelope Valley.

Model structure error estimated agriculturalpumpageAgricultural pumpage is an uncertain component ofgroundwater discharge that was assumed known duringmodel calibration. Conceivably, underestimation of theagricultural pumpage used in this study could result in, forexample, smaller values of hydraulic conductivity in orderto match observed water levels. As a result, attempts atachieving the appropriate initial condition (pre-development orsteady state) may result in an underestimation of naturalrecharge. The converse is also true for overestimation ofpumpage. The exact temporal and spatial uncertainty associatedwith 19162005 agricultural pumpage cannot be quantied.Without knowing the details of this uncertainty, magnitudes ofagricultural pumpage were explored uniformly in both spaceand time. Therefore, in this study, predictive uncertaintyresulting from agricultural pumpage uncertainty was addressedby uniformly perturbing the agricultural pumpage, in space andtime, followed by recalibration of the model.

The estimates of natural recharge made in this studyare much smaller than that made by the Court; therefore,we only consider the potential for underestimation ofpumpage. Therefore, two increases of 10 and 25% inagricultural pumpage were considered for this study. Therecalibrated models for the 10 and 25% increase inagricultural pumpage estimated a natural recharge rate of36.6 and 38.4 GL/year, respectively. As expected, in-creases in estimated natural recharge were observed as aresult of increased agricultural pumpage rates. However,these increases in recharge are relatively small, indicatingthat uncertainty in agricultural pumpage likely has littleeffect on the estimates of natural recharge.

This is only a very basic exploration of the effects ofunderestimated agricultural pumpage on the prediction ofnatural recharge. Indeed, the regularization targets fornatural recharge were set to the same values as duringcalibration, i.e., the targets are based on both the BCMresults and the natural recharge estimate by Leighton andPhillips (2003). Therefore, the results of this analysis only

indicate that it is possible to calibrate the model withapproximately 37 GL/year of natural recharge andincreased agricultural pumpage. This does not indicatethat larger volumes of natural recharge are impossible withlarger values of agricultural pumpage. Furthermore, theagricultural pumpage in this basic analysis is increaseduniformly for only two scenarios. A more sophisticatedinvestigation could be performed in which agriculturalpumpage is treated as a random variable, but such ananalysis is beyond the scope of this study.

Conclusions

A numerical groundwater-ow and land-subsidence modelhas been developed based on the model published byLeighton and Phillips (2003) to estimate the naturalrecharge and its associated uncertainty in the AntelopeValley, California. Mountain-front recharge is assumed tobe the primary source of natural recharge. The numericalsolution procedure known as the Newton Solver inMODFLOW-NWT has been employed. This solversenhanced capability improves the overall numericalstability of MODFLOW with particular improvements insimulating model cells that transition from wet to dry andvice versa. In this study, this is especially importantbecause most of the natural recharge that requiresestimation occurs along the mountain front where thereis often only a single active layer that could become dry orwet quite easily, depending on the parameter values beingimplemented by PEST.

The updated model was calibrated using the parameter-estimation and predictive-uncertainty software suite PESTand prior information was incorporated using theTikhonov regularization functionality in PEST. All modelparameters were regularized such that they tend towardexpected parameter values, which were based on profes-sional judgment, geologic knowledge of the area, BCMmodel results, and the results of Leighton and Phillips(2003). The resulting average annual natural rechargeestimated in this study is about 36 GL/year, which is veryclose to the previous estimate by Leighton and Phillips(2003); however, this value is much smaller than theCourt-determined value of 74 GL/year.

The estimate of natural recharge was considered as theprediction of the model. Predictive uncertainty analysiswas conducted based on both parameter uncertainty andthe model-structure errors associated with underestimatedagricultural pumpage estimates. The NSMC method wasused to explore the likely range of natural recharge giventhe conceptual model used and observations available forcalibration. The mean value result of the NSMC methodwas about 40 GL/year with a standard deviation of about2.5 GL/year. Pareto trade-off concepts were also used toshow the degradation to model calibration that accom-panies increases in natural recharge values. These resultsindicated that when the natural recharge reaches about 54GL/year, the model t degrades dramatically.

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The effects on predictive uncertainty resulting fromunderestimated agricultural pumpage were also consid-ered. The agricultural pumpage was increased uniformlyin time and space by 10 and 25% and recalibrated forboth cases. The resulting estimated natural recharge for anincrease of 10 and 25% was 36.6 and 38.4 GL/year,respectively. This indicates that the uncertainty associatedwith agricultural pumpage has little effect on the estimateof natural recharge.

Acknowledgements This work was supported by the US Geolog-ical Survey Cooperative Water Program, Los Angeles CountyDepartment of Public Works, Antelope ValleyEast Kern WaterAgency, Palmdale Water District, and Edwards Air Force Base.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use,distribution, and reproduction in any medium, provided you giveappropriate credit to the original author(s) and the source, provide a linkto the Creative Commons license, and indicate if changes were made.

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Natural...AbstractIntroductionDescription of study areaHydrogeologyPre-development recharge and dischargePost-development recharge and dischargeLand subsidence and aquifer-system compaction

Groundwater flow and subsidence modelModel discretizationModel boundaries

Model calibrationModel parameterizationObservation dataTikhonov regularization

Calibrated model simulation resultsUncertainty of natural-recharge estimatesNull-space Monte Carlo analysisPareto trade-off uncertainty analysisNatural recharge feasibility testsModel structure errorestimated agricultural pumpage

ConclusionsReferences

natural recharge estimation and uncertainly analysis of an adjudicated groundwater basin using a...

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