Stochastic Rainfall Model Research Articles

Representing low‐frequency variability in continuous rainfall simulations: A hierarchical random Bartlett Lewis continuous rainfall generation model

AbstractLow‐frequency variability, in the form of the El Niño‐Southern Oscillation, plays a key role in shaping local weather systems. However, current continuous stochastic rainfall models do not account for this variability in their simulations. Here a modified Random Pulse Bartlett Lewis stochastic generation model is presented for continuous rainfall simulation exhibiting low‐frequency variability. Termed the Hierarchical Random Bartlett Lewis Model (HRBLM), the model features a hierarchical structure to represent a range of rainfall characteristics associated with the El Niño‐Southern Oscillation with parameters conditioned to vary as functions of relevant climatic states. Long observational records of near‐continuous rainfall at various locations in Australia are used to formulate and evaluate the model. The results indicate clear benefits of using the hierarchical climate‐dependent structure proposed. In addition to accurately representing the wet spells characteristics and observed low‐frequency variability, the model replicates the interannual variability of the antecedent rainfall preceding the extremes, which is known to be of considerable importance in design flood estimation applications.

TROPICAL DAILY RAINFALL AMOUNT MODELLING USING MARKOV CHAIN-MIXED EXPONENTIAL (MCME)

This study is concerned with the development of a stochastic rainfall model that can generate many sequences of synthetic daily rainfall series with the similar properties as those of the observed. The proposed model is Markov chain-mixed exponential (MCME). This model is based on a combination of rainfall occurrence (represented by the first-order two-state Markov chain) and the distribution of rainfall amounts on wet days (described by the mixed exponential distribution). The feasibility of the MCME model is assessed using daily rainfall data from four rainfall stations (station S02, S05, S07 and S11) in Johor, Malaysia. For all the rainfall stations, it was found that the proposed MCME model was able to describe adequately rainfall occurrences and amounts. Various statistical and physical properties of the daily rainfall processes also considered. However, the validation results show that the models’ predictive ability was not as accurate as their descriptive ability. The model was found to have fairly well ability in predicting the daily rainfall process at station S02, S05 and S07. Nonetheless, it was able to predict the daily rainfall process at station S11 accurately.

Stochastic Models for the Disaggregation of Precipitation Time Series on Sub-Daily Scale: Identiication of Parameters by Global Optimization

Stochastic disaggregation model, based on coupling of the modified version of the Bartlett-Lewis Rectangular Pulse stochastic rainfall model and proportional adjusting procedure, is shown to disaggregate daily observed precipitation to hourly scale. Furthermore synthetic hourly time series are generated.This model requires the identification of a set of parameters that allow to reproduce, as well as possible, the statistical properties of the observed precipitation. The identification is formulated as a global optimization problem. A comparison between observed and modeled statistics of the precipitation time series is presented for the weather station of San Martino Valle Caudina (Southern Italy).

Derivation and evaluation of landslide-triggering thresholds by a Monte Carlo approach

Abstract. Assessment of landslide-triggering rainfall thresholds is useful for early warning in prone areas. In this paper, it is shown how stochastic rainfall models and hydrological and slope stability physically based models can be advantageously combined in a Monte Carlo simulation framework to generate virtually unlimited-length synthetic rainfall and related slope stability factor of safety data, exploiting the information contained in observed rainfall records and field-measurements of soil hydraulic and geotechnical parameters. The synthetic data set, dichotomized in triggering and non-triggering rainfall events, is analyzed by receiver operating characteristics (ROC) analysis to derive stochastic-input physically based thresholds that optimize the trade-off between correct and wrong predictions. Moreover, the specific modeling framework implemented in this work, based on hourly analysis, enables one to analyze the uncertainty related to variability of rainfall intensity within events and to past rainfall (antecedent rainfall). A specific focus is dedicated to the widely used power-law rainfall intensity–duration (I–D) thresholds. Results indicate that variability of intensity during rainfall events influences significantly rainfall intensity and duration associated with landslide triggering. Remarkably, when a time-variable rainfall-rate event is considered, the simulated triggering points may be separated with a very good approximation from the non-triggering ones by a I–D power-law equation, while a representation of rainfall as constant–intensity hyetographs globally leads to non-conservative results. This indicates that the I–D power-law equation is adequate to represent the triggering part due to transient infiltration produced by rainfall events of variable intensity and thus gives a physically based justification for this widely used threshold form, which provides results that are valid when landslide occurrence is mostly due to that part. These conditions are more likely to occur in hillslopes of low specific upslope contributing area, relatively high hydraulic conductivity and high critical wetness ratio. Otherwise, rainfall time history occurring before single rainfall events influences landslide triggering, determining whether a threshold based only on rainfall intensity and duration may be sufficient or it needs to be improved by the introduction of antecedent rainfall variables. Further analyses show that predictability of landslides decreases with soil depth, critical wetness ratio and the increase of vertical basal drainage (leakage) that occurs in the presence of a fractured bedrock.

