Spatial Stream Network | Poisson Consulting

Spatial Stream Network Analysis of Skeena Watershed Stream Temperatures 2025

Wed, 10 Sep 2025 00:00:00 +0000

The suggested citation for this analytic appendix is:

Hill, N.E., Thorley, J.L., & Irvine, A. (2025) Spatial Stream Network Analysis of Skeena Watershed Stream Temperatures 2025. A Poisson Consulting Analytic Appendix. URL: https://www.poissonconsulting.ca/f/1130667589.

Background

The primary goal of the current analyses is to answer the following questions:

How can we model stream temperature to include spatial correlation through a stream network, and predict growing season degree days?

Data Preparation

Hourly water temperature data collected in the Skeena watershed in northern British Columbia between 2015 and 2025 were downloaded from water-temp-bc, which stores a combination of historic and recent real-time hydrometric data from Environment and Climate Change Canada.

Hourly air temperature data (at two metres above ground level) for the Skeena watershed for the years 2015-2025 were downloaded from the ERA-5-Land simulation using the Copernicus Climate Change Service (C3S) Climate Data Store as NetCDF files (Muñoz Sabater 2019).

The data were extracted, cleaned (i.e., checked for errors and corrected if possible), and tidied (i.e., manipulated into a consistent format) using using R version 4.5.1 (R Core Team 2022).

Key assumptions of the data preparation included:

For each site, the time series of stream temperature values were classified as “unreasonable” if they were \(<\) 0˚C or \(>\) 30˚C, changed more than 2˚C from one hour to the next, were adjacent to another unreasonable value, or within a 5-hour gap between two unreasonable value; these were excluded from the analysis.
The simulated air temperature from the modeled simulation is a reasonable approximation of the truth.
Discharge data were not included in this analysis because Pacific Climate Impacts Consortium’s Gridded Hydrologic Model Output does not currently have output for the Skeena watershed.
Seven sites with few observed data points proved insufficient to capture the annual-scale fluctuations in stream temperature; these sites (station numbers 08EC014, 08ED004, 08EE008, 08EE025, 08EF001, 08EG012, and 08EG018) were excluded from the analysis.

Statistical Analysis

Model parameters were estimated using Bayesian methods. Where sufficient information was available, causal parameters were estimated accounting for biases due to correlation by embedding the assumed causal pathways into the model in the form of a causal network (Arif and MacNeil 2023) which is represented visually as a directed acyclic graph (DAG). The estimates were produced using Stan (Carpenter et al. 2017). For additional information on Bayesian estimation the reader is referred to McElreath (2020).

Unless stated otherwise, the Bayesian analyses used weakly informative prior distributions (Gelman et al. 2017). The posterior distributions were estimated from 1500 Markov Chain Monte Carlo (MCMC) samples thinned from the second halves of 3 chains (Kery and Schaub 2011, 38–40). Model convergence was confirmed by ensuring that the potential scale reduction factor \(\hat{R} \leq 1.05\) (Kery and Schaub 2011, 40) and the effective sample size (Brooks et al. 2011) \(\textrm{ESS} \geq 150\) for each of the monitored parameters (Kery and Schaub 2011, 61).

Model adequacy was assessed via posterior predictive checks (Kery and Schaub 2011). More specifically, the proportion of zeros in the data and the first four central moments (mean, variance, skewness, and kurtosis) of the deviance residuals were compared to the expected values by simulating new data based on the posterior distribution and assumed sampling distribution and calculating the deviance residuals.

The sensitivity of the posteriors to the choice of prior distributions was evaluated by separately power-scaling the priors and likelihood, estimating the properties of the perturbed posteriors using Pareto smoothed importance sampling (Vehtari et al. 2017), and evaluating the distance between the base and perturbed posteriors using the cumulative Jensen-Shannon (CJS) distance (Kallioinen et al. 2023). A threshold CJS distance of 0.05 was used to indicate sensitivity; a prior sensitivity above this value suggests a strong prior, and a likelihood sensitivity below this value suggests that the likelihood is non-informative (Kallioinen et al. 2023). For hierarchical effects, only the top-level parameter was power-scaled to avoid perturbing the priors multiple times (Kallioinen et al. 2023).

The parameters are summarized in terms of the point estimate, lower and upper 95% compatibility limits (Rafi and Greenland 2020), and the surprisal s-value (Greenland 2019). Together a pair of lower and upper compatibility limits (CLs) are referred to as a compatibility interval (CI). The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The s-value indicates how surprising it would be to discover that the true value of the parameter is in the opposite direction to the estimate (Greenland 2019). An s-value of \(>\) 4.32 bits, which is equivalent to a p-value \(<\) 0.05 (Kery and Schaub 2011; Greenland and Poole 2013), indicates that the surprise would be equivalent to throwing at least 4.3 heads in a row on a fair coin.

Variable selection was based on the heuristic of directional certainty (Kery and Schaub 2011; Murtaugh 2014; Castilho and Prado 2021). Fixed effects were included if their s-value was \(>\) 4.32 bits (Kery and Schaub 2011). Based on a similar argument, random effects were included if their standard deviation had a lower 95% CL \(>\) 5% of the median estimate.

