How to Correlate Turbidity with Total Suspended Solids Using Regression
Turbidity and total suspended solids (TSS) are closely related water-quality measurements, but they are not interchangeable. Turbidity describes how strongly suspended particles scatter or absorb light, while TSS measures the dry mass of material retained by a filter. Regression analysis provides a practical way to translate an optical turbidity signal into an estimated concentration of suspended solids.
A useful regression model depends on the water body, sediment source, particle-size distribution, sensor configuration, and sampling method. A relationship developed in a clear, mineral-rich river may perform poorly in a wastewater stream containing organic flocs or in a coastal area where fine sediment changes with tides. Calibration therefore has to be treated as a site-specific measurement process rather than a universal conversion.
The objective is to pair reliable turbidity observations with laboratory TSS results, fit a defensible model, and then verify its performance under the range of conditions in which the sensor will operate. When this process is documented carefully, an optical monitor can provide high-frequency estimates of suspended sediment between laboratory sampling events.
Understand What Each Measurement Represents
Turbidity is an optical property. A sensor sends light into the water and measures scattered or transmitted light, commonly reporting the result in nephelometric turbidity units (NTU) or a comparable instrument-specific unit. The reading responds to the number, size, shape, color, and refractive properties of particles in the optical path. Bubbles, fouling, ambient light, and sensor geometry can also influence the signal.
TSS is a gravimetric measurement. A known volume of water is passed through a filter, the retained material is dried under controlled conditions, and the increase in filter mass is divided by the sample volume. Results are usually reported in milligrams per liter (mg/L). This method measures mass directly, but it represents only the collected sample and can be affected by filter handling, drying temperature, sample homogeneity, and laboratory detection limits.
Because the two variables respond to different physical properties, a strong correlation is common but not guaranteed. A small quantity of highly scattering clay may produce substantial turbidity, whereas a larger mass of dark organic particles may create a weaker optical response. Understanding this distinction prevents a regression equation from being treated as a fixed physical law.
Optical instruments used for sediment monitoring can vary in wavelength, beam arrangement, path length, detector angle, and signal processing. These details matter when comparing data from different devices or combining historical records. The technical background on optical sensing technologies helps explain why sensor design should be considered during calibration.
Build A Representative Calibration Dataset
The quality of the regression is controlled primarily by the quality and range of the paired dataset. Each calibration record should contain a turbidity measurement and a TSS result collected from the same location and as close together in time as practical. In fast-changing flows, even a short delay can create a mismatch because a sediment plume may rise or fall rapidly.
Collect samples across the full expected operating range rather than concentrating on normal conditions. Include low, medium, and high turbidity events, as well as rising and falling limbs of storms, dredging operations, discharge cycles, or tidal movement. A model built from only low-flow samples may underestimate high-concentration events and produce misleading sediment-load calculations.
The sample location should represent the water seen by the sensor. If the instrument is mounted near the bank while the laboratory bottle is collected from the channel center, the two measurements may describe different particle populations. Depth, flow velocity, stratification, and local sediment resuspension should be considered when selecting both the monitoring point and the sampling method.
Record metadata with every observation. Useful fields include date and time, sensor serial number, deployment depth, water temperature, conductivity, flow rate, weather, recent disturbance, sample volume, filter type, laboratory result, and any maintenance performed. These records make it easier to identify outliers and determine whether a change in correlation reflects sediment behavior or an instrument problem.
Prepare The Data Before Fitting
Begin by checking units, timestamps, duplicate records, missing values, and laboratory quality-control results. Confirm that turbidity readings are within the instrument’s calibrated range and that TSS samples were processed consistently. A single transcription error can have a disproportionate effect when the dataset is small.
Plot TSS against turbidity before choosing a model. A scatter plot often reveals whether the relationship is linear, curved, clustered, or affected by a few influential observations. Plotting the data on logarithmic axes can also be informative when concentrations span several orders of magnitude. However, a visually straight line on log axes does not automatically justify a log-transformed model; the residual behavior and prediction purpose must be evaluated as well.
Outliers require investigation rather than automatic removal. A high residual may result from a genuine sediment pulse, a poorly mixed sample, a trapped air bubble, an obstructed optical window, or a laboratory handling error. Retain a point when it represents a real condition the monitoring program must capture. Exclude or correct it only when there is documented evidence of invalid measurement.
If several samples are collected during one event, avoid treating all of them as fully independent without considering serial correlation. Closely spaced observations may contain less independent information than samples gathered across separate hydrological events. Splitting validation data by event, rather than randomly by individual row, provides a more realistic test of future performance.
Select And Interpret A Regression Model
For an initial model, ordinary least squares regression often takes the form:
[ \widehat{\mathrm{TSS}} = a + b(\mathrm{Turbidity}) ]
Here, (a) is the intercept and (b) is the slope. The slope estimates the change in TSS associated with a one-unit increase in turbidity, while the intercept represents the predicted TSS when turbidity is zero. A negative intercept is not automatically a flaw, but it may produce physically impossible negative predictions at low turbidity and should be examined carefully.
A log-log model can be useful when the relationship is multiplicative or when variance increases with concentration:
[ \log(\mathrm{TSS}) = a + b\log(\mathrm{Turbidity}) ]
This model can improve residual behavior across a wide range, but back-transforming predictions requires care because the arithmetic mean of transformed predictions is not necessarily the mean on the original scale. A correction for retransformation bias may be needed, especially when the model is used for mass-load estimates.