Regionalization of the Modified Bartlett–Lewis rectangular pulse stochastic rainfall model across the Korean Peninsula

The six parameters of the Modified Bartlett–Lewis Rectangular Pulse (MBLRP) model were regionalized across the Korean Peninsula for all 12 calendar months. The parameters of the MBLRP model were estimated at each of the 59 rain gauges and they were spatially interpolated using the Ordinary Kriging method in order to produce maps. The parameter search space used in the parameter estimation process was repetitively narrowed through cross-validation in order to remove the impact of the multi-modality of the MBLRP model. The synthetic rainfall time series generated based on the parameter maps successfully reproduced the various statistical properties of the observed rainfall, such as mean, variance, lag-1 autocorrelation, and probability of zero rainfall at a wide range of time accumulation levels (e.g. hourly through daily). The maps representing the general rainfall characteristics, such as the average rainfall depth per rain storm, the average rain storm duration, the average number of rain cells per rain storm, and the average rain cell duration were also produced based on the estimated parameters. Lastly, some helpful tips in regionalizing the parameters of the Poisson cluster rainfall models are discussed.

Statistical intercomparison of idealized rainfall measurements using a stochastic fractional dynamics model

AbstractA statistical method is developed for comparing precipitation data from measurements performed by (hypothetical) perfect instruments using a recently developed stochastic model of rainfall. The stochastic dynamical equation that describes the underlying random process naturally leads to a consistent spectrum and incorporates the subtle interdependence of the length and time scales governing the statistical fluctuations of the rain rate field. The main attraction of such a model is that the complete set of second‐moment statistics embodied in the space‐time covariance of both the area‐averaged instantaneous rain rate (represented by radar or passive microwave data near the ground) and the time‐averaged point rain rate (represented by rain gauge data) can be expressed as suitable integrals over the spectrum. With the help of this framework, the model allows one to carry out a faithful intercomparison of precipitation estimates derived from radar or passive microwave remote sensing over an area with direct observations by rain gauges or disdrometers, assuming all the measuring instruments to be ideal. A standard linear regression analysis approach to the intercomparison of radar and gauge rain rate estimates is formulated in terms of the appropriate observed and model‐derived quantities. We also estimate the relative sampling error as well as separate absolute sampling errors for radar and gauge measurements of rainfall from the spectral model.

Application of a stochastic weather generator to assess climate change impacts in a semi-arid climate: The Upper Indus Basin

Summary Assessing local climate change impacts requires downscaling from Global Climate Model simulations. Here, a stochastic rainfall model (RainSim) combined with a rainfall conditioned weather generator (CRU WG) have been successfully applied in a semi-arid mountain climate, for part of the Upper Indus Basin (UIB), for point stations at a daily time-step to explore climate change impacts. Validation of the simulated time-series against observations (1961–1990) demonstrated the models’ skill in reproducing climatological means of core variables with monthly RMSE of Differences between a control (1961–1990) and future (2071–2100) regional climate model (RCM) time-slice experiment were then used to provide change factors which could be applied within the rainfall and weather models to produce perturbed ‘future’ weather time-series. These project year-round increases in precipitation (maximum seasonal mean change:+27%, annual mean change: +18%) with increased intensity in the wettest months (February, March, April) and year-round increases in mean temperature (annual mean +4.8 °C). Climatic constraints on the productivity of natural resource-dependent systems were also assessed using relevant indices from the European Climate Assessment (ECA) and indicate potential future risk to water resources and local agriculture. However, the uniformity of projected temperature increases is in stark contrast to recent seasonally asymmetrical trends in observations, so an alternative scenario of extrapolated trends was also explored. We conclude that interannual variability in climate will continue to have the dominant impact on water resources management whichever trajectory is followed. This demonstrates the need for sophisticated downscaling methods which can evaluate changes in variability and sequencing of events to explore climate change impacts in this region.