The results are displayed graphically by plotting the modeled relationships between individual variables and the response with the remaining variables held constant. In general, continuous and discrete fixed variables are held constant at their arithmetic mean and first level values, respectively, while random effects are held constant at their typical value (Kery and Schaub 2011, 77–82). Unless stated otherwise the typical value is the arithmetic mean. When informative the influence of a particular variable is expressed in terms of the effect size (i.e., relative change in the response variable) with the 95% CI (Bradford et al. 2005).

The analyses were implemented using R version 4.5.1 (R Core Team 2022) and the embr family of packages.

Model Descriptions

Stream Temperature

The data were analyzed using a Spatial Stream Network model (Ver Hoef and Peterson 2010; Peterson and Hoef 2010), with code adapted from the SSNbayes package (Santos-Fernandez et al. 2022). The necessary stream network distances and connectivity were calculated using the BC Freshwater Atlas and the SSNbler (Peterson et al. 2024) and SSN2 (Dumelle et al. 2024) R packages. Air and stream temperature data were averaged by site and week; modeling was done on this weekly time scale.

The expected stream temperatures were modeled using the 3-parameter version of the air2stream model (Toffolon and Piccolroaz 2015). The average stream temperature (in ˚C) in the first week, \(W_{s,j=1}\) was estimated by the model and assumed to be the same for all sites.

For all subsequent weeks (i.e., \(j > 1\)), the change in the stream temperature (in ˚C) between week \(j - 1\) and week \(j\) for the \(s^{th}\) site, \(\Delta W_{s,j}\), was modeled as follows:

\[\begin{equation} \Delta W_{s,j} = a1_s + a2_s A_{s,j} - a3_s W_{s,j - 1}, \end{equation}\]

where \(a1_s\), \(a2_s\), and \(a3_s\) are the parameters of the air2stream model for the \(s^{th}\) site, \(A_{s,j}\) is the air temperature (in ˚C) for the \(s^{th}\) site in the \(j^{th}\) week and \(W_{s, j - 1}\) is the expected stream temperature at the \(s^{th}\) site in the previous week.

The expected stream temperature for the \(s^{th}\) site in the \(j^{th}\) week was then calculated:

\[\begin{equation} W_{s,j} = W_{s,j - 1} + \Delta W_{s,j}. \end{equation}\]

Growing Season Degree Days (GSDD) are the accumulated thermal units (in ˚C) during the growing season based on the mean daily water temperature values, which is a useful predictor of age-0 rainbow and westslope cutthroat trout size at the beginning of winter. The start and end of the growing season were based on the definitions of Coleman and Fausch (2007):

Start: the beginning of the first week that average stream temperatures exceeded and remained above 5˚C for the season.
End: the last day of the first week that average stream temperature dropped below 4˚C.

GSDD were derived for each site and year by assuming that the daily stream temperatures at each site were the predicted weekly mean stream temperature for every day in the given week.

Key assumptions of the model include:

The stream network is dendritic, not braided.
The expected stream temperatures were set to 0˚C if they were estimated to be negative by the model.
The stream temperature in the first week is the same for all sites.
The parameters of the air2stream model (\(a1\), \(a2\), and \(a3\)) vary randomly by site.
The residual variation is multivariate normally distributed.
The covariance structure of the residual variation combines the following covariance components:
- Nugget (allows for variation at a single location)
- Exponential tail-down (allows for spatial dependence between flow-connected and flow-unconnected locations)

Preliminary analysis found that:

Allowing the initial stream temperature to vary randomly by site produced unrealistic stream temperatures for January (> 10˚C).

Model Templates

Stream Temperature

data {
  int nsite;
  int nweek;

  int <lower=0> N_y_obs; // number observed values
  int <lower=0> N_y_mis; // number missing values
  array[N_y_obs] int<lower=1> i_y_obs;
  array[N_y_mis] int<lower=1> i_y_mis;
  vector [N_y_obs] y_obs;  // matrix[N_y_obs,1] y_obs[T];
  array[nsite * nweek] real air_temp;
  array[nsite * nweek] int<lower=0> site;
  array[nsite * nweek] int<lower=0> week;

  matrix [nsite, nsite] D;
  matrix [nsite, nsite] I;
  matrix [nsite, nsite] H;
  matrix [nsite, nsite] flow_con_mat;

parameters {
  vector<lower=0, upper=30>[N_y_mis] y_mis; // declaring the missing y

  real<lower=0> sigma_nug; // sd of nugget effect
  real<lower=0> sigma_td; // sd of tail-down
  real<lower=0> alpha_td; // range of the tail-down model
  real<lower=0> bInitialTemp;
  real<lower=0> s1;
  real<lower=0> s2;
  real<lower=0> s3;
  real<lower=-5, upper=15> m1;
  real<lower=-5, upper=1.5> m2;
  real<lower=-5, upper=5> m3; 
  array[nsite] real<lower=-5, upper=15> a1;
  array[nsite] real<lower=-5, upper=1.5> a2;
  array[nsite] real<lower=-5, upper=5> a3;