Other options include polynomial regression, segmented regression, robust regression, and generalized additive models. These approaches can describe curvature or reduce sensitivity to unusual observations, but added complexity should be justified by physical behavior and validation results. An elaborate model that fits one deployment perfectly may be less stable than a simple linear equation when conditions change.
The coefficient of determination, or (R^2), is useful for describing explained variation, but it should not be the only performance measure. Report root mean square error (RMSE), mean absolute error (MAE), bias, prediction intervals, and the number and range of samples. A model can have a high (R^2) while still producing unacceptable errors at the low end or during important sediment events.
| Model approach | Useful when | Main limitation | Important check |
|---|---|---|---|
| Linear raw-scale | Turbidity and TSS show a roughly straight relationship | Unequal variance can weaken predictions | Inspect residuals across concentration |
| Log-log | Data cover a wide range and variability grows with concentration | Back-transformed estimates may be biased | Apply and report retransformation method |
| Polynomial | The scatter plot shows consistent curvature | Can behave unrealistically outside calibration range | Restrict predictions to observed conditions |
| Segmented | Different sediment sources or regimes create distinct slopes | Requires enough data around the breakpoint | Confirm breakpoint with field evidence |
| Robust regression | A few valid extreme points strongly affect least squares | May underweight important high events | Compare with ordinary least squares |
| Event-specific model | Seasonal or operational conditions produce separate relationships | Requires more samples and model management | Validate each regime independently |
Validate The Equation In Field Conditions
Validation measures how well the equation predicts observations that were not used to estimate its coefficients. With a small dataset, k-fold cross-validation can provide a useful estimate of predictive error. With a larger monitoring program, hold out complete storms, seasons, dredging campaigns, or discharge periods. This tests whether the calibration transfers across events rather than merely reproducing neighboring samples.
Examine residuals, defined as observed TSS minus predicted TSS. Randomly distributed residuals around zero support the model structure. A curved pattern suggests that a linear equation is inadequate, while increasing spread indicates heteroscedasticity. A consistent positive or negative pattern indicates bias that may affect reported sediment concentrations or cumulative loads.
Prediction intervals should accompany individual estimates whenever decisions depend on uncertainty. A regression confidence interval describes uncertainty in the average fitted relationship; a prediction interval is wider because it also includes the variability of a future observation. For compliance, dredging control, or sediment-budget applications, the prediction interval is often more relevant than a confidence interval around the mean.
Validation should include practical checks on the sensor itself. Compare the optical signal with independent field observations, inspect the instrument window, review zero checks, and confirm that readings are not saturated during peak events. If the sensor has multiple gain ranges or automatic range switching, verify that the regression remains continuous across transitions.
Account For Changing Sediment Conditions
A turbidity–TSS relationship can shift when the source or composition of suspended material changes. Rainfall may introduce fine mineral particles, while low-flow conditions may favor organic debris or algae. Dredging can generate a plume with a different particle-size distribution from naturally transported sediment. Seasonal bed changes, vegetation growth, and upstream construction can all alter the optical response per unit mass.
Where these changes are predictable, use stratified calibration. Separate models may be appropriate for wet-season and dry-season conditions, different tidal phases, or distinct operational states. Another approach is to add explanatory variables such as conductivity, water level, flow, or particle-size indicators. However, additional predictors should be included only when they are measured reliably and improve independent validation.
Sensor installation also affects the relationship. A monitor positioned in a quiescent zone may experience settling or localized accumulation, while a high-velocity position can expose it to air entrainment and rapidly varying particle concentrations. The sampling design should describe the hydraulic setting and ensure that the calibration samples represent the instrument’s measurement volume.
Long-term deployments need recalibration triggers. Establish review points after major floods, equipment relocation, changes in dredging method, persistent fouling, or evidence of model drift. Trending the residuals over time can reveal a gradual loss of performance before the reported TSS estimates become unusable.
Apply A Defensible Calibration Workflow
A repeatable workflow makes regression results easier to audit and maintain. Start with a written sampling plan, then preserve raw sensor files and laboratory records separately from processed datasets. Keep the final equation, calibration range, date of fit, validation method, and known limitations with the deployment documentation.
The following practices improve the reliability of an estimated TSS time series:
- Collect paired samples over the complete expected turbidity and sediment-concentration range.
- Use event-based validation so the test data represent future field conditions.
- Report MAE, RMSE, bias, prediction intervals, and calibration limits alongside (R^2).
- Investigate outliers using field notes, sensor diagnostics, and laboratory quality-control records.
- Reassess the model after major changes in sediment source, hydraulics, sensor position, or deployment purpose.
Once validated, apply the equation only within its observed calibration range unless additional evidence supports extrapolation. Flag negative, saturated, or otherwise implausible predictions rather than silently replacing them. If the resulting TSS estimates will be multiplied by discharge to calculate suspended-sediment load, propagate uncertainty through that calculation and identify periods when either concentration or flow data are unreliable.
For organizations selecting equipment, deployment support, or product-management information, the D & A Instruments website provides access to background material associated with turbidity monitoring and suspended-solids applications. Technical context is especially valuable when deciding whether a calibration should be general, event-specific, or tied to a particular optical configuration.
A regression equation becomes useful when it is treated as part of a measurement system rather than as a standalone spreadsheet formula. Pair representative samples with stable sensor data, document the sediment conditions, validate predictions by event, and review performance as the site changes. These steps turn turbidity monitoring into a practical, transparent estimate of suspended solids that can support environmental research, dredging oversight, hydrology, and long-term water-quality records.