Stochastic Modelling of the Daily Rainfall Frequency and Amount

Daily rainfall is an important origin of water supply to the dry land in cultivated areas of Pakistan where the inhabitants primarily rely on farming. Stochastic rainfall models are concerned with generation of the occurrence and amount of daily rainfall. To generate the sequence of rainfall occurrence, first-order Markov chain model was employed by using the method of transitional probability matrices, while rainfall amount was generated by applying gamma distribution. The parameters of model were estimated from the archives of daily rainfall records for the period 1981–2010. The shape and scale parameters were estimated by method of moments and hence it became possible to find the parametric values at the study areas and then generate synthetic sequences according to the gamma distribution. The essential parameters for the stochastic generation include the means, variance or standard deviation and conditional and unconditional probabilities of wet and dry days.

Statistical downscaling of precipitation using a stochastic rainfall model conditioned on circulation patterns – an evaluation of assumptions

ABSTRACTFor climate impact assessment regarding hydrology, the availability of long precipitation time series with high temporal and spatial resolution is essential. A possible approach to obtain this data is the statistical downscaling of precipitation simulated by a global climate model (GCM) using a stochastic rainfall model with parameters conditioned on circulation patterns (CP). This approach requires: (1) the existence of a strong relationship between CP and precipitation, (2) the sufficient reproduction of CPs by the GCM, (3) the adequate simulation of precipitation by the rainfall model and (4) either stationarity of the relationship between precipitation and CPs or an approach to account for non‐stationarity. The objective of this research is the careful evaluation and discussion of the above stated four hypotheses. For this purpose, a case study for the Aller–Leine river basin in Northern Germany has been created. It has been found that CPs can be defined which show significant differences in precipitation behaviour. The CPs derived from re‐analysis data are well reproduced by the GCM simulations. In addition, the hourly stochastic rainfall model simulates the observed precipitation characteristics well, except for a certain overestimation of the extremes. However, the change in rainfall between past and future time periods as predicted by a regional climate model could not be explained by the change in CP frequency, due to the non‐stationarity of the relationship between rainfall and CP. This can be best accounted for by re‐estimating the parameters of the stochastic rainfall model for future conditions based on corrected observations using a delta change approach regarding simulated rainfall from a regional climate model.

A comparative study of mixed exponential and Weibull distributions in a stochastic model replicating a tropical rainfall process

A stochastic rainfall model is presented for the generation of hourly rainfall data in an urban area in Malaysia. In view of the high temporal and spatial variability of rainfall within the tropical rain belt, the Spatial–Temporal Neyman–Scott Rectangular Pulse model was used. The model, which is governed by the Neyman–Scott process, employs a reasonable number of parameters to represent the physical attributes of rainfall. A common approach is to attach each attribute to a mathematical distribution. With respect to rain cell intensity, this study proposes the use of a mixed exponential distribution. The performance of the proposed model was compared to a model that employs the Weibull distribution. Hourly and daily rainfall data from four stations in the Damansara River basin in Malaysia were used as input to the models, and simulations of hourly series were performed for an independent site within the basin. The performance of the models was assessed based on how closely the statistical characteristics of the simulated series resembled the statistics of the observed series. The findings obtained based on graphical representation revealed that the statistical characteristics of the simulated series for both models compared reasonably well with the observed series. However, a further assessment using the AIC, BIC and RMSE showed that the proposed model yields better results. The results of this study indicate that for tropical climates, the proposed model, using a mixed exponential distribution, is the best choice for generation of synthetic data for ungauged sites or for sites with insufficient data within the limit of the fitted region.

Analysis of the Applicability of Parameter Estimation Methods for a Stochastic Rainfall Model