transformed parameters {
  vector[nsite * nweek] y; // long vector of y
  array[nweek] vector[nsite] Y;
  matrix[nsite, nsite] C_td; // tail-down cov
  real <lower=0> var_nug; // nugget
  real <lower=0> var_td; // partial sill tail-down

  vector[nsite * nweek] eTempDiff;
  vector<lower=0, upper=30>[nsite * nweek] eTemp; 
  array[nweek] vector[nsite] mu;
  y[i_y_obs] = y_obs;
  y[i_y_mis] = y_mis;
  var_nug = sigma_nug^2; // variance nugget
  var_td = sigma_td^2; // variance tail-down
  // Place observations into matrices
  for (t in 1:nweek){
    Y[t] = y[((t - 1) * nsite + 1):(t * nsite)];
  }
  eTemp[1:nsite] = rep_vector(bInitialTemp, nsite);
  for (i in (nsite + 1):(nweek * nsite)) {
    eTempDiff[i] = (a1[site[i]] + a2[site[i]] * air_temp[i] - a3[site[i]] * eTemp[i - nsite]);
    
    eTemp[i] = eTemp[i - nsite] + eTempDiff[i];
    if (eTemp[i] < 0) {
      eTemp[i] = 0.0;
    }
  }
  // Define 1st mu
  mu[1] = eTemp[1:nsite];
  // Define rest of mu; ----
  for (t in 2:nweek){
    mu[t] = eTemp[((t - 1) * nsite + 1):(t * nsite)];
  }
  // Covariance matrices ----
  // Tail-down exponential model
    for (i in 1:nsite) {
    for (j in 1:nsite) {
      if (flow_con_mat[i, j] == 1) { // if points are flow connected
        C_td[i, j] = var_td * exp(-3 * H[i, j] / alpha_td);
      }
      else{ // if points are flow unconnected
        C_td[i, j] = var_td * exp(-3 * (D[i, j] + D[j, i]) / alpha_td);
      }
    }
  }

model {
  sigma_nug ~ exponential(0.05); // sd nugget
  sigma_td ~ exponential(2); // sd tail-down
  alpha_td ~ normal(20000, 20000); // range tail-down
  bInitialTemp ~ normal(1, 1);

  s1 ~ exponential(1);
  s2 ~ exponential(1);
  s3 ~ exponential(1);
  m1 ~ normal(0.8, 1);
  m2 ~ normal(0.4, 1);
  m3 ~ normal(0.4, 1);
  a1 ~ normal(m1, s1);
  a2 ~ normal(m2, s2);
  a3 ~ normal(m3, s3);

  for (t in 1:nweek) {
    target += multi_normal_cholesky_lpdf(Y[t] | mu[t], cholesky_decompose(C_td + var_nug * I + 1e-6));
  }

generated quantities {
  array[nweek * nsite] real ePrediction;
  array[nweek] real log_lik;
  real lprior;
  for (t in 1:nweek) {
    log_lik[t] = multi_normal_cholesky_lpdf(Y[t] | mu[t], cholesky_decompose(C_td + var_nug * I + 1e-6));
    ePrediction[((t - 1) * nsite + 1):(t * nsite)] = to_array_1d(multi_normal_cholesky_rng(mu[t], cholesky_decompose(C_td + var_nug * I + 1e-6)));
  }
  lprior = exponential_lpdf(sigma_nug | 0.05) +
           exponential_lpdf(sigma_td | 2) +
           (normal_lpdf(alpha_td | 20000, 20000) - normal_lccdf(0 | 20000, 20000)) + // truncated normal
           (normal_lpdf(bInitialTemp | 1, 1) - normal_lccdf(0 | 1, 1)) + // truncated normal
           normal_lpdf(m1 | 0.8, 1) +
           normal_lpdf(m2 | 0.4, 1) +
           normal_lpdf(m3 | 0.4, 1) +
           exponential_lpdf(s1 | 1) +
           exponential_lpdf(s2 | 1) +
           exponential_lpdf(s3 | 1);

Block 1. Model description.

Results

Tables

Locations

Table 1. Stream temperature site locations.

site	station name	longitude	latitude
08EB003	SKEENA RIVER AT GLEN VOWELL	-127.673	55.3011
08EB004	KISPIOX RIVER NEAR HAZELTON	-127.716	55.4338
08EB005	SKEENA RIVER ABOVE BABINE RIVER	-127.687	55.7171
08EB007	KITWANGA RIVER NEAR KITWANGA	-128.054	55.1164
08EC001	BABINE RIVER AT BABINE	-126.63	55.3225
08EC004	PINKUT CREEK NEAR TINTAGEL	-125.43	54.4054
08EC013	BABINE RIVER AT OUTLET OF NILKITKWA LAKE	-126.698	55.4265
08ED001	NANIKA RIVER AT OUTLET OF KIDPRICE LAKE	-127.452	53.9303
08ED002	MORICE RIVER NEAR HOUSTON	-127.427	54.1168
08EE003	BULKLEY RIVER NEAR HOUSTON	-126.719	54.3994
08EE004	BULKLEY RIVER AT QUICK	-126.9	54.6186
08EE005	BULKLEY RIVER NEAR SMITHERS	-127.133	54.7697
08EE012	SIMPSON CREEK AT THE MOUTH	-127.204	54.8099
08EE013	BUCK CREEK AT THE MOUTH	-126.65	54.3961
08EE020	TELKWA RIVER BELOW TSAI CREEK	-127.497	54.6074
08EF005	ZYMOETZ RIVER ABOVE O.K. CREEK	-128.325	54.4936
08EG019	KITSUMKALUM RIVER BELOW ALICE CREEK	-128.744	54.6793

Stream Temperature

Table 2. Parameter descriptions.

Parameter	Description
`C_td`	Covariance matrix of the tail-down exponential model
`D`	Downstream hydrologic distance matrix
`H`	Total hydrologic distance matrix
`I`	The identity matrix
`N_y_mis`	Number of missing water temperature values
`N_y_obs`	Number of observed water temperature values
`Y[t]`	Vector of water temperature values for all sites in the `t`^th week
`a1[i]`	Intercept-type parameter of the air2stream model for the `i`^th site
`a2[i]`	Effect of `air_temp[i]` on `eTempDiff[i]` for the `i`^th site
`a3[i]`	Effect of the previous week’s expected water temperature (`eTemp[i - nsite]`) on `eTempDiff[i]`, for the `i`^th site
`air_temp[i]`	The `i`^th air temperature value (˚C)
`alpha_td`	The variance of spatially independent points
`bInitialTemp`	Expected average water temperature for the week starting 01-01-2015 for all sites
`eTempDiff[i]`	Expected difference in average water temperature from the previous week
`eTemp[i]`	Expected value of `water_temp[i]`
`flow_con_mat`	Site connectivity matrix
`i_y_mis`	Indexes of missing water temperature values
`i_y_obs`	Indexes of observed water temperature values
`m1`	Mean of the site-wise random effect for the `a1` parameter
`m2`	Mean of the site-wise random effect for the `a2` parameter
`m3`	Mean of the site-wise random effect for the `a3` parameter
`mu[t]`	Vector of `eTemp` values for all sites in the `t`^th week
`nsite`	Number of sites
`nweek`	Number of weeks
`s1`	Standard deviation of the site-wise random effect for the `a1` parameter
`s2`	Standard deviation of the site-wise random effect for the `a2` parameter
`s3`	Standard deviation of the site-wise random effect for the `a3` parameter
`sigma_nug`	Standard deviation of the nugget effect
`sigma_td`	Standard deviation of the exponential tail-down covariance model
`site[i]`	The `i`^th site
`var_nug`	Variance of the nugget effect
`var_td`	Variance of the exponential tail-down covariance model
`week[i]`	The `i`^th week
`y[i]`	The `i`^th water temperature value (˚C)
`y_mis`	Vector of missing water temperature values
`y_obs`	Vector of observed water temperature values

Table 3. Model coefficients.

term	estimate	lower	upper	svalue
alpha_td	1.93e+04	1.03e+04	4.35e+04	10.6
bInitialTemp	1.68	3.33e-01	2.78	10.6
m1	9.40e-01	7.51e-01	1.13	10.6
m2	2.60e-01	2.01e-01	3.24e-01	10.6
m3	3.19e-01	2.65e-01	3.77e-01	10.6
s1	3.56e-01	2.57e-01	5.23e-01	10.6
s2	1.29e-01	9.34e-02	1.91e-01	10.6
s3	1.10e-01	7.93e-02	1.63e-01	10.6
sigma_nug	8.06e-01	7.87e-01	8.24e-01	10.6
sigma_td	8.35e-01	7.79e-01	8.95e-01	10.6

Table 4. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged	num_divergent	max_treedepth	ebfmi
9401	3802	3	500	2	482	1.014	TRUE	0	0	1.13

Table 5. Model sensitivity.

information	prior_cjs	lik_cjs	nterms	insensitive
strong prior	0.088	0.611	2	FALSE
weak prior & informative data	0.032	0.224	13250	TRUE

Table 6. Model sensitivity by parameter.

parameter	information	prior_cjs	lik_cjs	nterms	insensitive
a1	weak prior & informative data	0.006	0.295	17	TRUE
a2	weak prior & informative data	0.007	0.281	17	TRUE
a3	weak prior & informative data	0.008	0.292	17	TRUE
alpha_td	strong prior	0.078	1.016	1	FALSE
bInitialTemp	strong prior	0.088	0.611	1	FALSE
ePrediction	weak prior & informative data	0.032	0.224	9401	TRUE
m1	weak prior & informative data	0.006	0.34	1	TRUE
m2	weak prior & informative data	0.006	0.431	1	TRUE
m3	weak prior & informative data	0.006	0.598	1	TRUE
s1	weak prior & informative data	0.017	0.665	1	TRUE
s2	weak prior & informative data	0.007	0.822	1	TRUE
s3	weak prior & informative data	0.013	0.558	1	TRUE
sigma_nug	weak prior & informative data	0.012	4.76	1	TRUE
sigma_td	weak prior & informative data	0.006	0.846	1	TRUE
y_mis	weak prior & informative data	0.029	0.239	3792	TRUE