추계학적 강우모형 NSRPM (Neyman-Scott Rectangular Pulse Model)은 RPM (Rectangular Pulse Model)에서 반영하지 못하는 강우의 군집특성을 잘 반영하여 시간 스케일의 강우를 생성함으로서 수문학적 적용성이 뛰어난 강우모형이다. NSRPM은 5개의 모형 매개변수로 이루어져 있으며 매개변수 추정을 위하여 최적화 기법으로 널리 쓰이고 있는 수치해석 기법인 DFP (Davidon-Fletcher-Powell)기법, 직접적 탐색 기법인 유전자 알고리즘을 사용하고 있다. 그러나 DFP 기법은 입력 초기값에 민감하며 국소 최저치에 수렴하는 확률이 높은 단점이 있으며 유전자 알고리즘기법은 탐색에 소요되는 시간이 많이 걸린다는 단점이 있다. 본 연구에서 사용된 Nelder-Mead기법은 순차적 탐색기법으로 연산 속도가 빠르며 입력 초기값이 필요하지 않아 사용하기 쉬운 장점을 가지고 있다. 본 연구는 전국 지상기상관측소 59개소를 대상으로 1973-2011년 39년 동안의 시간강우 자료를 수집하고 최적화 기법 DFP 기법, 유전자 알고리즘, Nelder-Mead 기법을 이용하여 NSRPM의 매개변수를 추정하여 지속시간 1시간, 6시간, 12시간, 24시간 별 평균, 분산, 공분산에 대해서 각 기법의 정확성을 평가하였다. 본 연구결과 전반적으로 Nelder-Mead기법이 가장 높은 정확도를 보였으며 유전자 알고리즘, DFP 기법 순으로 나타났다. A stochastic rainfall model, NSRPM (Neyman-Scott Rectangular Pulse Model), is able to reflect the cluster characteristics of rainfall events which is unable in the RPM (Rectangular Pulse Model). Therefore NSRPM has advantage in the hydrological applications. The NSRPM consists of five model parameters and the parameters are estimated using optimization techniques such as DFP (Davidon-Fletcher-Powell) method and genetic algorithm. However the DFP method is very sensitive in initial values and is easily converge to local minimum. Also genetic algorithm has disadvantage of long computation time. Nelder-Mead method has several advantages of short computation time and no need of a proper initial value. In this study, the applicability of parameter estimation methods was evaluated using rainfall data of 59 national rainfall networks from 1973-2011. Overall results demonstrated that accuracy in parameter estimation is in the order of Nelder-Mead method, genetic algorithm, and DFP method.

World’s Greatest Observed Point Rainfalls: Jennings (1950) Scaling Law

Abstract The observed relation of worldwide precipitation maxima P versus duration d follows the Jennings scaling law, P ≈ db, with scaling coefficient b ≈ 0.5. This scaling is demonstrated to hold for single-station rainfall extending over three decades. A conceptual stochastic rainfall model that reveals similar scaling behavior is introduced as a first-order autoregressive process [AR(1)] to represent the lower tropospheric vertical moisture fluxes, whose upward components balance the rainfall while the downward components are truncated and defined as no rain. Estimates of 40-yr ECMWF Re-Analysis (ERA-40) vertical moisture flux autocorrelations (at grids near the rainfall stations) provide estimates for the truncated AR(1). Subjected to maximum depth-duration analysis, the scaling coefficient b ≈ 0.5 is obtained extending for about two orders of magnitude, which is associated with a wide range of vertical moisture flux autocorrelations 0.1 < a < 0.7.

A rainfall model for drought risk analysis in south-east UK

Drought risk assessment ideally requires long-term rainfall records especially where inter-annual droughts are of potential concern, and spatially consistent estimates of rainfall to support regional and inter-regional scale assessments. This paper addresses these challenges by developing a spatially consistent stochastic model of monthly rainfall for south-east UK. Conditioned on 50 gauged sites, the model infills the historic record from 1855–2011 in both space and time, and extends the record by synthesising droughts which are consistent with the observed rainfall statistics. The long record length allows more insight into the variability of rainfall and potentially a stronger basis for risk assessment than is generally possible. It is shown that, although localised biases exist in both space and time, the model results are generally consistent with the observed record including for a range of inter-annual droughts and spatial statistics. Simulations show that some of the most severe inter-annual droughts on the record may recur, despite a trend towards generally wetter winters.

Quantifying effects of catchments storage thresholds on step changes in the flood frequency curve

[1] Previous work has shown that non linear catchment response related to a storage threshold may translate into a step change in the flood frequency curve. The aim of this paper is to understand the controls of this step change for catchments where runoff is generated by the saturation excess mechanism and a clear separation between a permanently saturated region and a variably saturated region with spatially uniform storage deficits exists. The magnitude of the step change is quantified by the maximum of the second derivative (curvature) of the flood peaks with respect to their return period. Sensitivity analyses with a stochastic rainfall model and a simple rainfall runoff model show that the magnitude of the step change decreases with increasing temporal variability of antecedent soil storage, and increases with increasing area of the variably saturated region. The return period where the step change occurs is very similar to the return period of the rainfall volume that is needed to exceed the storage threshold. Diagrams are presented that show the joint effects of spatial and temporal storage variability on the magnitude and return period of the step change. The diagrams are useful for assessing whether step changes in the flood frequency curve are likely to occur in catchments where the runoff generation characteristics are as examined here and the flood records are too short to indicate a step change.