Table 7. Mean GSDD by year (with 95% prediction intervals).

year	estimate	lower	upper
2015	2031	1944	2117
2016	2120	2034	2211
2017	1906	1829	1992
2018	2044	1958	2125
2019	2042	1962	2137
2020	1815	1741	1896
2021	1925	1847	2016
2022	1984	1910	2060
2023	2233	2157	2315
2024	2006	1922	2093

Table 8. Mean GSDD by site (with 95% prediction intervals).

site	estimate	lower	upper
08EE020	1029	964	1098
08EE012	1365	1310	1418
08EB005	1653	1605	1706
08EF005	1743	1683	1804
08EB003	1869	1817	1921
08EB007	1910	1841	1978
08ED001	1937	1884	1987
08EE013	2030	1983	2077
08EG019	2056	1992	2119
08EB004	2062	2011	2113
08ED002	2139	2079	2200
08EE005	2309	2251	2367
08EE003	2312	2265	2359
08EC001	2330	2272	2385
08EE004	2364	2313	2416
08EC013	2449	2401	2499
08EC004	2639	2588	2688

Figures

Stream Temperature

Figure 1. Tail-down covariance by hydrologic distance (with 95% CIs).

Figure 2. Predicted water temperature by date (with 95% prediction intervals). The points are the observed data.

Figure 3. Predicted GSDD by year and site (with 95% prediction intervals).

Figure 4. GSDD median estimate and width of 95% prediction interval (PI) in 2015 by site. The black lines are the stream network.

Figure 5. GSDD median estimate and width of 95% prediction interval (PI) in 2016 by site. The black lines are the stream network.

Figure 6. GSDD median estimate and width of 95% prediction interval (PI) in 2017 by site. The black lines are the stream network.

Figure 7. GSDD median estimate and width of 95% prediction interval (PI) in 2018 by site. The black lines are the stream network.

Figure 8. GSDD median estimate and width of 95% prediction interval (PI) in 2019 by site. The black lines are the stream network.

Figure 9. GSDD median estimate and width of 95% prediction interval (PI) in 2020 by site. The black lines are the stream network.

Figure 10. GSDD median estimate and width of 95% prediction interval (PI) in 2021 by site. The black lines are the stream network.

Figure 11. GSDD median estimate and width of 95% prediction interval (PI) in 2022 by site. The black lines are the stream network.

Figure 12. GSDD median estimate and width of 95% prediction interval (PI) in 2023 by site. The black lines are the stream network.

Figure 13. GSDD median estimate and width of 95% prediction interval (PI) in 2024 by site. The black lines are the stream network.

Acknowledgements

The organizations and individuals whose contributions have made this analytic appendix possible include:

Hillcrest Geographics
- Simon Norris
Poisson Consulting
- Priscilla Durojaiye

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Rafi, Zad, and Sander Greenland. 2020. “Semantic and Cognitive Tools to Aid Statistical Science: Replace Confidence and Significance by Compatibility and Surprise.” BMC Medical Research Methodology 20 (1): 244. https://doi.org/10.1186/s12874-020-01105-9.

Santos-Fernandez, Edgar, Jay M. Ver Hoef, Erin E. Peterson, James McGree, Daniel Isaak, and Kerrie Mengersen. 2022. “Bayesian Spatio-Temporal Models for Stream Networks.” Computational Statistics & Data Analysis 170 (June): 107446. https://doi.org/10.1016/j.csda.2022.107446.

Toffolon, Marco, and Sebastiano Piccolroaz. 2015. “A Hybrid Model for River Water Temperature as a Function of Air Temperature and Discharge.” Environmental Research Letters 10 (11): 114011. https://doi.org/10.1088/1748-9326/10/11/114011.

Vehtari, Aki, Andrew Gelman, and Jonah Gabry. 2017. “Practical Bayesian Model Evaluation Using Leave-One-Out Cross-Validation and WAIC.” Statistics and Computing 27 (5): 1413–32. https://doi.org/10.1007/s11222-016-9696-4.

Ver Hoef, Jay M., and Erin E. Peterson. 2010. “A Moving Average Approach for Spatial Statistical Models of Stream Networks.” Journal of the American Statistical Association 105 (489): 6–18. https://doi.org/10.1198/jasa.2009.ap08248.

Spatial Stream Network Analysis of Nechako Watershed Stream Temperatures 2022b

Fri, 25 Oct 2024 00:00:00 +0000

The suggested citation for this analytic appendix is:

Hill, N.H., Thorley, J.L., & Irvine, A. (2024) Spatial Stream Network Analysis of Nechako Watershed Stream Temperatures 2022b. A Poisson Consulting Analysis Appendix. URL: https://www.poissonconsulting.ca/f/1295467017.

Background

The primary goal of the current analyses is to answer the following question:

How can we model stream temperature to include spatial correlation through a stream network?

Data Preparation

Sub-hourly water temperature data collected in the Nechako Watershed in northern British Columbia between 2019 and 2021 were downloaded as csv files from Zenodo (Morris et al. 2022).

Hourly air temperature data (at two metres above ground level) for the Nechako Watershed for the years 2019-2021 were downloaded from the ERA-5-Land simulation using the Copernicus Climate Change Service (C3S) Climate Data Store as NetCDF files (Muñoz Sabater 2019).

Daily baseflow and surface runoff data for the Nechako Watershed for the years 2019-2021 using the ACCESS1-0_rcp85 climate scenario were downloaded from the Pacific Climate Impacts Consortium’s Gridded Hydrologic Model Output as NetCDF files (Victoria Pacific Climate Impacts Consortium 2020).

The data were prepared for analysis using R version 4.4.1 (R Core Team 2022).

Key assumptions of the data preparation included:

All stream temperature data are correct, except those flagged “Fail”, which were excluded from analysis (see Gilbert et al. (2022) for details).
The simulated air temperature and discharge data from their respective modeled simulations are reasonable approximations of the truth.
The daily discharge at each stream temperature site was calculated by taking the weighted average of the sum of the baseflow and runoff for the watershed area upstream of the site; these were then averaged over weekly periods for each site.
One site that corresponded to just four consecutive observed data points proved insufficient to capture the annual-scale fluctuations in stream temperature; this site (WHC) was dropped from the analysis.
There were 146 observations with negative stream temperatures measurements, which were all set to 0˚C.

Statistical Analysis

Model parameters were estimated using Bayesian methods. The estimates were produced using Stan (Carpenter et al. 2017). For additional information on Bayesian estimation the reader is referred to McElreath (2020).

The sensitivity of the posteriors to the choice of prior distributions was evaluated by doubling the standard deviations of the priors and then using \(\hat{R}\) to evaluate whether the samples were drawn from the same posterior distribution (Thorley and Andrusak 2017).

The parameters are summarized in terms of the point estimate, lower and upper 95% compatibility limits (Rafi and Greenland 2020) and the surprisal s-value (Greenland 2019). Together a pair of lower and upper compatibility limits (CLs) are referred to as a compatibility interval (CI). The estimate is the median (50th percentile) of the MCMC samples while the 95% CLs are the 2.5th and 97.5th percentiles. The s-value indicates how surprising it would be to discover that the true value of the parameter is in the opposite direction to the estimate (Greenland 2019). An s-value of \(>\) 4.32 bits, which is equivalent to a p-value \(<\) 0.05 (Kery and Schaub 2011; Greenland and Poole 2013), indicates that the surprise would be equivalent to throwing at least 4.3 heads in a row on a fair coin.

The analyses were implemented using R version 4.4.1 (R Core Team 2022) and the embr family of packages.

Model Descriptions

Stream Temperature

The data were analyzed using a Spatial Stream Network model (Ver Hoef and Peterson 2010; Peterson and Hoef 2010), with code adapted from the SSNbayes package (Santos-Fernandez et al. 2022). The necessary stream network distances and connectivity were calculated using the BC Freshwater Atlas. Air and stream temperature data were averaged by site and week; modeling was done on this weekly time scale.

The expected stream temperatures were modeled using the 4-parameter version of the air2stream model (Toffolon and Piccolroaz 2015). The average stream temperature (in ˚C) in the first week, \(W_{s,j=1}\) was estimated by the model and assumed to be the same for all sites.

For all subsequent weeks (i.e., \(j > 1\)), the change in the stream temperature (in ˚C) between week \(j - 1\) and week \(j\) for the \(s^{th}\) site, \(\Delta W_{s,j}\), was modeled as follows:

\[\begin{equation} \Delta W_{s,j} = \frac{1}{(\theta_{s,j})^{a4_s}}(a1_s + a2_s A_{s,j} - a3_s W_{s,j - 1}), \end{equation}\]

where \(a1_s\), \(a2_s\), \(a3_s\), and \(a4_s\) are the parameters of the air2stream model for the \(s^{th}\) site, \(A_{s,j}\) is the air temperature (in ˚C) for the \(s^{th}\) site in the \(j^{th}\) week, \(W_{s, j - 1}\) is the expected stream temperature at the \(s^{th}\) site in the previous week, and \(\theta_{s,j}\) is the dimensionless discharge for the \(s^{th}\) site in the \(j^{th}\) week. \(\theta_{s,j}\) was calculated as follows:

\[\begin{equation} \theta_{s,j} = \frac{d_{s,j}}{\bar{d_s}} \end{equation}\]

where \(d_{s,j}\) is the discharge for the \(s^{th}\) site in the \(j^{th}\) week, and \(\bar{d_s} = \frac{\sum_{j = 1}^{J}(d_{s,j})}{J}\) is the mean discharge across all \(J\) weeks for the \(s^{th}\) site.

The expected stream temperature for the \(s^{th}\) site in the \(j^{th}\) week was then calculated:

\[\begin{equation} W_{s,j} = W_{s,j - 1} + \Delta W_{s,j}. \end{equation}\]

Start: the beginning of the first week that average stream temperatures exceeded and remained above 5˚C for the season.
End: the last day of the first week that average stream temperature dropped below 4˚C.