Downscaling transient climate change with a stochastic weather generator for the Geer catchment, Belgium

CR Climate Research Contact the journal Facebook Twitter RSS Mailing List Subscribe to our mailing list via Mailchimp HomeLatest VolumeAbout the JournalEditorsSpecials CR 57:95-109 (2013) - DOI: https://doi.org/10.3354/cr01170 Downscaling transient climate change with a stochastic weather generator for the Geer catchment, Belgium S. Blenkinsop1,*, C. Harpham2, A. Burton1, P. Goderniaux3, S. Brouyère4, H. J. Fowler1 1Water Resource Systems Research Laboratory, School of Civil Engineering and Geosciences, Newcastle University, Cassie Building, Newcastle upon Tyne NE1 7RU, UK 2Climatic Research Unit, School of Environmental Sciences, University of East Anglia, Norwich NR47TJ, UK 3Geology and Applied Geology, University of Mons, 7000 Mons, Belgium 4Group of Hydrogeology and Environmental Geology—Aquapôle, University of Liège, 4000 Liège, Belgium *Email: stephen.blenkinsop@ncl.ac.uk ABSTRACT: The coarse resolution of climate models creates the need for future scenarios which are downscaled to an appropriate spatial scale. Considerable effort has been devoted to the development of downscaling methods, but a number of important issues remain to meet users’ needs. These include the assessment of uncertainty for future scenarios, and the production of scenarios at time scales relevant to stakeholders. This paper describes a methodology which addresses these issues by producing a multi-model ensemble of transient climate-change scenarios. The method couples an existing stochastic rainfall model to a new, transient implementation of a weather generator, using changes projected by an ensemble of regional climate model experiments. The methodology is demonstrated by the generation of transient scenarios of daily rainfall, temperature and potential evapotranspiration for the Geer catchment in Belgium for the period 2010-2085. The utility of these scenarios is demonstrated by assessing the changes projected by the simulated time series of several temperature indices. The Geer is projected to experience a decrease in the occurrence of frost days with a corresponding shortening of the frost season and lengthening of the growing season. By examining an ensemble of transient scenarios, the range of uncertainty in these projections is assessed, but, further, it is suggested that additional information on the projected timing of specified threshold events or system responses may be provided. This could aid stakeholders in assessing the likely time scales of required interventions and adaptation responses. KEY WORDS: Climate change · Transient scenarios · Weather generator · Downscaling · RCMs Full text in pdf format NextCite this article as: Blenkinsop S, Harpham C, Burton A, Goderniaux P, Brouyère S, Fowler HJ (2013) Downscaling transient climate change with a stochastic weather generator for the Geer catchment, Belgium. Clim Res 57:95-109. https://doi.org/10.3354/cr01170 Export citation RSS - Facebook - Tweet - linkedIn Cited by Published in CR Vol. 57, No. 2. Online publication date: August 20, 2013 Print ISSN: 0936-577X; Online ISSN: 1616-1572 Copyright © 2013 Inter-Research.

A new daily weather generator to preserve extremes and low-frequency variability

This paper addresses deficiencies of stochastic Weather Generators (WGs) in terms of reproduction of low-frequency variability and extremes, as well as the unanticipated effects of changes to precipitation occurrence under climate change scenarios on secondary variables. A new weather generator (named IWG) is developed in order to resolve such deficiencies and improve WGs performance. The proposed WG is composed of three major components, including a stochastic rainfall model able to reproduce realistic rainfall series containing extremes and inter-annual monthly variability, a multivariate daily temperature model conditioned to the rainfall occurrence, and a suitable multi-variate monthly generator to fit the low-frequency variability of daily maximum and minimum temperature series. The performance of IWG was tested by comparing statistical characteristics of the simulated and observed weather data, and by comparing statistical characteristics of the simulated runoff outputs by a daily rainfall-runoff model fed by the generated and observed weather data. Furthermore, IWG outputs are compared with those of the well-known LARS-WG weather generator. The tested characteristics are a variety of different daily statistics, low-frequency variability, and distribution of extremes. It is concluded that the performance of the IWG is acceptable, better than LARS-WG in the majority of tests, especially in reproduction of extremes and low-frequency variability of weather and runoff series.