GSDD were derived for each site and year by assuming that the daily stream temperatures at each site were the predicted weekly mean stream temperature for every day in the given week.

Key assumptions of the model include:

The stream network is dendritic, not braided.
The expected stream temperatures were set to 0˚C if they were estimated to be negative by the model.
The stream temperature in the first week is the same for all sites.
The parameters of the air2stream model (\(a1\), \(a2\), \(a3\), and \(a4\)) vary randomly by site.
The residual variation is multivariate normally distributed.
The covariance structure of the residual variation combines the following covariance components:
- Nugget (allows for variation at a single location)
- Exponential tail-down (allows for spatial dependence between flow-connected and flow-unconnected locations)

Preliminary analysis found that:

The exponential tail-down model was better at explaining the spatial correlation in the data than exponential tail-up or euclidean distance models (Ver Hoef and Peterson 2010; Peterson and Hoef 2010).
The full 8-parameter air2stream model did not converge.
Preliminary analysis found that allowing the initial stream temperature to vary randomly by site produced unrealistic stream temperatures for January (> 10˚C).

Model Templates

Stream Temperature

.data {
  int nsite;
  int nweek;

  int <lower=0> N_y_obs; // number observed values
  int <lower=0> N_y_mis; // number missing values
  int <lower=1> i_y_obs [N_y_obs] ;  // [N_y_obs,T]
  int <lower=1> i_y_mis [N_y_mis] ;  // [N_y_mis,T]
  vector [N_y_obs] y_obs;  // matrix[N_y_obs,1] y_obs[T];
  real discharge [nsite * nweek];
  real air_temp [nsite * nweek];
  int<lower=0> site[nsite * nweek];
  int<lower=0> week[nsite * nweek];

  matrix [nsite, nsite] D;
  matrix [nsite, nsite] I;
  matrix [nsite, nsite] H;
  matrix [nsite, nsite] flow_con_mat;

parameters {
  vector<lower=0, upper=30>[N_y_mis] y_mis; // declaring the missing y

  real<lower=0> sigma_nug; // sd of nugget effect
  real<lower=0> sigma_td; // sd of tail-down
  real<lower=0> alpha_td; // range of the tail-down model
  real<lower=0> bInitialTemp;
  real<lower=0> s1;
  real<lower=0> s2;
  real<lower=0> s3;
  real<lower=0> s4;
  real<lower=-5, upper=15> m1;
  real<lower=-5, upper=1.5> m2;
  real<lower=-5, upper=5> m3; 
  real<lower=-1, upper=1> m4;
  real<lower=-5, upper=15> a1[nsite];
  real<lower=-5, upper=1.5> a2[nsite]; 
  real<lower=-5, upper=5> a3[nsite]; 
  real<lower=-1, upper=1> a4[nsite];

transformed parameters {
  vector[nsite * nweek] y; // long vector of y
  vector[nsite] Y[nweek]; // array of y
  matrix[nsite, nsite] C_td; // tail-down cov
  real <lower=0> var_nug; // nugget
  real <lower=0> var_td; // partial sill tail-down

  vector[nsite * nweek] eTempDiff;
  vector<lower=0, upper=30>[nsite * nweek] eTemp;
  vector[nsite] mu [nweek];
  y[i_y_obs] = y_obs;
  y[i_y_mis] = y_mis;
  var_nug = sigma_nug^2; // variance nugget
  var_td = sigma_td^2; // variance tail-down
  // Place observations into matrices
  for (t in 1:nweek){
    Y[t] = y[((t - 1) * nsite + 1):(t * nsite)];
  }
  eTemp[1:nsite] = rep_vector(bInitialTemp, nsite);
  for (i in (nsite + 1):(nweek * nsite)) {
    eTempDiff[i] = (1/(discharge[i]^a4[site[i]])) * (a1[site[i]] + a2[site[i]] * air_temp[i] - a3[site[i]] * eTemp[i - nsite]);
    
    eTemp[i] = eTemp[i - nsite] + eTempDiff[i];
    if (eTemp[i] < 0) {
      eTemp[i] = 0.0;
    }
  }
  // Define 1st mu
  mu[1] = eTemp[1:nsite];
  // Define rest of mu; ----
  for (t in 2:nweek){
    mu[t] = eTemp[((t - 1) * nsite + 1):(t * nsite)];
  }
  // Covariance matrices ----
  // Tail-down exponential model
    for (i in 1:nsite) {
    for (j in 1:nsite) {
      if (flow_con_mat[i, j] == 1) { // if points are flow connected
        C_td[i, j] = var_td * exp(-3 * H[i, j] / alpha_td);
      }
      else{ // if points are flow unconnected
        C_td[i, j] = var_td * exp(-3 * (D[i, j] + D[j, i]) / alpha_td);
      }
    }
  }

model {
  sigma_nug ~ exponential(0.05); // sd nugget
  sigma_td ~ exponential(2); // sd tail-down
  alpha_td ~ normal(0, 20000) T[0, ]; // range tail-down
  bInitialTemp ~ normal(0, 0.1) T[0, ];

  s1 ~ exponential(50);
  s2 ~ exponential(50);
  s3 ~ exponential(50);
  s4 ~ exponential(50);
  m1 ~ normal(0.8, 1);
  m2 ~ normal(0.4, 1);
  m3 ~ normal(0.4, 1);
  m4 ~ normal(0.1, 1);
  a1 ~ normal(m1, s1);
  a2 ~ normal(m2, s2);
  a3 ~ normal(m3, s3);
  a4 ~ normal(m4, s4);

  for (t in 1:nweek) {
    target += multi_normal_cholesky_lpdf(Y[t] | mu[t], cholesky_decompose(C_td + var_nug * I + 1e-6));
  }

Block 1. Model description.

Results

Tables

Stream Temperature

Table 1. Parameter descriptions.

Parameter	Description
`C_td`	Covariance matrix of the tail-down exponential model
`D`	Downstream hydrologic distance matrix
`H`	Total hydrologic distance matrix
`I`	The identity matrix
`N_y_mis`	Number of missing water temperature values
`N_y_obs`	Number of observed water temperature values
`Y[t]`	Vector of water temperature values for all sites in the `t`^th week
`a1[i]`	Intercept-type parameter of the air2stream model for the `i`^th site
`a2[i]`	Effect of `air_temp[i]` on `eTempDiff[i]` for the `i`^th site
`a3[i]`	Effect of the previous week’s expected water temperature (`eTemp[i - nsite]`) on `eTempDiff[i]`, for the `i`^th site
`a4[i]`	Effect of `discharge[i]` on `eTempDiff[i]` for the `i`^th site
`air_temp[i]`	The `i`^th air temperature value (˚C)
`alpha_td`	The variance of spatially independent points
`bInitialTemp`	Expected average water temperature for the week starting 01-01-2019 for all sites
`discharge[i]`	Dimensionless discharge for the `i`^th observation (discharge for that observation divided by the mean discharge across all observations for that site)
`eTempDiff[i]`	Expected difference in average water temperature from the previous week
`eTemp[i]`	Expected value of `water_temp[i]`
`flow_con_mat`	Site connectivity matrix
`i_y_mis`	Indexes of missing water temperature values
`i_y_obs`	Indexes of observed water temperature values
`m1`	Mean of the site-wise random effect for the `a1` parameter
`m2`	Mean of the site-wise random effect for the `a2` parameter
`m3`	Mean of the site-wise random effect for the `a3` parameter
`m4`	Mean of the site-wise random effect for the `a4` parameter
`mu[t]`	Vector of `eTemp` values for all sites in the `t`^th week
`nsite`	Number of sites
`nweek`	Number of weeks
`s1`	Standard deviation of the site-wise random effect for the `a1` parameter
`s2`	Standard deviation of the site-wise random effect for the `a2` parameter
`s3`	Standard deviation of the site-wise random effect for the `a3` parameter
`s4`	Standard deviation of the site-wise random effect for the `a4` parameter
`sigma_nug`	Standard deviation of the nugget effect
`sigma_td`	Standard deviation of the exponential tail-down covariance model
`site[i]`	The `i`^th site
`var_nug`	Variance of the nugget effect
`var_td`	Variance of the exponential tail-down covariance model
`week[i]`	The `i`^th week
`y[i]`	The `i`^th water temperature value (˚C)
`y_mis`	Vector of missing water temperature values
`y_obs`	Vector of observed water temperature values

Table 2. Model coefficients.

term	estimate	lower	upper	svalue
alpha_td	9.77e+03	6.16e+03	1.69e+04	10.6
bInitialTemp	7.14e-02	4.01e-03	2.36e-01	10.6
m1	6.30e-01	4.65e-01	7.96e-01	10.6
m2	3.22e-01	2.67e-01	3.76e-01	10.6
m3	3.21e-01	2.73e-01	3.74e-01	10.6
m4	2.79e-01	1.80e-01	3.93e-01	10.6
s1	2.67e-01	2.07e-01	3.45e-01	10.6
s2	1.18e-01	9.08e-02	1.59e-01	10.6
s3	1.08e-01	8.46e-02	1.45e-01	10.6
s4	2.01e-01	1.55e-01	2.66e-01	10.6
sigma_nug	4.20e-01	3.85e-01	4.58e-01	10.6
sigma_td	1.37e+00	1.19e+00	1.59e+00	10.6

Table 3. Model convergence.

n	K	nchains	niters	nthin	ess	rhat	converged
3297	2126	3	500	5	543	1.013	TRUE

Table 4. Model sensitivity.

all	analysis	sensitivity	bound
all	1.013	1.076	1.132

Figures

Stream Temperature

Figure 1. Tail-down covariance by hydrologic distance (with 95% CIs).

Figure 2. Predicted water temperature by date (with 95% CIs). The points are the observed data.

Figure 3. Predicted GSDD by year and site (with 95% CIs).

Figure 4. GSDD median estimate and width of 95% CI by year and site. The black lines are the stream network.

Acknowledgements

The organisations and individuals whose contributions have made this analytic appendix possible include:

Hillcrest Geographics
- Simon Norris

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