Further developments of a transient Poisson-cluster model for rainfall

Stochastic point processes for rainfall are known to be able to preserve the temporal variability of rainfall on several levels of aggregation (e.g. hourly, daily), especially when the cluster approach is used. One major assumption in most of the applications todate is the stationarity of the rainfall properties in time, which must be reconsidered under a climate change hypothesis. Here, we propose new theoretical developments of a Poisson-based model with cluster, namely the Neyman–Scott Rectangular Pulses Model, which treats storm frequency as a nonstationary function. In this paper, storm frequency is modelled as a linear function of time in order to reproduce an increase (or decrease) of the mean annual precipitation. The basic theory is reconsidered and the moments are derived up to the third order. Then, a calibration method based on the generalized method of moments is proposed and discussed. An application to a rainfall time series from Uccle illustrates how this model can reproduce a trend for the average rainfall. This work opens new avenues for future developments on transient stochastic rainfall models and highlights the major challenges linked to this approach.

Step changes in the flood frequency curve: Process controls

Empirical distribution functions of flood peaks in small catchments sometimes show discontinuities in the slope; that is, the largest flood peaks are significantly larger than the rest of the record. The aim of this paper is to understand whether these discontinuities, or step changes, can be a consistent effect of hydrological processes. We conducted field surveys in two Austrian alpine catchments 73 km2in size to map the spatial patterns of surface runoff generation and hydrogeologic storage. On the basis of this information, we selected the parameters of a distributed continuous runoff model, which is designed to simulate well the point when the storage capacity of the catchment is exhausted. Then we calibrated a stochastic rainfall model and performed Monte Carlo simulations of runoff to generate flood frequency curves for the two catchments. The curves exhibit a step change around a return period of 30 years. An analysis of the storage capacities suggests that this step change is due to a threshold of storage capacity being exceeded, which causes fast surface runoff in large parts of the catchments. The threshold occurs when the storage within the catchment is spatially rather uniform. To identify step changes, reliable estimates of the catchment storage capacity are needed on the basis of detailed hydrogeological information. The occurrence of a step change is of importance for estimating low‐probability floods since the flood estimates with the step change accounted for can be significantly different from those based on commonly used distribution functions. We therefore suggest that step changes in the flood frequency curve of small catchments can be real and their possible presence should be taken into account in design flood estimation.

Identifying the dependency pattern of daily rainfall of Dhaka station in Bangladesh using Markov chain and logistic regression model

Bangladesh is a subtropical monsoon climate characterized by wide seasonal variations in rainfall, moderately warm temperatures, and high humidity. Rainfall is the main source of irrigation water everywhere in the Bangladesh where the inhabitants derive their income primarily from farming. Stochastic rainfall models were concerned with the occurrence of wet day and depth of rainfall for different regions to model the daily occurrence of rainfall and achieved satisfactory results around the world. In connection to the Markov chain of different order, logistic regression is conducted to visualize the dependence of current rainfall upon the rainfall of previous two-time period. It had been shown that wet day of the previous two time period compared to the dry day of previous two time period influences positively the wet day of current time period, that is the dependency of dry-wet spell for the occurrence of rain in the rainy season from April to September in the study area. Daily data are collected from meteorological department of about 26 years on rainfall of Dhaka station during the period January 1985-August 2011 to conduct the study. The test result shows that the occurrence of rainfall follows a second order Markov chain and logistic regression also tells that dry followed by dry and wet followed by wet is more likely for the rainfall of Dhaka station and also the model could perform adequately for many applications of rainfall data satisfactorily.

Estimating functions and the generalized method of moments

Estimating functions provide a very general framework for statistical inference, and are particularly useful when one is either unable or unwilling to specify a likelihood function. This paper aims to provide an accessible review of estimating function theory that has potential for application to the analysis and modelling of a wide range of complex systems. Assumptions are given in terms that can be checked relatively easily in practice, and some of the more technical derivations are relegated to an online supplement for clarity of exposition. The special case of the generalized method of moments is considered in some detail. The main points are illustrated by considering the problem of inference for a class of stochastic rainfall models based on point processes, with simulations used to demonstrate the performance of the methods.